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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf,len=1078
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='01005v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='AG]  3 Jan 2023  MUMFORD TATE GROUPS AND THE HODGE CONJECTURE  ANANYO DAN AND INDER KAUR  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' In this article we study the (cohomological) Hodge conjecture for singular varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  We prove the conjecture for simple normal crossing varieties that can be embedded in a family  where the Mumford-Tate group remains constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  We show how to produce such families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Furthermore, we show for varieties with worse singularities the conjecture can be expressed  solely in terms of the algebraic classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Introduction  The underlying field will always be C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Recall, the classical Hodge conjecture claims that  given a smooth projective variety X, every (rational) Hodge class in X is the cohomology class  of an algebraic cycle in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' The conjecture is known in some cases (see [20, 32] for a survey  of known results and [6, 30] for related results), but is open in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' A typical strategy has  been to consider smooth, projective low dimensional varieties that are birational to already  known cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This is primarily because the exceptional divisors arising from the resolution of  the indeterminacy locus satisfy the Hodge conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' However, this strategy fails in higher  dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Another approach is to consider families of varieties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' in the case of abelian  varieties) and then use a Noether-Lefschetz-type argument to conclude that the Hodge classes  in a very general fiber in the family are powers of the first Chern class of a line bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This  implies the Hodge conjecture for a very general fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' In this article, we combine ideas from  both these approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  It is well-known that any smooth projective variety X is birational to a hypersurface Xhyp in  a projective space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This hypersurface Xhyp is almost always singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Note that there is homo-  logical version of the Hodge conjecture for singular varieties given by Jannsen [13, Conjecture  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2] (see also [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' He proved that the classical Hodge conjecture is equivalent to the singular  version (see [13, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='9], see also [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Therefore, proving the singular Hodge conjecture  for Xhyp would imply the Hodge conjecture for X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  In the present article, we give a cohomological formulation of the Hodge conjecture for singular  varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' There are obvious reasons why this interpretation has so far been unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Firstly  for X singular, the classical Chow group is not compatible with pull-back morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' In [9,  Chapter 17] (see also [10, Proposition 4]), Fulton and MacPherson developed the operational  Chow group, denoted Ap(X) which is compatible with pull-back morphisms and for smooth  varieties coincides with the classical Chow group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' However, even for the operational Chow group,  we know by [29] that in general, there is no map Ap(X) → H2p(X, Q) with good properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Date: January 4, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' 14C15, 14C30, 32S35, 32G20, 14D07, 14C05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Hodge conjecture, Limit mixed Hodge structures, Operational Chow group, Cycle  class map, flag Hilbert schemes, singular varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' is funded by EPSRC grant number EP/T019379/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' was funded by the DFG, TRR 326 Geometry  and Arithmetic of Uniformized Structures, project number 444845124 and is currently funded by EPSRC grant  number EP/W026554/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  1  2  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' DAN AND I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' KAUR  Nevertheless, by the work of Bloch-Gillet-Soul´e (see [2]) there is a (functorial) cycle class map:  clp : Ap(X) ⊗ Q → GrW  2pH2p(X, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Using this we formulate the cohomological singular Hodge conjecture as follows:  Singular Hodge conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let X be a projective variety such that the dimension of the  singular locus is at most p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Then, the image of the cycle class map clp coincides with  H2p  Hdg(X) := GrW  2pH2p(X, Q) ∩ F pGrW  2pH2p(X, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  If X is of dimension n and the above conjecture holds for X, then we say that X satisfies  SHC(p, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Of course, if X is non-singular then the singular Hodge conjecture is the same as  the classical Hodge conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' In this case, we say that X satisfies HC(p, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' The Lefschetz  (1, 1)-theorem implies HC(1, n) holds true, for any n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Recall, a very general hypersurface of any dimension satisfies the Hodge conjecture (as the  cohomology ring is generated by the class of the hyperplane section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Therefore we can always  embed Xhyp in a one parameter family of hypersurfaces such that a general fibre satisfies the  Hodge conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' One then expects that the Hodge classes on Xhyp “spread out” to Hodge  classes in the family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Since a general member of the family satisfies the Hodge conjecture, we  know that the Hodge class away from the centre is the cohomology class of an algebraic cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  By the simple operation of taking closure, one can then extend the algebraic cycles on the  general fiber to the central fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' One needs to check that the cohomology class of this “new”  algebraic cycle on the central fiber coincides with the Hodge class we started with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' However,  there are several technical problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Heuristically, the specialization map is not injective and  hence Hodge classes need not “spread out”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Even if a Hodge class does spread out, it might  not restrict to a Hodge class on the general fibre!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' In this article we study these problems and  give several examples of families of varieties where these problems can be circumvented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let us  make this precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Let X be a singular, projective variety of dimension n and π : X → ∆ be a flat family of  projective varieties, smooth over ∆∗ with the central fiber X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Fix an integer p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Denote by h  the universal cover for ∆∗ and by X∞ the pull-back of X to h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' By Ehresmann’s theorem, for  every u ∈ h there is an isomorphism of cohomology groups H2p(X∞, Q) and H2p(Xu, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' The  natural Hodge filtration on H2p(Xu, Q) induces a filtration F p  u on H2p(X∞, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' The limit Hodge  filtration on H2p(X∞, Q) arises as the limit of this filtration as the imaginary part of u tends to  ∞ (see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='3 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' However, there may be rational points H2p(X∞, Q) ∩ F pH2p(X∞, C)  of the limit Hodge filtration that do not come from the rational points of the filtration F p  u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  The Noether-Lefschetz locus gives examples of this phenomena even for smooth families (see  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' As a result, H2p(X∞, Q) may contain more Hodge classes than that on a general  fiber!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This means that although a Hodge class on X0 maps to a Hodge class on X∞ via the  specialization map, it need not extend to a Hodge class on the family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  The jump in the rank of the Hodge lattice is captured by Mumford-Tate groups (see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1  for the definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We call π a Mumford-Tate family if the rank of the Mumford-Tate group  remains “constant in the limit” (see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2 for precise definitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Moreover, we call a singular,  projective variety MT-smoothable if it can be embedded as a central fiber of a Mumford-Tate  family where the general fiber satisfies the Hodge conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We prove the following:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let X be a projective variety of dimension 4 with strict normal crossings sin-  gularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' If X is MT-smoothable, then X satisfies SHC(p, 4) for every p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  In Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2 below, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1 for any dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Clearly Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1 leads  to the following questions:  MUMFORD TATE GROUPS AND THE HODGE CONJECTURE  3  Question 1: How to find Mumford-Tate families?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Question 2: Can we generalize Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1 to varieties with worse singularities?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  For an exhaustive answer of Question 1 one would need a complete description of the Noether-  Lefschetz locus for families of hypersurfaces in all dimensions greater than 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This problem  is largely open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  However in §6, we give a general method to obtain Mumford-Tate families  from known ones using the theory of correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Recall, that given a coherent sheaf E  on a product of two smooth, projective varieties X × Y , the i-th Chern class of E induces a  morphism of pure Hodge structures from H2m−k(X) to H2i−k(Y ) for all integers i and k, where  m = dim(X) (see §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let us denote such a morphism by Φ(i,k)  E  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We say Y is cohomologically  generated by (X, E) if the cohomology ring H∗(Y ) is generated (as a ring) by the images of  morphisms of the form Φ(i,k)  E  as i and k varies over all integers (see Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Note  that several examples of cohomologically generated varieties appear in existing literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' For  example, in [23] Mumford and Newstead proved that the moduli space of stable rank 2 bundles  with odd degree determinant over a curve C is cohomologically generated by the pair (C, U),  where U is the universal bundle associated to the moduli space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' In [21,22] Markmann showed a  similar result for moduli spaces of sheaves over certain surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' In §6 we show how this notion  of cohomologically generated leads to producing more Mumford-Tate families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let π1 : X ∗ → ∆∗ and π2 : Y∗ → ∆∗ be two smooth, projective families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Assume  that there exists a coherent sheaf U over X ∗ ×∆∗ Y∗ such that it is flat over ∆∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Suppose that for  general t ∈ ∆∗, Yt is cohomologically generated by (Xt, Ut), where Ut := U|Xt×Yt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' If the family  π1 is (strictly) Mumford-Tate family, then so is the family π2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  See Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='5 for the precise formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' An obvious choice for π1 is a family of smooth  curves degenerating to a singular curve (with arbitrary singularities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' See Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1 for a  proof in the case when the singular curve is nodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Let us turn to Question 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Suppose X is a singular projective variety of dimension n and p be  an integer such that dim(Xsing) ≤ p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Suppose φ : �  X → X is any resolution of singularities  and E is the exceptional divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' By [25, Corollary-Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='37], we have an exact sequence  on cohomology  H2p(X) → H2p( �  X) → H2p(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  We conjecture that taking algebraic cohomology groups preserves the exactness of the sequence:  Conjecture A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' The following sequence is exact:  H2p  A (X) → H2p  A ( �  X) → H2p  A (E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Note that, this conjecture does not involve Hodge classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Surprisingly, we prove that if  X is MT-smoothable, then this conjecture is equivalent to the singular Hodge conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' In  particular,  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let X be as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' If X satisfies SHC(p, n), then X satisfies Conjecture A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Conversely, if HC(p−1, n−1) holds true, X is MT-smoothable and satisfies Conjecture A, then  X satisfies SHC(p, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  See Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='5 for the precise statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Outline: The paper is organised as follows: in §2 we briefly recall the necessary preliminaries  on limit mixed Hodge structures and flag Hilbert schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' In §3 we recall the definition of a  Mumford-Tate group and introduce Mumford-Tate families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We give both examples and non-  examples of such families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' In §4, we define limit algebraic cohomology groups and limit Hodge  classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We recall the preliminaries on Operational Chow group and the Bloch-Gillet-Soul´e cycle  4  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' DAN AND I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' KAUR  class map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We give the singular Hodge conjecture and prove some of the preliminary results  which we use later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' In §5, we prove the main results of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Finally, in §6 we give a  method to produce Mumford-Tate families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Preliminaries  In this section we briefly recall some of the basics on limit mixed Hodge structures and flag  Hilbert schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Limit mixed Hodge structures play an important role throughout this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  See [25, §11] for a detailed treatment of the topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Consider a flat family of projective varieties,  π : X → ∆,  smooth over ∆∗ of relative dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Suppose the central fiber X0 := π−1(0) is a reduced,  simple normal crossings divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Denote by π′ : X∆∗ → ∆∗ the restriction of π to the punctured  disc ∆∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Denote by X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=', Xr the irreducible components of the central fiber X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' For m ≥ 2,  denote by X(m) the disjoint union of the intersections of m number of irreducible components  of X0 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=',  X(m) :=  �  |I|=m  I=(1≤i1<i2<.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='<im≤r)  � m  �  k=1  Xik  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Let e : h → ∆∗ be the exponential map from the upper half plane h to the punctured disc  ∆∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Denote by X∞ := X∆∗ ×∆∗ h the base change of X∆∗ to h via the exponential map e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Monodromy operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Since h is simply connected, the natural inclusion  is : Xe(s) ֒→ X∞  for any s ∈ h, induces an isomorphism of cohomology groups:  i∗  s : H2p(X∞, Z) ∼  −→ H2p(Xe(s), Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Note that, the morphism i∗  s changes even if e(s) does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' In particular, we have the monodromy  operator associate to the family π given by the composition:  T : H2p(X∞, Z)  i∗  s+1  −−→  ∼  H2p(Xe(s), Z)  (i∗  s)−1  −−−−→  ∼  H2p(X∞, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  See [16, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' 67, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='13)] for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Denote by N := −(1/2πi) log(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Using this operator  N we will recall the limit Hodge filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Limit Hodge filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Denote by  F •  s H2p(X∞, C) := (i∗  s)−1(F •H2p(Xe(s), C))  the preimage of the Hodge filtration on H2p(Xe(s), C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' The dimension of F k  s H2p(X∞, C), denoted  mk, does not depend on the choice of s ∈ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Consider the Grassmann variety parameterizing  mk-dimensional subspaces of H2p(X∞, C), denoted Grass(mk, H2p(X∞, C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' There is a natural  map:  h → Grass(mk, H2p(X∞, C)) sending s ∈ h to exp(2πisN)F k  s H2p(X∞, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  This map is invariant under the translation s �→ s + 1 and tends to a limit F kH2p(X∞, C) as  the imaginary part of s tends to ∞ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=',  F kH2p(X∞, C) :=  lim  Im(s)→∞ exp(2πisN)F k  s H2p(X∞, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  MUMFORD TATE GROUPS AND THE HODGE CONJECTURE  5  See [16, §I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='6] or [26, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' 254, 255] for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Clearly,  lim  Im(s)→∞ exp(2πisN)(F p  s H2p(X∞, C) ∩ H2p(X∞, Q)) ⊂ F pH2p(X∞, C) ∩ H2p(X∞, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1)  This inclusion will play an important role in the definition of the Mumford-Tate family in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Limit weight filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' One can observe that the decreasing filtration  F 0H2p(X∞, C) ⊇ F 1H2p(X∞, C) ⊇ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' ⊇ F 2pH2p(X∞, C) ⊇ 0  need not be a Hodge filtration i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=', F k ∩ F  2p+1−k need not be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' It was observed by Schmid  that H2p(X∞, Q) can be equipped with an increasing limit weight filtration W•, arising from  the monodromy action by T, such that the two filtrations F • and W• together define a mixed  Hodge structure on H2p(X∞, Q) (see [26, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Steenbrink in [28] retrieved the limit  weight filtration using a spectral sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We recall the E1-terms of the spectral sequence:  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1 ( [25, Corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' The spectral sequence  ∞Ep,q  1  :=  �  k≥max{0,p}  Hq+2p−2k(X(2k − p + 1), Q)(p − k)  with the differential map d : ∞Ep−1,q  1  → ∞Ep,q  1  being a combination of the restriction morphism  and the Gysin morphism, degenerates at E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Moreover, ∞Ep,q  1  ⇒ Hp+q(X∞, Q) with the weight  filtration given by ∞Ep,q  2  = GrW  q Hp+q(X∞, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Specialization map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' By the identification between H2p(X∞, Z) and H2p(Xs, Z) men-  tioned above, we get a specialization morphism (see [1, §2]) which is a morphism of mixed  Hodge structures:  sp : H2p(X0, Z) → H2p(X∞, Z),  where H2p(X∞, Q) is equipped with the limit mixed Hodge structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Using the Mayer-Vietoris  sequence observe that the weight filtration on H2p(X0, Q) arises from the spectral sequence with  E1-terms:  Ep,q  1  = Hq(X(p + 1), Q) ⇒ Hp+q(X0, Q)  where the differential d : Ep−1,q  1  → Ep,q  1  is the restriction morphism (see [28, Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Note that, the spectral sequence degenerates at E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' By the definition of Ej,q  1  and ∞Ej,q  1  given above, we have a natural morphism  from Ej,q  1  to ∞Ej,q  1 , which commutes with the respective differential maps d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' As a result, this  induces a morphism of spectral sequences:  φ : Ep,q  2  → ∞Ep,q  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2)  We now compute the kernel over the weight graded pieces of the specialization morphism:  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' For p ≥ 0, we have an exact sequence of the form:  Hq−2(X(p + 2), Q) → Ep,q  2  φ−→  ∞Ep,q  2  where the first morphism is induced by the Gysin morphism  Hq−2(X(p + 2), Q) → Hq(X(p + 1), Q) = Ep,q  1  and φ is as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  6  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' DAN AND I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' KAUR  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Note that the composed morphism  Hq−2(X(p + 2), Q) → Hq(X(p + 1), Q) → Hq(X(p + 2), Q) is the zero map,  where the first morphism is simply the Gysin morphism and the second morphism is the restric-  tion map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Therefore, there is a natural map from Hq−2(X(p + 2), Q) to Ep,q  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' The difference  between the spectral sequences Ep,q  1  and ∞Ep,q  1  is that the differential map in the latter case also  allows Gysin morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Therefore, the kernel of the morphism φ is isomorphic to the image of  the Gysin map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This proves the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  □  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Flag Hilbert schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We refer the reader to [27, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='5] for a detailed study of flag Hilbert  schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let  π : X∆∗ → ∆∗  be a smooth, projective morphism over the punctured disc ∆∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Fix a relative polarization L on  X∆∗ inducing a closed immersion of X∆∗ into a relative projective space PN  ∆∗ for some integer  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' By the constancy of Hilbert polynomials in flat, projective families, every fiber of π has the  same Hilbert polynomial (with respect to the polarization L), say Q (see [12, Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Recall, given a Hilbert polynomial P, there exists a projective scheme, denoted HilbP,Q, called  a flag Hilbert scheme parameterizing pairs of the form (Y ⊂ X ⊂ PN), where Y (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' X) is of  Hilbert polynomial P (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  The flag Hilbert scheme HilbP,Q is equipped with an universal family Y ⊂ Xuniv with Y, Xuniv  flat over HilbP,Q and for every s ∈ HilbP,Q, the corresponding fiber Ys (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Xs) has Hilbert  polynomial P (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Q) satisfying the universal property: if there exists a closed subscheme  Z ⊂ X∆∗, flat over ∆∗ with fibers having Hilbert polynomial P, then there exists an unique  morphism f : ∆∗ → HilbP,Q such that the pull-back of the universal family Y ⊂ Xuniv to ∆∗ is  isomorphic to Z ⊂ X∆∗ (see [27, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' For every 0 < ǫ ∈ R small enough, there exists sǫ ∈ ∆∗ of distance less than  ǫ from the origin, such that every closed subvariety Zsǫ of codimension p in Xsǫ extends to a  ∆∗-flat closed subscheme Z ⊂ X∆∗ such that the fiber Z ∩ Xsǫ over sǫ is isomorphic to Zsǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Since the Hilbert polynomial of the fibers of π is Q, by the universal property of Hilbert  schemes there is a natural morphism  f : ∆∗ → HilbQ  such that the pull-back of the universal family on HilbQ to ∆∗ is isomorphic to X∆∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let S be  the set of Hilbert polynomials P of degree n − p such that the image of the natural projection  morphism from HilbP,Q to HilbQ does not contain the image of f i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=', intersects properly the  image of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Clearly, S is a countable set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Note that the union of countably many proper closed  subsets in ∆∗ does not contain any open subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Hence, for every 0 < ǫ ∈ R small enough, there  exists sǫ ∈ ∆∗ of distance less than ǫ from the origin, such that f(sǫ) does not lie in the image  of the projection from HilbP,Q to HilbQ, as P varies in the set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' In other words, every closed  subscheme in Xsǫ extends to to a ∆∗-flat closed subscheme of X∆∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This proves the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Mumford-Tate families  In this section we introduce the concept of Mumford-Tate families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' These are smooth families  of projective varieties such that the associated limit mixed Hodge structure has “as many” Hodge  classes as a general fiber in the family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' The motivation behind the name is that Mumford-Tate  groups are determined uniquely by the set of Hodge classes in the associated tensor algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let  us first recall the definition of the Mumford-Tate group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  MUMFORD TATE GROUPS AND THE HODGE CONJECTURE  7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Mumford-Tate groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Denote by S the Weil restriction of scalars for the field extension  C/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let V be a Q-vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' A pure Hodge structure of weight n on V is given by a non-  constant homomorphism of R-algebraic groups  φ : C∗ = S(R) → GL(V )(R)  such that φ(r) = rnId for all r ∈ R∗ ⊂ S(R) = C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Let VC := V ⊗Q C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  To this group  homomorphism one associates the Hodge decomposition:  VC =  �  p+q=n  V p,q where V p,q := {v ∈ VC| φ(z)v = zpzqv for all z ∈ C∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  The Mumford-Tate group associated to the pure Hodge structure (V, φ), denoted MT(V, φ),  is the smallest Q-algebraic subgroup of GL(V ) whose set of real points contain the image of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Denote by  T m,n(V ) := V ⊗m ⊗ Hom(V, Q)⊗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Note that, the Hodge structure on V induces a pure Hodge structure on T m,n(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Elements of  F 0(T m,n(VC)) ∩ T m,n(V )  are called Hodge tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' The Mumford-Tate group as the largest subgroup of GL(VQ) which  fixes the Hodge tensors (see [11, §I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='B]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We now recall some well-known examples of Mumford-Tate groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  (1) Let X be an abelian variety and V = H1(X, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' The Mumford-Tate group associated  to the pure Hodge structure on V will be denoted by MT(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' The polarization on X  corresponds to a non-degenerate alternating form φ : V ⊗ V → Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Denote by GSp(V, φ)  the group of symplectic simplitudes with respect to the symplectic form φ:  GSp(V, φ) := {g ∈ GL(V ) | ∃ λ ∈ C∗ such that φ(gv, gw) = λφ(v, w) ∀ v, w ∈ V }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Recall, for any abelian variety X, the Mumford-Tate group of X is contained in the  group of symplectic simplitudes i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' MT(X) ⊆ GSp(V, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' An abelian variety is called  simple if it does not contain an abelian subvariety other than 0 and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' If X is simple  and dim(X) = p, where p is a prime number, then MT(X) = GSp(V, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  (2) Let c be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let X be a general complete intersection subvariety contained  in P2m+c of codimension c, for some m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Assume that the degree of X is at least 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Denote by V := Hn(X, Q)prim and φ : V ⊗ V → Q the polarization on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let GO(V, φ)  be the group of orthogonal simplitudes with respect φ:  GO(V, φ) := {g ∈ GL(V ) | ∃ λ ∈ C∗ such that φ(gv, gw) = λφ(v, w) ∀ v, w ∈ V }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Then the Mumford-Tate group of X, MT(X) = GO(V, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Mumford-Tate families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Keep setup as in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Given any s ∈ h, recall the exponential  map e from h to ∆∗ and the natural inclusion is from Xe(s) into X∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Recall,  π : X∆∗ → ∆∗  the family of smooth, projective varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' For any s ∈ h, H2p(Xe(s), Q) is equipped with a natural  pure Hodge structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Denote by MTp(Xe(s)) the Mumford-Tate group associated to this pure  Hodge structure on H2p(Xe(s), Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We say that π is a Mumford-Tate family of weight p if for any  class γ ∈ F pH2p(X∞, C) ∩ H2p(X∞, Q) satisfying Nγ = 0, the pullback i∗  s(γ) ∈ H2p(Xe(s), Q) is  fixed by MTp(Xe(s)) for a general s ∈ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We say that π is Mumford-Tate if it is Mumford-Tate  of all weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We now give some examples of Mumford-Tate families:  8  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' DAN AND I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' KAUR  (1) By Lefschetz hyperplane section theorem, for any smooth hypersurface X in P2m for  m ≥ 2, we have H2p(X, Q) ∼= Q for any 0 ≤ p ≤ 2m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This implies if π parametrizes  smooth, hypersurfaces in P2m, then π is Mumford-Tate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  (2) Let π : X → ∆ be a smooth family of prime dimensional abelian varieties such that the  central fiber π−1(0) is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Then π is a Mumford-Tate family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Indeed, since π is a  smooth family, the local system Vp := R2pπ∗Q has no monodromy over the punctured  disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Hence, H2p(X∞, Q) ∼= H2p(X0, Q) as pure Hodge structures, for all p and the local  system Vp is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' By the same argument, R1π∗Q is a trivial local system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' A choice  of the trivialization fixes an identification:  ψt : V0  ∼  −→ Vt, where Vt := H1(Xt, Q) for any t ∈ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Note that the natural polarizations on V0 and Vt commutes with the identification ψt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  This induces an isomorphism:  GSp(Vt, φt) ∼  −→ GSp(V0, φ0) sending  �  Vt  g−→  ∼ Vt  �  to  �  V0  ψt  −→  ∼ Vt  g−→  ∼ Vt  ψ−1  t  −−→  ∼  V0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1)  Now, γ0 ∈ H2p(X∞, Q) = H2p(X0, Q) is a Hodge class if and only if it is fixed by the  Mumford-Tate group MT(X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Since X0 is simple, MT(X0) = GSp(V0, φ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Using the  identification (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1), since the Hodge class γ0 is fixed by GSp(V0, φ0), i∗  s(γ) = φs(γ) is  fixed by GSp(Vs, φs) for any s ∈ ∆∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Since MT(Xs) is contained in GSp(Vs, φs), φs(γ)  is fixed by MT(Xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Hence, φs(γ) is a Hodge class in H2p(Xs, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This proves the claim  that π is a Mumford-Tate family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  (3) Let π : X → ∆ be a smooth family of complex intersection subvarieties of codimension  c and let π−1(0) = X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Suppose that MT(X0) = GO(Hn(X0, Q)prim, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Then π is a  Mumford-Tate family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' The proof for this is the same as that of (2) above with GSp  replaced by GO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' (Examples of non Mumford-Tate families) Recall for d ≥ 4, the Noether-  Lefschetz theorem states that a very general smooth, degree d surface in P3 has Picard number  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' The Noether-Lefschetz locus parametrizes smooth degree d surfaces in P3 with Picard number  at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' See [3–5] for some its geometric properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This means that there are smooth families  π : X → ∆ of hypersurfaces in P3 such that 0 ∈ ∆ lies on the Noether-Lefschetz locus and ∆∗  does not intersect the Noether-Lefschetz locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Since π is a smooth family, the local system  R2π∗Q does not have any monodromy over the punctured disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Then, H2(X∞, Q) ∼= H2(X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='Q)  as pure Hodge structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' In particular, by the condition on the central fiber X0, the rank of  the Hodge lattice in H2(X∞, Q) is at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' But the rank of the Hodge lattice in H2(Xs, Q) is  1 for any s ∈ ∆∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Since the pullback morphism i∗  s is an isomorphism, this implies that there is  a Hodge class on H2(X∞, Q) that does not pullback to a Hodge class on H2(Xs, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Hence, π  cannot be a Mumford-Tate family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' A cohomological version of the Hodge conjecture for singular varieties  In this section we define limit algebraic cohomology classes and limit Hodge classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We show  that the limit algebraic cohomology classes are contained in the monodromy invariant limit  Hodge classes and the converse holds for Mumford-Tate families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' In subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='4 we  recall the necessary preliminaries for the Operational Chow group and the Bloch-Gillet-Soul´e  cycle class map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' In 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='5 we state the Singular Hodge conjecture and in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='6 we show that the  cohomology classes of algebraic cycles on a simple normal crossings variety are contained in the  Hodge classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  We begin by recalling the classical Hodge conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  MUMFORD TATE GROUPS AND THE HODGE CONJECTURE  9  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' The classical Hodge conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let X be a smooth, projective variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Given an  integer p > 0, denote by Zp(X) the free abelian group generated by codimension p algebraic  subvarieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' There is a natural cycle class map:  clp : Zp(X) → H2p(X, Z)  which associates to an algebraic subvariety W ⊂ X of codimension p, the fundamental class  [W] ∈ H2p(X, Z) (see [31, §11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2] for further details) and extend linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Furthermore, by [31,  Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='20], the image of the cycle class map clp lies in Hp,p(X, C) ∩ H2p(X, Z) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=', the  cohomology class of an algebraic variety is a Hodge class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Tensoring the cycle class map by  rationals gives:  clp : Zp(X) ⊗Z Q → H2p(X, Q) ∩ Hp,p(X, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  We denote by H2p  Hdg(X) := H2p(X, Q) ∩ Hp,p(X, C) the space of Hodge classes and the space of  algebraic classes H2p  A (X) ⊂ H2p(X, Q) is the image of the (rational) cycle class map clp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' The  (rational) Hodge conjecture claims that the (rational) cycle class map clp is surjective for all p  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=', the natural inclusion H2p  A (X) ⊂ H2p  Hdg(X) is an equality for all p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let X be a smooth, projective variety of dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We say that X satisfies  HC(p, n) if the natural inclusion H2p  A (X) ⊂ H2p  Hdg(X) is an equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We say that X satisfies  the Hodge conjecture if it satisfies HC(p, n) for every p ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We say that HC(p, n) holds true to  mean that every smooth, projective variety of dimension n satisfies HC(p, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Relative cycle class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let  π : X∆∗ → ∆∗  be a smooth, projective morphism of relative dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let Z ⊂ X∆∗ be a closed subscheme  of X∆∗, flat over ∆∗ and of relative dimension n − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' The fundamental class of Z defines a  global section γZ of the local system H2p := R2pπ∗Z such that for every t ∈ ∆∗, the value  γZ(t) ∈ H2p(Xt, Z) of γZ at the point t is simply the fundamental class of Zt := Z ∩ Xt in Xt  (see [9, §19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2] and [25, §B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='9] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' The pull-back of the local system H2p under the  exponential map e : h → ∆∗ is a trivial local system with fiber H2p(X∞, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' The global section  γZ defines an element of H2p(X∞, Z), which we again denote by γZ, such that for every s ∈ h,  the image i∗  s(γZ) is the fundamental class of Z ∩ Xe(s) in Xe(s), where is is the natural inclusion  of Xe(s) into X∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Denote by H2p  A (X∞) the sub-vector space of H2p(X∞, Q) generated by all such  elements of the form γZ arising from a ∆∗-flat closed subscheme of relative dimension n − p  in X∆∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We call H2p  A (X∞) the limit algebraic cohomology group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We define the limit Hodge  cohomology group  H2p  Hdg(X∞) := F pH2p(X∞, C) ∩ W2pH2p(X∞, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Note that, H2p  Hdg(X∞) need not be monodromy invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Recall, N is a morphism of mixed  Hodge structures from H2p(X∞, Q) to H2p(X∞, Q)(−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We denote by H2p  Hdg(X∞)inv the mon-  odromy invariant part of H2p  Hdg(X∞) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=',  H2p  Hdg(X∞)inv := ker  �  H2p  Hdg(X∞) ֒→ H2p(X∞, Q) N  −→ H2p  Hdg(X∞, Q)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  We now prove that the limit algebraic cohomology group lies in the limit Hodge cohomology  group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This is the asymptotic version of a classical result in Hodge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' The limit algebraic cohomology group is contained in the monodromy invari-  ant part of the limit Hodge cohomology group i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=', the natural inclusion H2p  A (X∞) ⊂ H2p(X∞, Q)  factors through H2p  Hdg(X∞)inv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  10  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' DAN AND I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' KAUR  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Take γ ∈ H2p  A (X∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' By construction, there exist ∆∗-flat closed subschemes Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=', Zr of  relative dimension n − p in X∆∗ such that γ = � aiγZi for ai ∈ Q and γZi ∈ H2p  A (X∞) is as  defined above, arising from the fundamental class of Zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' By construction, each γZi arises from  a global section of the local system H2p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Hence, γZi is monodromy invariant i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=', T(γZi) = γZi  for 1 ≤ i ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This implies NγZi = 0 for 1 ≤ i ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  As the cohomology class of Zi ∩ Xe(s) lies in F pH2p(Xe(s), Q), we have γZi ∈ F p  s H2p(X∞, Q)  for all s ∈ h (notations as in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This implies γZi lies in exp(2πisN)F p  s H2p(X∞, Q) for every  s ∈ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Recall from §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='3 that F pH2p(X∞, Q) contains the limit of exp(2πisN)F p  s H2p(X∞, C)  as Im(s) approaches ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Hence, γZi ∈ F pH2p(X∞, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' As γZi is monodromy invariant and a  rational class, it must lie in W2pH2p(X∞, Q) (use the invariant cycle theorem along with the  fact that the degree 2p cohomology of the central fiber is of weight at most 2p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Therefore,  γ ∈ H2p  Hdg(X∞)inv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This proves the first part of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  □  We now ask when is H2p  A (X∞) isomorphic to H2p  Hdg(X∞)inv?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' One can naively guess that if the  general fibers in the family π satisfy the Hodge conjecture then this happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' However, this is  not enough (see Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='3 above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' In particular, one needs to additionally assume that the  family π is Mumford-Tate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We prove:  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Suppose that π is a Mumford-Tate family of weight p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' If a general fiber of π  satisfies HC(p, n), then the inclusion from H2p  A (X∞) to H2p  Hdg(X∞)inv is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Note that, by general in the statement of the proposition, we mean the complement of finitely  many proper, closed subvarieties of the punctured disc ∆∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We need to show that every element in H2p  Hdg(X∞)inv lies in H2p  A (X∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Since π is a  Mumford-Tate family, we have  H2p  Hdg(X∞)inv =  lim  Im(s)→∞(F p  s H2p(X∞, Q) ∩ H2p(X∞, Q)inv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1)  It therefore suffices to show that  lim  Im(s)→∞(F p  s H2p(X∞, Q) ∩ H2p(X∞, Q)inv)  is contained in H2p  A (X∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='4 for every 0 < ǫ ∈ R small enough, there exists sǫ ∈ ∆∗ of distance less than ǫ  from the origin, such that Xsǫ satisfies HC(p, n) and every closed subvariety Zsǫ of codimension  p in Xsǫ extends to a ∆∗-flat closed subscheme Z ⊂ X∆∗ such that the fiber Z ∩ Xsǫ over sǫ  is isomorphic to Zsǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  As observed before Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2, the fundamental class of Z defines  a section γZ ∈ H2p  A (X∞) and is monodromy invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Since F pH2p(Xsǫ, Q) is isomorphic to  H2p  A (Xsǫ), this implies  H2p(X∞, Q)inv ∩ F p  sǫH2p(X∞, Q) = (i∗  sǫ)−1(H2p  A (Xsǫ)) ⊆ H2p  A (X∞),  where isǫ is the natural inclusion of Xe(sǫ) into X∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Therefore, the limit as Im(s) tends to ∞,  of H2p(X∞, Q)inv ∩ F p  s H2p(X∞, Q) is contained in H2p  A (X∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This proves the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Operational Chow group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let Y be a quasi-projective variety (possibly singular), of  dimension say n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Consider a non-singular hyperenvelope of a compactification of Y (see [10,  §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1] for the definition and basic properties of hyperenvelopes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' The hyperenvelope gives rise  to a cochain complex of motives (see [10, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' For any positive integer p, one can then obtain  an abelian group R0CHp(Y ) arising as the cohomology group after applying the functor CHp(−)  MUMFORD TATE GROUPS AND THE HODGE CONJECTURE  11  to the cochain complex of motives (see [10, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Observe that R0CHp(Y ) does not depend  on the choice of the compactification or the hyperenvelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Note that,  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Fix a positive integer p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' the following holds true for R0CHp(Y ):  (1) if Y is projective,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' then R0CHp(Y ) is the operational Chow group Ap(Y ) defined by  Fulton and MacPherson (see [9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Chapter 17]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  (2) if Y is non-singular (but not necessarily projective),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' then Ap(Y ) is the free abelian group  generated by the codimension p subvarieties in Y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' upto rational equivalence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  (3) if Y is non-singular and Y is a compactification of Y with boundary Z := Y \\Y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' we then  have the exact sequence:  0 → R0CHp(Y ) → R0CHp(Y ) → R0CHp(Z)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2)  (4) if Y is the union of two proper closed subvarieties Y1 and Y2, then we have the exact  sequence:  0 → R0CHp(Y ) → R0CHp(Y1) ⊕ R0CHp(Y2) → R0CHp(Y1 ∩ Y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='3)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  (1) This is [10, Proposition 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  (2) This is [9, Proposition 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1 and Corollary 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  (3) This is [10, Theorem 2(iii) and §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  (4) This is [10, Theorem 2(iv) and §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  □  Notation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' If Y is quasi-projective but not projective, we denote by Ap  c(Y ) := R0CHp(Y ),  the compactly supported operational Chow cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Given any compactification Y of Y ,  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='5 implies that we have the following exact sequence  0 → Ap  c(Y ) → Ap(Y ) → Ap(Y \\Y )  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='4)  For Y a projective variety, there are natural functorial cycle class maps (see [2] or [17, §2]):  clp : Ap(Y ) → GrW  2pH2p(Y, Q) and clc  p : Ap  c(Ysm) → GrW  2pH2p  c (Ysm, Q)  which agree with the usual cycle class map (see [31, §11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2]) if Y is non-singular (here Ysm  denotes the smooth locus of Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' For Y projective, define the algebraic cohomology group denoted  by H2p  A (Y ) ⊂ GrW  2pH2p(Y, Q) to be the image of the cycle class map clp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Bloch-Gille-Soul´e Cycle class map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let Y be a scheme and φ : U → Y , γ : V → U×Y U  be envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let pi : V → U denote the compositions of γ with the projections U ×Y U → U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' ( [2, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='3]) There is a left-exact sequence of Chow cohomology groups  0 → CH∗(Y )  φ∗  −→ CH∗(U)  p∗  1−p∗  2  −−−−→ CH∗(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Using the cycle map over smooth, quasi-projective varieties U and V , Bloch-Gillet-Soul´e uses  the above theorem to conclude:  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' ( [2, Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='4]) On the category of varieties over C, there is a “cycle class”  natural transformation of contravariant functors to the category of commutative, graded rings:  �  p  clp :  �  p  CHp(−) →  �  p  GrW  0 H2p(− , Q(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  12  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' DAN AND I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' KAUR  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Singular Hodge conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We are now ready to give a formulation of the Hodge  conjecture for singular varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let Y be a projective variety of dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Fix a positive  integer p ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We say that Y satisfies SHC(p, n) if the singular locus of Y is of dimension at  most p − 1 and the algebraic cohomology group H2p  A (Y ) coincides with  H2p  Hdg(Y ) := GrW  2pH2p(Y, Q) ∩ F 2pGrW  2pH2p(Y, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  In the case when Y is non-singular and projective, this simply is the classical Hodge conjecture  (in weight p), which we already denote by HC(p, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Algebraic cycles on simple normal crossings divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We now prove that the coho-  mology classes of algebraic cycles on a simple normal crossings variety are Hodge classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This  is a generalization to the singular case of a classical result in Hodge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Recall, X0 is called a  simple normal crossings variety if X0 is connected, X0 = X1 ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' ∪ Xr with Xi irreducible, non-  singular for all i and the intersection of any p of the irreducible components of X0 is non-singular  of codimension p, for any p ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let X0 be a simple normal crossings variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Then, the cycle class map clp from  Ap(X0) to GrW  2pH2p(X0, Q) factors through  H2p  Hdg(X0) := F pGrW  2pH2p(X0, C) ∩ GrW  2pH2p(X0, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We use recursion on the components of X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let X0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=', Xr be the irreducible components  of X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Denote by Zi := X0\\(X1 ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' ∪ Xi), the complement of the components X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=', Xi for  i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let Z0 := X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Since Xi, Xj and Xi ∩ Xj are non-singular for all i, j, they have pure  Hodge structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Moreover by [25, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='39], H2p−1(Xi ∩Zi, Q) is of weight at most 2p−1  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=', GrW  2pH2p−1(Xi ∩ Zi, Q) = 0 for all 1 ≤ i ≤ r − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Therefore for all 1 ≤ i ≤ r − 1, we have  the following exact sequence of pure Hodge structures:  0 → GrW  2pH2p(Zi−1, Q) → H2p(Xi, Q) ⊕ GrW  2pH2p(Zi, Q) → GrW  2pH2p(Xi ∩ Zi, Q)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='5)  Moreover, by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='5, we have the exact sequence:  0 → Ap(Zi−1) → Ap(Xi) ⊕ Ap(Zi) → Ap(Xi ∩ Zi)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='6)  By the functoriality of the cycle class maps clp, we have the following diagram  0  ✲ Ap(Zi−1)  ✲ Ap(Xi) ⊕ Ap(Zi)  ✲ Ap(Xi ∩ Zi)  0  ✲ GrW  2pH2p(Zi−1, Q)  clp  ❄  ✲ H2p(Xi, Q) ⊕ GrW  2pH2p(Zi, Q)  clp  ❄  ✲ GrW  2pH2p(Xi ∩ Zi, Q)  clp  ❄  For the base case, consider i = r−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Note that, Zr−1 = Xr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Since Xr is non-singular, Ap(Zr−1)  is the usual Chow group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Therefore, clp(Ap(Zr−1)) ⊂ H2p  Hdg(Zr−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Now for the recursion step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Assume that clp(Ap(Zi)) ⊂ H2p  Hdg(Zi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Since the exact sequence  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='5) is a morphism of pure Hodge structures, the commutativity of the left hand square implies  that clp(Ap(Zi−1)) ⊂ H2p  Hdg(Zi−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This proves the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  □  MUMFORD TATE GROUPS AND THE HODGE CONJECTURE  13  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Main results  In this section we introduce the concept of MT-smoothable varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Consider a simple normal  crossings variety X (in the sense of §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Denote by X(2) the disjoint union of intersection of  any 2 irreducible components of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We prove that if X is MT-smoothable and X(2) satisfies  HC(p − 1, n − 1) then X satisfies SHC(p, n) (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  This is a generalization of  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1 in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Moreover, if there is an irreducible component Xi of X such  that the restriction morphism on cohomology is surjective, then Xi satisfies the classical Hodge  conjecture (see Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Finally, if the variety has worse singularities than simple normal  crossings, then we reduce the singular Hodge conjecture to a question solely on the algebraic  classes (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let X be a singular projective variety of dimension n and p be an integer such  that dim(Xsing) ≤ p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We say that X is MT-smoothable of weight p if there exists a flat,  projective, Mumford-Tate family  π0 : Y → ∆  smooth over ∆∗, containing X as a central fiber and a general fiber satisfying HC(p, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We call  π0 a MT-smoothing of weight p of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Given a normal crossings variety X, We prove:  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let X be a simple normal crossings variety of dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Assume that every  irreducible component of X(2) satisfies HC(p − 1, n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' If X is MT-smoothable of weight p,  then X satisfies SHC(p, n) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=',  H2p  A (X, Q) ∼= H2p  Hdg(X, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Moreover, for every irreducible component Xi of X, the image of the restriction morphism from  H2p  Hdg(X, Q) to H2p  Hdg(Xi, Q) are cohomology classes of algebraic cycles i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=', the image  Im(H2p  Hdg(X, Q) → H2p  Hdg(Xi, Q))  is contained in H2p  A (Xi, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Since X is MT-smoothable of weight p, there exists a Mumford-Tate family of weight p  π : X → ∆  with central fiber X and general fibers satisfying HC(p, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='4 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='9,  we have a morphism spA from H2p  A (X) to H2p  A (X∞) given by the composition:  spA : H2p  A (X) ֒→ H2p  Hdg(X)  sp  −→ H2p  Hdg(X∞)inv ∼= H2p  A (X∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  We claim that spA is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Recall from Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2, H2p  A (X∞) is generated as a Q-vector  space by classes γZ where Z ⊂ X∆∗ is a ∆∗-flat closed subscheme of relative dimension n − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Denote by Z the closure of Z in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' By [9, §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1], the intersection product Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='Xi of Z with Xi  is of codimension p in Xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Denote by γi ∈ H2p(Xi, Q) the cohomology class of the intersection  product Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='Xi for 1 ≤ i ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' By the associativity of intersection product (see [9, Proposition  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1 or Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='3]), for any pair of integers 1 ≤ i < j ≤ r, the image of γi (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' γj) under  the restriction morphisms from H2p(Xi, Q) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' H2p(Xj, Q)) to H2p(Xi ∩ Xj, Q) coincides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='5) one can observe that there exists an algebraic cohomology class γ ∈ H2p  A (X) such  that the image of γ under the restriction morphism from H2p  A (X) to H2p  A (Xi) is γi for 1 ≤ i ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  In other words, the cohomology class of Z in H2p(X, Q) (see [25, §B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='9]) pulls back to γ in  H2p(X, Q) and to the cohomology class [Z ∩ Xt] ∈ H2p(Xt, Q) over Xt, for any t ∈ ∆∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This  14  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' DAN AND I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' KAUR  means that under the specialization morphism sp from H2p(X, Q) to H2p(X∞, Q), γ maps to  γZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This proves our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='3, the kernel of the specialization morphism  GrW  2pH2p(X, Q) = E0,2p  2  sp  −→ ∞E0,2p  2  = GrW  2pH2p(X∞, Q)  is isomorphic to the image of the Gysin morphism from H2p−2(X(2), Q) to H2p(X, Q) (as X(2)  is non-singular, H2p−2(X(2), Q) has a pure Hodge structure of weight 2p − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' By assumption,  every irreducible component of X(2) satisfies HC(p − 1, n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Then, we get the following  commutative diagram of exact sequences:  H2p  A (X(2))  ✲ H2p  A (X)  spA✲ H2p  A (X∞)  ✲ 0  ⟲  ⟲  H2p  Hdg(X(2))  ∼=  ❄  ✲ H2p  Hdg(X)  ❄  ∩  sp✲ H2p  Hdg(X∞)inv  ∼=  ❄  By diagram chase (or using four lemma for the diagram of exact sequences), we conclude that the  middle morphism from H2p  A (X) to H2p  Hdg(X) is surjective, hence an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This proves  the first part of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' The second part of the theorem follows immediately from the  following commutative diagram, which arises from the Mayer-Vietoris sequence:  H2p  A (X) ⊂  ✲ H2p  A (Xi) ⊕ H2p  A (X\\Xi)  ⟲  H2p  Hdg(X)  ∼=  ❄  ⊂✲ H2p  Hdg(Xi) ⊕ H2p  Hdg(X\\Xi)  ❄  ∩  This proves the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  □  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Notations and hypothesis as in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let X1 be an irreducible com-  ponent in X such that the complement Xc  1 := X\\X1 (the closure of X\\X1 in X) satisfies:  Im(H2p  Hdg(X1) → H2p  Hdg(Xc  1 ∩ X1)) ⊂ Im(H2p  Hdg(Xc  1) → H2p  Hdg(Xc  1 ∩ X1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1)  Then, X1 satisfies HC(p, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Using the Mayer-Vietoris sequence we have the following commutative diagram:  0  ✲ H2p  A (X)  ✲ H2p  A (X1) ⊕ H2p  A (Xc  1)  ⟲  0  ✲ H2p  Hdg(X)  ∼=  ❄  ⊂✲ H2p  Hdg(X1) ⊕ H2p  Hdg(Xc  1)  ❄  ∩  ✲ H2p  Hdg(Xc  1 ∩ X1)  where the isomorphism of the first vertical arrow follows from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2 and the bottom row  is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' If (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1) is satisfied then for any γ ∈ H2p  Hdg(X1), there exists γ′ ∈ H2p  Hdg(Xc  1) such that  their restrictions to X1 ∩ Xc  1 agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' In other words, γ ⊕ γ′ maps to zero in H2p  Hdg(Xc  1 ∩ X1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  By diagram chase, one observes that there exists γA ∈ H2p  A (X1) which maps to γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This proves  H2p  A (X1) ∼= H2p  Hdg(X1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' In other words, X1 satisfies HC(p, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This proves the corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  □  MUMFORD TATE GROUPS AND THE HODGE CONJECTURE  15  One immediately asks whether there are examples where (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1) is satisfied?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let X be a projective variety of dimension n with only ordinary double point  singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Suppose also that X is smoothable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Then, there exists a flat, projective family  π0 : Y → ∆  smooth over ∆∗, X as the central fiber and Y is a regular variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Moreover, there exists a  semi-stable reduction of π0:  π : X → ∆  such that the central fiber X0 := �  X ∪ E, where E is a disjoint union of quadric hypersurfaces in  Pn+1 and E ∩ �  X0 is the intersection of E by hyperplanes in copies of Pn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' If n = 2p for some  p, then the n-th rational cohomology of a quadric hypersurface in Pn is isomorphic to Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This  implies the natural restriction morphism from H2p(E) to H2p(E ∩ �  X) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' In this case,  taking X1 := �  X, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  A natural conjecture arises from our observations:  Conjecture A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let X be a singular projective variety, φ :  �  X → X be any resolution of  singularities and E be the exceptional divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let p be an integer such that dim(Xsing) ≤ p−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  We then have an exact sequence on cohomology (see [25, Corollary-Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='37]):  H2p(X) → H2p( �  X) → H2p(E)  We conjecture that taking algebraic cohomology groups preserves the exactness of the sequence  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=', the following sequence is exact:  H2p  A (X) → H2p  A ( �  X) → H2p  A (E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  We now observe that this conjecture is closely related to the singular Hodge conjecture (which  is equivalent to the Hodge conjecture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let X be a singular projective variety of dimension n and p be an integer such  that dim(Xsing) ≤ p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' If X satisfies SHC(p, n), then X satisfies Conjecture A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Conversely, if  HC(p − 1, n − 1) holds true, X is MT-smoothable of weight p and satisfies Conjecture A, then  X satisfies SHC(p, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' If X satisfies the SHC(p, n), then H2p  A (X) ∼= H2p  Hdg(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let  φ : �  X → X  be a resolution of X and E be the exceptional divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We then have the following commutative  diagram:  H2p  A (X)  ✲ H2p  A ( �  X)  ✲ H2p  A (E)  ⟲  ⟲  H2p  Hdg(X)  ∼=  ❄  ⊂✲ H2p  Hdg( �  X)  ❄  ∩  ✲ H2p  Hdg(E)  ❄  ∩  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2)  where the bottom row is exact, injective on the left and the top row is a complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' To prove  Conjecture A, we need to show that the top row is exact in the middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' For this, take γ ∈ H2p  A ( �  X)  which maps to zero in H2p  A (E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' By diagram chase it is easy to check that there exists γ′ ∈ H2p  A (X)  which maps to γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' In other words, the top row of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2) is exact in the middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This proves the  first part of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  16  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' DAN AND I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' KAUR  We now assume that X satisfies Conjecture A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let π0 : Y → ∆ be a MT-smoothing of weight  p of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' By the semi-stable reduction theorem (see [15, Chapter II]) there exists a flat, projective  family π : X → ∆ which has the same fiber over ∆∗ as π0, X is regular, the central fiber X0 is  a reduced simple normal crossings divisor with one of the irreducible components, say �  X being  proper birational to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Furthermore, the complement �  Xc := X0\\ �  X satisfies:  X0\\ �  Xc ∼= �  X\\( �  Xc ∩ �  X) ∼= X\\Xsing  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=', X is isomorphic to Y away from Xsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Using the Mayer-Vietoris sequence and Conjecture  A we have the following commutative diagram of exact sequences:  H2p  A (X)  ✲ H2p  A ( �  X)  ✲ H2p  A ( �  X ∩ �  Xc)  ⟲  ⟲  H2p  A (X0)  ❄  ∩  ✲ H2p  A ( �  X) ⊕ H2p  A ( �  Xc)  ❄  ∩  ✲ H2p  A ( �  X ∩ �  Xc)  ∼=  ❄  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='3)  where the first vertical morphism is induced by the pullback from X to X0 and the second one  is the natural inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' By snake lemma, this gives rise to the exact sequence:  0 → H2p  A (X) → H2p  A (X0) → H2p  A ( �  Xc)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='4)  Since Xsing is of dimension at most p − 1, Hi(Xsing) = 0 for i ≥ 2p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Then, the long exact  sequences in cohomology associated to the pairs (X, Xsing) and (X0, �  Xc) (see [25, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='46  and Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='14])) implies GrW  2pH2p  c (U) ∼= GrW  2pH2p(X) where U := X\\Xsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Furthermore,  0 → GrW  2pH2p  c (U, Q) → GrW  2pH2p(X0, Q) → GrW  2pH2p( �  Xc, Q)  is an exact sequence of pure Hodge structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This gives rise to the exact sequence:  0 → H2p  Hdg(X) → H2p  Hdg(X0) → H2p  Hdg( �  Xc)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='5)  of Q-vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Then, there is a natural morphism of exact sequences from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='4) to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='5):  0  ✲ H2p  A (X)  ✲ H2p  A (X0)  ✲ H2p  A ( �  Xc)  ⟲  ⟲  0  ✲ H2p  Hdg(X)  ❄  ∩  ✲ H2p  Hdg(X0)  ∼=  ❄  ✲ H2p  Hdg( �  Xc)  ❄  ∩  where the isomorphism of the middle vertical arrow follows from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Applying snake  lemma once again we conclude that the first vertical morphism is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' In other words, X  satisfies SHC(p, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This proves the converse and hence the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  □  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Examples of Mumford-Tate families  In §3 we introduced Mumford-Tate families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  For such families, the central fiber displays  interesting properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' For example, if the central fiber is smooth, then it is easy to check that it  satisfies the Hodge conjecture if a general fiber satisfies the Hodge conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' More generally,  if the central fiber is a reduced, simple normal crossings divisor, then it satisfies the singular  Hodge conjecture if the general fiber satisfies the Hodge conjecture (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' In this  section we use correspondences to give a general method to produce Mumford-Tate families (see  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We give examples in Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  MUMFORD TATE GROUPS AND THE HODGE CONJECTURE  17  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Strict Mumford-Tate families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let π1 : X ∗ → ∆∗ be a smooth, projective morphism  over the punctured disc ∆∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Recall that π1 is called a Mumford-Tate family if the pullback  of every monodromy invariant Hodge class on H2p(X∞, Q) to a general fiber is fixed by the  associated Mumford-Tate group, for every p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Here we generalize this condition to the tensor  algebra of the cohomology ring H∗(X∞, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This is a slightly stronger notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' In particular, it  is possible that wedge product of two elements from odd degree cohomology groups become a  Hodge class, although they are individually not Hodge classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This is a common phenomena  appearing in the cohomology of abelian varieties, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This will play a crucial role below  to produce new examples of Mumford-Tate families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  In order to study the tensor algebras more effectively, we separate the odd cohomology groups  from the even ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We take exterior algebra of the odd cohomology groups and the symmetric  algebra of the even ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This is done to preserve compatibility with cup-products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Given two  r-tuple of positive integers m := (m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=', mr) and k := (k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=', kr), denote by  Tk  m :=  k1  �  Hm1(X∞, Q) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' ⊗  kr  �  Hmr(X∞, Q), if each mi is odd,  Tk  m := Symk1Hm1(X∞, Q) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' ⊗ SymkrHmr(X∞, Q), if each mi is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Given an r-tuple of even positive integers m := (m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=', mr), an l-tuple of odd positive integers  n := (n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=', nl) and an r (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' l) tuple of arbitrary positive integers k := (k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=', kr) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  k′ := (k′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=', k′  l)), denote by  T(k,k′)  (m,n) the pure part of Tk  m ⊗ Tk′  n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=', T(k,k′)  (m,n) := GrW  a Tk  m ⊗ Tk′  n ,  where a := �r  i=1 miki + �l  j=1 njk′  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Denote by  T(m,n) :=  �  (k,k′)  T(k,k′)  (m,n),  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1)  where k and k′ ranges over all k-tuple and l-tuple of positive integers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Denote by  Ts  (m,n) the same as T(m,n) with X∞ replaced by Xs for any s ∈ ∆∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Note that, the Hodge structure on Hm(Xs, Q) is pure for all m, so the “pure part” condition  is redundant in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let MTs  m be the Mumford-Tate group associated to the pure Hodge  structure Hm(Xs, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Then, the product of the Mumford-Tate groups  MTs  (m,n) := MTs  m1 × MTs  m2 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' × MTs  mr × MTs  n1 × MTs  n2 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' × MTs  nl  acts on Ts  (m,n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' The family π is called strictly Mumford-Tate with respect to (m, n) if for any  Hodge class γ ∈ T(m,n) and s ∈ h general, j∗  s(γ) is fixed by MTs  (m,n), where  j∗  s : T(m,n) → Ts  (m,n)  is induced by the pullback of the natural inclusion of Xs inside X∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let π1 : X → ∆ be a flat, projective family of genus g curves for g ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We  assume that π1 is smooth over ∆∗ and the central fiber is a very general irreducible nodal curve  (in the sense of [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Then, π1 is strictly Mumford-Tate with respect to ((0, 2), (1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Consider the family of Jacobians associated to the family of curves π1,  π2 : J → ∆∗ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=', for all t ∈ ∆∗, π−1  2 (t) = Jac(Xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  By the definition of cohomology of abelian varieties, there is a natural isomorphism of mixed  Hodge structures between H1(X∞, Q) and H1(J∞, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This induces an isomorphism of mixed  18  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' DAN AND I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' KAUR  Hodge structures,  ∗�  H1(X∞, Q) ∼  −→ H∗(J∞, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  By [7, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='3], we have  H∗  Hdg(J∞, Q) ∼= Q[θ]/(θg+1), where g = genus(Xt), t ∈ ∆∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Note that, Sym∗H0(X∞, Q) ∼= Q[T0] and Sym∗H2(X∞, Q) ∼= Q[T1] where T0 and T1 are Hodge  classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Consider the direct sum of vector spaces T(0,2),(1) as in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1) associated to the family π1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Then, the space of Hodge classes THdg in T(0,2),(1) is isomorphic to Q[T0, T1, θ]/(θg+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Similarly,  the set of Hodge class Ts  Hdg in Ts  (0,2),(1) contains Q[T s  0 , T s  1 , θs]/((θs)g+1), where (−)s := j∗  s(−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Hence, T s  0 , T s  1 and θs are fixed by the Mumford-Tate group MTs  (0,2),(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Therefore, π1 is strictly  Mumford-Tate with respect to ((0, 2), (1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This proves the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  □  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Cohomologies generated by Chern classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let X, Y be smooth, projective varieties  of dimension m and n, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Combining K¨unneth decomposition with Poincare duality,  we have for every i, k ≥ 0,  H2i−k(X × Y ) ≃  �  k  H2n−k(X) ⊗ H2i−k(Y )  ∨ ≃  �  k  Hom(H2m−k(X), H2i−k(Y )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2)  Let E be a coherent sheaf on the fibre product X ×Y and ci(E) be the i-th Chern class of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' De-  note by Φ(i,k)  E  the projection of ci(E) in H2i−k(Y ) to the component Hom(H2m−k(X), H2i−k(Y )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  By [31, Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='41], the induced morphism  Φ(i,k)  E  : H2m−k(X) → H2i−k(Y ) is a morphism of pure Hodge structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='3)  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let π1 : X ∗ → ∆∗ and π2 : Y∗ → ∆∗ be two smooth, projective families of  relative dimensions m and n, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Assume that there exists a coherent sheaf U over  X ∗ ×∆∗ Y∗ such that it is flat over ∆∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Then the morphism  Φ(i,k)  Ut  : H2m−k(Xt) → H2i−k(Yt)  induces a morphism of (limit) mixed Hodge structures:  Φ(i,k)  U,∞ : H2m−k(X∞) → H2i−k(Y∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Furthermore, the morphisms Φ(i,k)  U,∞ and Φ(i,k)  Ut  commute with pullback to closed fibers i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=', for  any u ∈ h with e(u) = t (where e is the exponential map) we have the following commutative  diagram:  H2m−k(X∞)  Φ(i,k)  U,∞  ✲ H2i−k(Y∞)  ⟲  H2m−k(Xt)  (ju)∗ ∼=  ❄  Φ(i,k)  Ut ✲ H2i−k(Yt)  (j′  u)∗ ∼=  ❄  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='4)  where ju : Yt ֒→ Y∞ and j′  u : Xt ֒→ X∞ are natural inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Consider the natural projective morphisms:  π : X ∗ ×∆∗ Y∗ → ∆∗, π1 : X ∗ → ∆∗ and π2 : Y∗ → ∆∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Consider the local system H2i := R2iπ∗Z over ∆∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We denote by  Hi  X ∗ := Riπ1∗Z and Hi  Y∗ := Riπ2∗Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  MUMFORD TATE GROUPS AND THE HODGE CONJECTURE  19  By K¨unneth decomposition in families (see [14, Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='18]), we have  H2i ≃  �  k  (Hk  X ∗ ⊗ H2i−k  Y∗ )  Applying Poincare duality to the local system Hk  X ∗ (see [16, §I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='6]), we get:  H2i ≃  �  k  (H2m−k  X ∗  )∨ ⊗ H2i−k  Y∗  ≃  �  k  Hom(H2m−k  X ∗  , H2i−k  Y∗ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  For any i, the i-th Chern class ci(U) defines a global section of H2i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Consider the projection  φ of ci(U) to Hom(H2m−k  X ∗  , H2i−k  Y∗ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Pulling back the morphism φ of local systems on ∆∗ to the  upper half plane h and taking global sections, we get the morphism  Φ(i,k)  U,∞ : H2m−k(X∞) → H2i−k(Y∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Restricting the morphism to the fiber over u ∈ h gives us the morphism Φ(i,k)  Ut  , where t := e(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  In particular, we have commutative diagram (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  It remains to check that Φ(i,k)  U,∞ is a morphism of limit mixed Hodge structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' By (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='3), Φ(i,k)  Ut  is a morphism of pure Hodge structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Since the limit Hodge filtrations on X∞ and Y∞ arise  simply as a limit of these Hodge filtrations, we conclude that Φ(i,k)  U,∞ preserves the limit Hodge  filtrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' It remains to check that Φ(i,k)  U,∞ preserves the limit weight filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Equivalently,  using the diagram (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='4) we need to prove that Φ(i,k)  Ut  preserves the weight filtration where the  weight filtration on Xt and Yt is induced by X∞ and Y∞, respectively (via the isomorphisms j∗  u  and j′  u  ∗, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Recall, the weight filtration on Xt and Yt is induced by the log of the  monodromy operators (see [25, Lemma-Definition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='9]):  NX := log(TX ) and NY := log(TY).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  So, it suffices to check that for all γ ∈ H2m−k(Xt), we have Φ(i,k)  Ut  (NX (γ)) = NYΦ(i,k)  Ut  (γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Since  ci(U) is a global section of the local system, it is monodromy invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This means the induced  morphism φ from H2m−k  X ∗  to H2i−k  Y∗  commutes with the monodromy operators i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=', for every  t ∈ ∆∗, we have following commutative diagram:  H2m−k(Xt)  Φ(i,k)  Ut✲ H2i−k(Yt)  ⟲  H2m−k(Xt)  TX  ❄  Φ(i,k)  Ut✲ H2i−k(Yt)  TY  ❄  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='5)  where TX and TY are the monodromy operators and Φ(i,k)  Ut  is as in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This implies for all  γ ∈ H2m−k(Xt), we have Φ(i,k)  Ut (TX (γ)) = TYΦ(i,k)  Ut  (γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Hence,  Φ(i,k)  Ut  (TX − Id)(γ) = Φ(i,k)  Ut  (TX (γ)) − Φ(i,k)  Ut  (γ) = TY(Φ(i,k)  Ut  (γ)) − Φ(i,k)  Ut  (γ) = (TY − id)Φ(i,k)  Ut  (γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  More generally, this implies for all m ≥ 1,  Φ(i,k)  Ut  (TX − Id)m(γ) = Φ(i,k)  Ut (TX − Id)(TX − Id)m−1(γ) = (TY − Id)Φ(i,k)  Ut  (TX − Id)m−1(γ)  Therefore, by recursion we have Φ(i,k)  Ut  (TX −Id)m(γ) = (TY −Id)mΦ(i,k)  Ut  (γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Using the logarithmic  expansion of NX and NY we conclude:  Φ(i,k)  Ut  (NX (γ)) = NYΦ(i,k)  Ut  (γ), for all γ ∈ H2m−k(Xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  20  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' DAN AND I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' KAUR  This implies that Φ(i,k)  Ut  preserves the limit weight filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This proves the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  □  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let X,Y be smooth, projective varieties of dimensions m and n, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Denote by E a coherent sheaf on X ×k Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' The variety Y is said to be cohomologically generated  by (X, E) if there is a collection SY (X, E) of pairs of integers (k, i) such that H∗(Y ) is generated  as a cohomology ring by the direct sum of the images of  Φ(i,k)  E  : H2m−k(X) → H2i−k(Y )  as the pair (k, i) varies over all the elements in SY (X, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Note that pr1(SY (X, E)) need not  contain all integers from 0 to 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We call SY (X, E) an associated indexing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Notations and Conventions 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We fix the following notations:  Seven := {(k, i) ∈ SY (X, E) | k even} and Sodd := {(k, i) ∈ SY (X, E) | k odd}  p(Seven) := {2m − k|(k, i) ∈ Seven} and p(Sodd) := {2m − k|(k, i) ∈ Sodd}  q(Seven) := {2i − k|(k, i) ∈ Seven} and q(Sodd) := {2i − k|(k, i) ∈ Sodd}  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let π1 : X ∗ → ∆∗ and π2 : Y∗ → ∆∗ be two smooth, projective families of  relative dimensions m and n, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Assume that there exists a coherent sheaf U over  X ∗ ×∆∗ Y∗ such that it is flat over ∆∗ and for general t ∈ ∆∗, Yt is cohomologically generated  by (Xt, Ut) by an indexing set SYt(Xt, Ut) such that π1 is strictly Mumford-Tate with respect to  (p(Seven), p(Sodd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Then, the family π2 is Mumford-Tate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let t ∈ ∆∗ be such that Yt is cohomologically generated by (Xt, Ut) with indexing set  SYt(Xt, Ut) such that π1 is strictly Mumford-Tate with respect to (p(Seven), p(Sodd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Using  Ehresmann’s theorem one can check that for any s ∈ ∆∗, Ys is cohomologically generated by  (Xs, Us) and we have an equality of indexing sets SYt(Xt, Ut) = SYs(Xs, Us).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Denote by  TX := T(p(Seven),p(Sodd)) and TY := T(q(Seven),q(Sodd)) with X∞ replaced by Y∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Recall, for any (k, i) ∈ SYt(Xt, Ut) we have the morphism Φ(i,k)  U,∞ of mixed Hodge structures from  H2m−k(X∞) to H2i−k(Y∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This induces a morphism of mixed Hodge structures:  φ : TX → TY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Recall, the cup-product morphism is a morphism of mixed Hodge structures [8, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  So, the composition of the cup-product morphism with φ:  Φ : TX  φ−→ TY  �  −→ H∗(Y∞, Q)  is a morphism of mixed Hodge structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Given s ∈ ∆∗, denote by (see §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1)  TXs := Ts  (p(Seven),p(Sodd)) and TYs := Ts  (q(Seven),q(Sodd)) with Xs replaced by Ys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  As before, we have the following composed morphism of Hodge structures:  Φs : TXs → TYs  �  −→ H∗(Ys, Q),  where the first morphism arises from Φ(i,k)  Us  as (k, i) ranges over entries in SYs(Xs, Us).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  By  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2 we then have the following commutative diagram:  TX  Φ✲ H∗(Y∞, Q)  ⟲  TXs  j∗  s  ❄  Φs✲ H∗(Ys, Q)  (j′  s)∗  ❄  MUMFORD TATE GROUPS AND THE HODGE CONJECTURE  21  where js (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' j′  s) is the natural inclusion of Xs (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Ys) into X∞ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Y∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Take γ ∈ F pH2p(Y∞, Q) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=', γ is a Hodge class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We need to prove that j′  s  ∗(γ) is a Hodge class  in H2p(Ys, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Since Ys is cohomologically generated by (Xs, Us) and Φ is a morphism of mixed  Hodge structures, there exists a Hodge class γ′ ∈ TX such that Φ(γ′) = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' As π1 is strictly  Mumford-Tate with respect to (p(Seven), p(Sodd)), we have j∗  s(γ′) is fixed by MTs  (p(Seven),p(Sodd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Hence, j∗  s(γ′) is a Hodge class in TXs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Since Φs is a morphism of Hodge structures, this means  (j′  s)∗(γ) = Φs ◦ j∗  s(γ′) is a Hodge class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Therefore, π2 is a Mumford-Tate family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This proves the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  □  We now use the above theorem to get an explicit example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let π1 : X → ∆ be a flat, projective family of curves satisfying the hypothesis  in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Fix an invertible sheaf L on X ∗ := π−1  1 (∆∗) of (relative) odd degree over the  punctured disc ∆∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Let  π2 : M(2, L) → ∆∗  be a relative moduli space of rank 2 semi-stable sheaves with fixed determinant L over X ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Then, π2 is a Mumford-Tate family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Consider the universal bundle U over X ∗ ×∆∗ M(2, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' It is well-known that for each  t ∈ ∆∗, the fiber M(2, L)t := π−1  2 (t) is cohomologically generated by (Xt, Ut) with the associated  indexing set (see [24, Theorem 1]):  {(0, 1), (0, 2), (1, 2), (2, 2)}  By Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='1, π1 is strictly Mumford-Tate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Then, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='5 implies that π2 is a  Mumford-Tate family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' This proves the corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  □  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' In fact, the relative moduli space M(2, L) mentioned in Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='6 degenerates  to a singular variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' A desingularization of this variety satisfies the classical Hodge conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  See [7, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='2] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content='  Acknowledgements  This article was motivated by some questions asked by Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Simpson, after the second  author gave a talk on the article [7] at the workshop ‘Moduli of bundles and related structures’  held at ICTS, Bengaluru, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Simpson for his interest and the organisers  for organising the workshop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' We also thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
+page_content=' Laterveer for his comments on an earlier  draft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf'}
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