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.

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.

Hodge Theory (MN-49)

2017 ◽  
Author(s):  
Eduardo Cattani ◽  
Fouad El Zein ◽  
Phillip A. Griffiths ◽  
Lê Dung Tráng
Keyword(s):  
Summer School ◽  
Hodge Structure ◽  
Hodge Theory ◽  
Algebraic Cycles ◽  
Mixed Hodge Structure ◽  
Hodge Structures ◽  
De Rham ◽  
New Treatment ◽  
Variations Of Hodge Structure ◽  
Mixed Hodge Structures

This book provides a comprehensive and up-to-date introduction to Hodge theory—one of the central and most vibrant areas of contemporary mathematics—from leading specialists on the subject. The topics range from the basic topology of algebraic varieties to the study of variations of mixed Hodge structure and the Hodge theory of maps. Of particular interest is the study of algebraic cycles, including the Hodge and Bloch–Beilinson Conjectures. Based on lectures delivered at the 2010 Summer School on Hodge Theory at the ICTP in Trieste, Italy, the book is intended for a broad group of students and researchers. The exposition is as accessible as possible and does not require a deep background. At the same time, the book presents some topics at the forefront of current research. The book is divided between introductory and advanced lectures. The introductory lectures address Kähler manifolds, variations of Hodge structure, mixed Hodge structures, the Hodge theory of maps, period domains and period mappings, algebraic cycles (up to and including the Bloch–Beilinson conjecture) and Chow groups, sheaf cohomology, and a new treatment of Grothendieck's algebraic de Rham theorem. The advanced lectures address a Hodge-theoretic perspective on Shimura varieties, the spread philosophy in the study of algebraic cycles, absolute Hodge classes (including a new, self-contained proof of Deligne's theorem on absolute Hodge cycles), and variation of 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 Variations of Mixed Hodge Structures of Multiple Polylogarithms

2004 ◽  
Vol 56 (6) ◽  
pp. 1308-1338 ◽  
Author(s):  
Jianqiang Zhao
Keyword(s):  
Number Field ◽  
Zeta Functions ◽  
Explicit Construction ◽  
Hodge Structures ◽  
Special Values ◽  
Multiple Polylogarithms ◽  
Real Analytic ◽  
Mixed Hodge Structures

AbstractIt is well known that multiple polylogarithms give rise to good unipotent variations of mixed Hodge-Tate structures. In this paper we shall explicitly determine these structures related to multiple logarithms and some other multiple polylogarithms of lower weights. The purpose of this explicit construction is to give some important applications. First we study the limit of mixed Hodge-Tate structures and make a conjecture relating the variations of mixed Hodge-Tate structures of multiple logarithms to those of general multiple polylogarithms. Then following Deligne and Beilinson we describe an approach to defining the single-valued real analytic version of the multiple polylogarithms which generalizes the well-known result of Zagier on classical polylogarithms. In the process we find some interesting identities relating single-valued multiple polylogarithms of the same weight k when k = 2 and 3. At the end of this paper, motivated by Zagier's conjecture we pose a problem which relates the special values of multiple Dedekind zeta functions of a number field to the single-valued version of multiple polylogarithms.

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 to Variations of Hodge Structure

2017 ◽  
Author(s):  
Eduardo Cattani
Keyword(s):  
Asymptotic Behavior ◽  
Hodge Structure ◽  
Flat Connection ◽  
Classifying Spaces ◽  
Hodge Structures ◽  
Projective Varieties ◽  
Local Systems ◽  
Period Mapping ◽  
Variations Of Hodge Structure ◽  
Smooth Projective Varieties

This chapter emphasizes the theory of abstract variations of Hodge structure (VHS) and, in particular, their asymptotic behavior. It first studies the basic correspondence between local systems, representations of the fundamental group, and bundles with a flat connection. The chapter then turns to analytic families of smooth projective varieties, the Kodaira–Spencer map, Griffiths' period map, and a discussion of its main properties: holomorphicity and horizontality. These properties motivate the notion of an abstract VHS. Next, the chapter defines the classifying spaces for polarized Hodge structures and studies some of their basic properties. Finally, the chapter deals with the asymptotics of a period mapping with particular attention to Schmid's orbit theorems.

Hodge Structures with Complex Multiplication

2012 ◽  
Author(s):  
Mark Green ◽  
Phillip Griffiths ◽  
Matt Kerr
Keyword(s):  
Abelian Variety ◽  
Number Field ◽  
Hodge Structure ◽  
Complex Multiplication ◽  
Classical Case ◽  
Number Fields ◽  
Hodge Structures ◽  
High Degree ◽  
Field Setting ◽  
Imaginary Number

This chapter describes Hodge structures with a high degree of symmetry, and specifically complex multiplication Hodge structures or CM Hodge structures. It broadens the notion of CM type by defining an n-orientation of a totally imaginary number field and constructs a precise correspondence between these and certain important kinds of CM Hodge structures. In the classical case of weight n = 1, the abelian variety associated to a CM type is recovered. The notion of the Kubota rank and reflex field associated to a CM Hodge structure is then generalized to the totally imaginary number field setting. When the Kubota rank is maximal, the CM Hodge structure is non-degenerate. The discussion also covers oriented imaginary number fields, Hodge structures with special endomorphisms, polarization and Mumford-Tate groups, and the Mumford-Tate group in the Galois case.

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