Polarized mixed Hodge structures and the local monodromy of a variation of Hodge structure

1982 ◽  
Vol 67 (1) ◽  
pp. 101-115 ◽  
Author(s):  
Eduardo Cattani ◽  
Aroldo Kaplan
Keyword(s):  
Hodge Structure ◽  
Hodge Structures ◽  
Variation Of Hodge Structure ◽  
Local Monodromy ◽  
Mixed Hodge Structures

scholarly journals Hodge correlators

2019 ◽  
Vol 2019 (748) ◽  
pp. 1-138
Author(s):  
Alexander B. Goncharov
Keyword(s):  
Fundamental Group ◽  
Hodge Structure ◽  
Euler System ◽  
Feynman Integral ◽  
Mixed Hodge Structure ◽  
Modular Curves ◽  
Hodge Structures ◽  
Motivic Cohomology ◽  
Linear Differential ◽  
Mixed Hodge Structures

Abstract Hodge correlators are complex numbers given by certain integrals assigned to a smooth complex curve. We show that they are correlators of a Feynman integral, and describe the real mixed Hodge structure on the pronilpotent completion of the fundamental group of the curve. We introduce motivic correlators, which are elements of the motivic Lie algebra and whose periods are the Hodge correlators. They describe the motivic fundamental group of the curve. We describe variations of real mixed Hodge structures on a variety by certain connections on the product of the variety by twistor plane. We call them twistor connections. In particular, we define the canonical period map on variations of real mixed Hodge structures. We show that the obtained period functions satisfy a simple Maurer–Cartan type non-linear differential equation. Generalizing this, we suggest a DG-enhancement of the subcategory of Saito’s Hodge complexes with smooth cohomology. We show that when the curve varies, the Hodge correlators are the coefficients of the twistor connection describing the corresponding variation of real MHS. Examples of the Hodge correlators include classical and elliptic polylogarithms, and their generalizations. The simplest Hodge correlators on the modular curves are the Rankin–Selberg integrals. Examples of the motivic correlators include Beilinson’s elements in the motivic cohomology, e.g. the ones delivering the Beilinson–Kato Euler system on modular curves.

scholarly journals MIXED HODGE STRUCTURES WITH MODULUS

2020 ◽  
pp. 1-35
Author(s):  
Florian Ivorra ◽  
Takao Yamazaki
Keyword(s):  
Hodge Structure ◽  
Mixed Hodge Structure ◽  
Hodge Structures ◽  
Classical Notion ◽  
Mixed Hodge Structures

We define a notion of mixed Hodge structure with modulus that generalizes the classical notion of mixed Hodge structure introduced by Deligne and the level one Hodge structures with additive parts introduced by Kato and Russell in their description of Albanese varieties with modulus. With modulus triples of any dimension, we attach mixed Hodge structures with modulus. We combine this construction with an equivalence between the category of level one mixed Hodge structures with modulus and the category of Laumon 1-motives to generalize Kato–Russell’s Albanese varieties with modulus to 1-motives.

scholarly journals Lowest weights in cohomology of variations of Hodge structure

2012 ◽  
Vol 206 ◽  
pp. 1-24
Author(s):  
Chris Peters ◽  
Morihiko Saito
Keyword(s):  
Open Subset ◽  
Hodge Structure ◽  
Analytic Space ◽  
Mixed Hodge Structure ◽  
Zariski Open Subset ◽  
Weight Part ◽  
Variation Of Hodge Structure ◽  
Local Monodromy ◽  
Variations Of Hodge Structure ◽  
Complex Analytic Space

AbstractLetXbe an irreducible complex analytic space withj:U ↪ Xan immersion of a smooth Zariski-open subset, and let 𝕍 be a variation of Hodge structure of weightnoverU. Assume thatXis compact Kähler. Then, provided that the local monodromy operators at infinity are quasi-unipotent,IHk(X, 𝕍) is known to carry a pure Hodge structure of weightk+n, whileHk(U, 𝕍) carries a mixed Hodge structure of weight at leastk+n. In this note it is shown that the image of the natural mapIHk(X, 𝕍) →Hk(U, 𝕍) is the lowest-weight part of this mixed Hodge structure. In the algebraic case this easily follows from the formalism of mixed sheaves, but the analytic case is rather complicated, in particular when the complementX — Uis not a hypersurface.

The Mumford-Tate Group of a Variation of Hodge Structure

2012 ◽  
Author(s):  
Mark Green ◽  
Phillip Griffiths ◽  
Matt Kerr
Keyword(s):  
Complex Manifold ◽  
Moduli Space ◽  
Hodge Structure ◽  
Integral Manifold ◽  
Equivalence Classes ◽  
Hodge Structures ◽  
Tate Group ◽  
Variation Of Hodge Structure ◽  
Definition Of ◽  
Variations Of Hodge Structure

This chapter deals with the Mumford-Tate group of a variation of Hodge structure (VHS). It begins by presenting a definition of VHS, which consists of a connected complex manifold and a locally liftable, holomorphic mapping that is an integral manifold of the canonical differential ideal. The moduli space of Γ‎-equivalence classes of polarized Hodge structures is also considered, along with a generic point for the VHS and the monodromy group of the VHS. Associated to a VHS is its Mumford-Tate group. The chapter proceeds by discussing the structure theorem for VHS, where S is a quasi-projective algebraic variety, referred to as global variations of Hodge structure. It concludes by describing an application of Mumford-Tate groups, along with the Noether-Lefschetz locus.

Mumford-Tate Groups

2012 ◽  
Author(s):  
Mark Green ◽  
Phillip Griffiths ◽  
Matt Kerr
Keyword(s):  
Hodge Structure ◽  
Algebraic Groups ◽  
Hodge Decomposition ◽  
Hodge Structures ◽  
The Third ◽  
Direct Sum ◽  
Hodge Filtration ◽  
Mixed Hodge Structures

This chapter provides an introduction to the basic definitions and properties of Mumford-Tate groups in both the case of Hodge structures and of mixed Hodge structures. Hodge structures of weight n are sometimes called pure Hodge structures, and the term “Hodge structure” then refers to a direct sum of pure Hodge structures. The chapter presents three definitions of a Hodge structure of weight n, given in historical order. In the first definition, a Hodge structure of weight n is given by a Hodge decomposition; in the second, it is given by a Hodge filtration; in the third, it is given by a homomorphism of ℝ-algebraic groups. In the first two definitions, n is assumed to be positive and the p,q's in the definitions are non-negative. In the third definition, n and p,q are arbitrary. For the third definition, the Deligne torus integers are used.

Introduction

2012 ◽  
Author(s):  
Mark Green ◽  
Phillip Griffiths ◽  
Matt Kerr
Keyword(s):  
Lie Group ◽  
Hodge Structure ◽  
Algebraic Groups ◽  
Complex Multiplication ◽  
Symmetry Groups ◽  
Hodge Structures ◽  
Fundamental Symmetry ◽  
Variation Of Hodge Structure ◽  
Geometric Origin

This book deals with Mumford-Tate groups, the fundamental symmetry groups in Hodge theory. Much, if not most, of the use of Mumford-Tate groups has been in the study of polarized Hodge structures of level one and those constructed from this case. In this book, Mumford-Tate groups M will be reductive algebraic groups over ℚ such that the derived or adjoint subgroup of the associated real Lie group M ℝ contains a compact maximal torus. In order to keep the statements of the results as simple as possible, the book emphasizes the case when M ℝ itself is semi-simple. The discussion covers period domains and Mumford-Tate domains, the Mumford-Tate group of a variation of Hodge structure, Hodge representations and Hodge domains, Hodge structures with complex multiplication, arithmetic aspects of Mumford-Tate domains, classification of Mumford-Tate subdomains, and arithmetic of period maps of geometric origin.

scholarly journals Arithmetic of Degenerating Principal Variations of Hodge Structure: Examples Arising From Mirror Symmetry and Middle Convolution

2016 ◽  
Vol 68 (2) ◽  
pp. 280-308 ◽  
Author(s):  
Genival da Silva ◽  
Matt Kerr ◽  
Gregory Pearlstein
Keyword(s):  
Number Field ◽  
Crucial Role ◽  
Mirror Symmetry ◽  
Hodge Structure ◽  
Hodge Structures ◽  
Tate Group ◽  
The Family ◽  
Variations Of Hodge Structure ◽  
Mixed Hodge Structures ◽  
Middle Convolution

AbstractWe collect evidence in support of a conjecture of Griffiths, Green, and Kerr on the arithmetic of extension classes of limiting mixed Hodge structures arising from semistable degenerations over a number field. After briefly summarizing how a result of Iritani implies this conjecture for a collection of hypergeometric Calabi–Yau threefold examples studied by Doran and Morgan, the authors investigate a sequence of (non-hypergeometric) examples in dimensions 1 ≤ d ≤ 6 arising from Katz's theory of the middle convolution. A crucial role is played by the Mumford-Tate group (which is G2) of the family of 6-folds, and the theory of boundary components of Mumford–Tate domains.

scholarly journals Lowest weights in cohomology of variations of Hodge structure

2012 ◽  
Vol 206 ◽  
pp. 1-24
Author(s):  
Chris Peters ◽  
Morihiko Saito
Keyword(s):  
Open Subset ◽  
Hodge Structure ◽  
Analytic Space ◽  
Mixed Hodge Structure ◽  
Zariski Open Subset ◽  
Weight Part ◽  
Variation Of Hodge Structure ◽  
Local Monodromy ◽  
Variations Of Hodge Structure ◽  
Complex Analytic Space

AbstractLetXbe an irreducible complex analytic space withj:U ↪ Xan immersion of a smooth Zariski-open subset, and let 𝕍 be a variation of Hodge structure of weightnoverU. Assume thatXis compact Kähler. Then, provided that the local monodromy operators at infinity are quasi-unipotent,IHk(X, 𝕍) is known to carry a pure Hodge structure of weightk+n, whileHk(U, 𝕍) carries a mixed Hodge structure of weight at leastk+n. In this note it is shown that the image of the natural mapIHk(X, 𝕍) →Hk(U, 𝕍) is the lowest-weight part of this mixed Hodge structure. In the algebraic case this easily follows from the formalism of mixed sheaves, but the analytic case is rather complicated, in particular when the complementX — Uis not a hypersurface.

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