On the locus of Hodge classes

Cattani, Eduardo; Deligné, Pierre; Kaplan, Aroldo

doi:10.1090/s0894-0347-1995-1273413-2

Cited by 107 publications

(134 citation statements)

References 7 publications

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“…We will explain the best evidence found up to now for the Hodge conjecture, namely the algebraicity of Hodge loci due to Cattani, Deligne and Kaplan [10], and some refinements obtained in [41] of their results, exploring the question whether Hodge loci are defined over the algebraic closure of Q, a necessary condition for the Hodge conjecture to hold true, and answering partially the following question, asked to us by V. Maillot and C. Soulé:…”

Section: Organisation Of the Papermentioning

confidence: 99%

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Some aspects of the Hodge conjecture

Voisin

2007

Jpn. J. Math.

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I will discuss positive and negative results on the Hodge conjecture. The negative aspects come on one side from the study of the Hodge conjecture for integral Hodge classes, and on the other side from the study of possible extensions of the conjecture to the general Kähler setting. The positive aspects come from algebraic geometry. They concern the structure of the so-called locus of Hodge classes, and of the Hodge loci.

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Section: Organisation Of the Papermentioning

confidence: 99%

“…The best evidence for the Hodge conjecture is thus due to Cattani, Deligne and Kaplan [10], who showed that the Hodge loci, and even the connected components of the locus of Hodge classes, are algebraic. We provide in [41] an improvement of their result by giving some criteria for the Hodge loci to be defined over Q.…”

Section: Introductionmentioning

confidence: 99%

Some aspects of the Hodge conjecture

Voisin

2007

Jpn. J. Math.

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“…In the remainder of this section, we prove the following generalization of Theorem 3.5; note the similarity with the main result of E. Cattani, P. Deligne, and A. Kaplan [8,Theorem 2.16]. By a slight abuse of notation, we also let Q denote the pairing between V C and sections (of the pullback to H n ) of F 0 H O , induced by the morphism V C → (F 0 H O ) ∨ described above.…”

Section: A Technical Resultsmentioning

confidence: 53%

“…, N n with coefficients that lie in a bounded interval [1, K]. By [8,Remark 4.65], the data in the SL 2 -Orbit Theorem depend real analytically on these coefficients; we can therefore use the convergence of the series as above to conclude that…”

Section: Lemma 312mentioning

confidence: 99%

Complex analytic Néron models for arbitrary families of intermediate Jacobians

Schnell

2011

Invent. math.

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Given a family of intermediate Jacobians (for a polarizable variation of integral Hodge structure of odd weight) on a Zariski-open subset of a complex manifold, we construct an analytic space that naturally extends the family. Its two main properties are: (a) the horizontal and holomorphic sections are precisely the admissible normal functions without singularities; (b) the graph of any admissible normal function has an analytic closure inside our space. As a consequence, we obtain a new proof for the zero locus conjecture of M. Green and P. Griffiths. The construction uses filtered D-modules and M. Saito's theory of mixed Hodge modules; it is functorial, and does not require normal crossing or unipotent monodromy assumptions.

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“…Predictably, the solution of Hodge's conjecture will find some basic internal relations among the three branches of mathematics, namely, analysis, topology, and algebraic geometry [5] [6] [7]. Using a nice phrase Atiyah said, "topologists used to study simple operators on complicated manifolds while analysts studied complicated operators on simple symbolic logic [12] [13] to construct the model of algebraic geometry problem (AGM), the middle step in the process consists of finding a image-mathematical solution to the image problem by algebraic geometry and the final to interpret for these complex and changing images of different matters for morphology physics by the equation as an interdisciplinary image mathematics model (IMM).…”

Section: Introductionmentioning

confidence: 99%

Morphological Group Theory of Material Structure

Zhou¹

2018

JAMP

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The correct formulation and understanding of micro-images is one of the difficulties that occur to microstructures science today, which need to develop a new appropriate mathematics for micro-images of matter system. Here I study the image mathematics and physics description of micro-images of material system by topology, set theory, symbolic logic and show that there is a naturally morphological equation, that is a law of qualitative structure of matter system, the law of the unity of two kinds of morphological structure (Jordan and hidden structure), which can be used to describe not only the common feature of different correlated matter, but also to correct classify the micro-images into different classes, so that to study the morphology groups for materials science and Algebraic geometry. The morphology equation can be found a number of applications for the observation and analysis of micro-images of material system and other natural sciences, some important basic concepts of algebraic geometry can also be newly explained by the morphology equation, such as: 1) To construct the image-mathematical language and to construct the image mathematics model (IMM) for microstructures; 2) To construct complex geometric structures (Concave polygon) then analyze these complex shape structure by analytic geometry and algebraic geometry, to study complicated operators on complicated spaces; 3) A new explanation for the logical basis, concept definition and proof way of algebraic geometry and uses it to analyze morphological structure of the new and parent phase and the problem of Hodge's theory and structure type, and points out that there may be a counterexamples for Hodge's conjecture.

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On the locus of Hodge classes

Abstract: Let S S be a nonsingular complex algebraic variety and V \mathcal {V} a polarized variation of Hodge structure of weight 2 p 2p with polarization form Q Q . Given an integer K K , let S ( K ) {S^{(K)}} be the space of pairs ( s , u ) (s,u) with … Show more

Cited by 107 publications

References 7 publications

Some aspects of the Hodge conjecture

Some aspects of the Hodge conjecture

Complex analytic Néron models for arbitrary families of intermediate Jacobians

Morphological Group Theory of Material Structure

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