Degeneration of Hodge Structures

Cattani, Eduardo; Kaplan, Aroldo; Schmid, Wilfried

doi:10.2307/1971333

Cited by 268 publications

(651 citation statements)

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“…As a preliminary step in our analysis of the zero locus of at infinity, we derive the local defining equations of ᐆ at an interior point of S. To this end, we begin with a review of mixed Hodge structures and their gradings, following [CKS86].…”

Section: The Zero Locus At a Smooth Pointmentioning

confidence: 99%

The zero locus of an admissible normal function

Brosnan

Pearlstein

2009

Ann. Math.

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Section: The Zero Locus At a Smooth Pointmentioning

confidence: 99%

The zero locus of an admissible normal function

Brosnan

Pearlstein

2009

Ann. Math.

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“…We then recall results of the SL(2/ -theory of [3] in a form suitable for our purposes. (A' ,B') is the image of (A, B) by some g E GL(V) close to the identity.…”

Section: Preliminariesmentioning

confidence: 99%

On the locus of Hodge classes

Cattani¹,

Deligné²,

Kaplan³

1995

J. Amer. Math. Soc.

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107

124

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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 s ∈ S s \in S , u ∈ V s u \in {\mathcal {V}_s} integral of type ( p , p ) (p,p) , and Q ( u , u ) ≤ K Q(u,u) \leq K . We show in Theorem 1.1 that S ( K ) {S^{(K)}} is an algebraic variety, finite over S S . When V \mathcal {V} is the local system H 2 p ( X s , Z ) {H^{2p}}({X_s},\mathbb {Z}) /torsion associated with a family of nonsingular projective varieties parametrized by S S , the result implies that the locus where a given integral class of type ( p , p ) (p,p) remains of type ( p , p ) (p,p) is algebraic.

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“…Saito originally developed the theory of Hodge modules with rational coefficients, but as explained in [Sai90a], everything works just as well with real coefficients, provided one relaxes the assumptions about local monodromy: the eigenvalues of the monodromy operator on the nearby cycles are allowed to be arbitrary complex numbers of absolute value one, rather than just roots of unity. This has already been observed several times in the literature [SV11]; the point is that Saito's theory rests on certain results about polarizable variations of Hodge structure [Sch73,Zuc79,CKS86], which hold in this generality.…”

Section: Bmentioning

confidence: 55%

Hodge modules on complex tori and generic vanishing for compact Kähler manifolds

2017

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Abstract. We extend the results of generic vanishing theory to polarizable real Hodge modules on compact complex tori, and from there to arbitrary compact Kähler manifolds. As applications, we obtain a bimeromorphic characterization of compact complex tori (among compact Kähler manifolds), semipositivity results, and a description of the Leray filtration for maps to tori. A. IntroductionThe term "generic vanishing" refers to a collection of theorems about the cohomology of line bundles with trivial first Chern class. The first results of this type were obtained by Green and Lazarsfeld in the late 1980s [GL87, GL91]; they were proved using classical Hodge theory and are therefore valid on arbitrary compact Kähler manifolds. About ten years ago, Hacon [Hac04] found a more algebraic approach, using vanishing theorems and the Fourier-Mukai transform, that has led to many additional results in the projective case; see also [PP11a,CJ13,PS13]. The purpose of this paper is to show that the newer results are in fact also valid on arbitrary compact Kähler manifolds.Besides [Hac04], our motivation also comes from a 2013 paper by Chen and Jiang [CJ13] in which they prove, roughly speaking, that the direct image of the canonical bundle under a generically finite morphism to an abelian variety is semi-ample. Before we can state more precise results, recall the following useful definitions (see §13 for more details).

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Degeneration of Hodge Structures

Cited by 268 publications

References 2 publications

The zero locus of an admissible normal function

The zero locus of an admissible normal function

On the locus of Hodge classes

Hodge modules on complex tori and generic vanishing for compact Kähler manifolds

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