Arithmetic of Period Maps of Geometric Origin

2012 ◽  
Author(s):  
Mark Green ◽  
Phillip Griffiths ◽  
Matt Kerr
Keyword(s):  
Hodge Structure ◽  
Complex Multiplication ◽  
Hodge Structures ◽  
Projective Varieties ◽  
Variation Of Hodge Structure ◽  
Cm Points ◽  
Geometric Origin ◽  
Isolated Points ◽  
Smooth Projective Varieties ◽  
Related Observation

This chapter considers some arithmetic aspects of period maps with a geometric origin. It focuses on the situation Φ‎ : S(ℂ) → Γ‎\D, where S parametrizes a family X → S of smooth, projective varieties defined over a number field k. The chapter recalls the notion of absolute Hodge classes (AH) and strongly absolute Hodge classes (SAH). The particular case when the Noether-Lefschetz locus consists of isolated points is alluded to in the discussion of complex multiplication Hodge structures (CM Hodge structures). A related observation is that one may formulate a variant of the “Grothendieck conjecture” in the setting of period maps and period domains. The chapter also describes a behavior of fields of definition under the period map, along with the existence and density of CM points in a motivic variation of Hodge structure.

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.

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.

scholarly journals Rank 3 rigid representations of projective fundamental groups

2018 ◽  
Vol 154 (7) ◽  
pp. 1534-1570 ◽  
Author(s):  
Adrian Langer ◽  
Carlos Simpson
Keyword(s):  
Irreducible Representation ◽  
Projective Variety ◽  
Fundamental Groups ◽  
Projective Varieties ◽  
Smooth Complex ◽  
Complex Projective Variety ◽  
Geometric Origin ◽  
Rank 2 ◽  
Complex Projective ◽  
Smooth Projective Varieties

Let$X$be a smooth complex projective variety with basepoint$x$. We prove that every rigid integral irreducible representation$\unicode[STIX]{x1D70B}_{1}(X\!,x)\rightarrow \operatorname{SL}(3,\mathbb{C})$is of geometric origin, i.e., it comes from some family of smooth projective varieties. This partially generalizes an earlier result by Corlette and the second author in the rank 2 case and answers one of their questions.

Fourier–Mukai Transforms

2007 ◽  
Author(s):  
D. Huybrechts
Keyword(s):  
Hodge Structure ◽  
Derived Categories ◽  
Projective Varieties ◽  
Central Notion ◽  
Left And Right ◽  
Smooth Projective Varieties

This chapter introduces the central notion of a Fourier-Mukai transform between derived categories. It is the derived version of the notion of a correspondence, which has been studied for all kinds of cohomology theories for many decades. In fact, Orlov's celebrated result, which is stated but not proved, says that any equivalence between derived categories of smooth projective varieties is of Fourier-Mukai type. Fourier-Mukai functors behave well in many respects: they are exact, admit left and right adjoints, can be composed, etc. The cohomological Fourier-Mukai transform behaves with respect to grading, Hodge structure, and Mukai pairing.

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.

On the Curves Associated to Certain Rings of Automorphic Forms

2001 ◽  
Vol 53 (1) ◽  
pp. 98-121 ◽  
Author(s):  
Kamal Khuri-Makdisi
Keyword(s):  
Ad Hoc ◽  
Automorphic Forms ◽  
Complex Multiplication ◽  
Graded Rings ◽  
Projective Varieties ◽  
Hecke Operator ◽  
Complex Tori ◽  
Hecke Correspondences ◽  
Cm Points ◽  
Projective Lines

AbstractIn a 1987 paper, Gross introduced certain curves associated to a definite quaternion algebra B over Q; he then proved an analog of his result with Zagier for these curves. In Gross’ paper, the curves were defined in a somewhat ad hoc manner. In this article, we present an interpretation of these curves as projective varieties arising from graded rings of automorphic forms on B×, analogously to the construction in the Satake compactification. To define such graded rings, one needs to introduce a “multiplication” of automorphic forms that arises from the representation ring of B×. The resulting curves are unions of projective lines equipped with a collection of Hecke correspondences. They parametrize two-dimensional complex tori with quaternionic multiplication. In general, these complex tori are not abelian varieties; they are algebraic precisely when they correspond to CM points on these curves, and are thus isogenous to a product E × E, where E is an elliptic curve with complex multiplication. For these CM points one can make a relation between the action of the p-th Hecke operator and Frobenius at p, similar to the well-known congruence relation of Eichler and Shimura.

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.

Sign in / Sign up

Export Citation Format

Share Document