Theory of variations of Hodge structure
Introduction
The theory of variations of Hodge structure is a sophisticated area in algebraic geometry and complex analysis, focusing on the study of how Hodge structures vary in families of complex algebraic varieties. This theory is pivotal in understanding the geometric and topological properties of complex manifolds and plays a crucial role in string theory and mirror symmetry. It extends the classical Hodge theory, which deals with the decomposition of the cohomology of a smooth projective variety over the complex numbers, into a more dynamic setting where these structures change continuously.
Background and Motivation
Hodge theory, named after the British mathematician W.V.D. Hodge, provides a bridge between the algebraic topology of a smooth projective variety and its complex geometry. A Hodge structure on a vector space is a decomposition that reflects the complex structure of the variety. The classical Hodge decomposition is static, applicable to a single variety. However, in many situations, one deals with families of varieties, such as in the study of moduli spaces, where it becomes essential to understand how these Hodge structures vary.
The motivation for studying variations of Hodge structure arises from several areas. In arithmetic geometry, understanding these variations is crucial for the study of Shimura varieties and automorphic forms. In string theory, variations of Hodge structure are fundamental in the study of Calabi-Yau manifolds and their moduli spaces, which are central to the formulation of mirror symmetry.
Basic Definitions
Hodge Structure
A Hodge structure of weight \( n \) on a finite-dimensional vector space \( V \) over the rational numbers \( \mathbb{Q} \) is a decomposition of its complexification \( V_{\mathbb{C}} = V \otimes_{\mathbb{Q}} \mathbb{C} \) into a direct sum of subspaces \( V^{p,q} \) such that \( \overline{V^{p,q}} = V^{q,p} \) and \( p + q = n \). This decomposition reflects the complex structure of the variety and is a key tool in understanding its geometric properties.
Variation of Hodge Structure (VHS)
A variation of Hodge structure is a family of Hodge structures parameterized by a complex manifold \( S \). Formally, it consists of a local system \( \mathcal{L} \) of \( \mathbb{Q} \)-vector spaces over \( S \), a holomorphic vector bundle \( \mathcal{H} \) with a flat connection \( \nabla \), and a filtration \( F^\bullet \) of \( \mathcal{H} \) by holomorphic subbundles, satisfying the Griffiths transversality condition: \( \nabla F^p \subset F^{p-1} \otimes \Omega_S^1 \).
Properties and Examples
Griffiths Transversality
The Griffiths transversality condition is a fundamental property of variations of Hodge structure. It ensures that the Hodge filtration changes in a controlled manner as one moves along the base manifold \( S \). This condition is crucial for the geometric interpretation of VHS and plays a significant role in the study of period mappings.
Period Domains and Period Mappings
A period domain is a complex manifold that parametrizes Hodge structures of a given type. The period mapping is a holomorphic map from the base manifold \( S \) to a period domain, reflecting how the Hodge structures vary. The image of this map is often a complex submanifold, and its study involves deep results in differential geometry and representation theory.
Examples
One of the simplest examples of a variation of Hodge structure is provided by the family of elliptic curves. As the complex structure of the elliptic curve varies, so does its Hodge structure, which can be described by the variation of its periods. More complex examples include families of K3 surfaces and Calabi-Yau manifolds, which are central objects in both mathematics and theoretical physics.
Applications
Arithmetic Geometry
In arithmetic geometry, variations of Hodge structure are used to study the arithmetic properties of algebraic varieties. They are closely related to the theory of motives and play a role in the formulation of the Hodge conjecture, one of the central problems in modern mathematics.
String Theory and Mirror Symmetry
In theoretical physics, particularly in string theory, variations of Hodge structure are essential in the study of compactifications and the formulation of mirror symmetry. The moduli space of Calabi-Yau manifolds, which are used to model extra dimensions in string theory, is described using variations of Hodge structure.
Advanced Topics
Mixed Hodge Structures
A mixed Hodge structure is a generalization of a pure Hodge structure, allowing for a more flexible decomposition that can accommodate singularities and degenerations. These structures are essential in the study of singular varieties and play a role in the theory of motivic cohomology.
Degenerations and Limit Mixed Hodge Structures
When studying families of varieties, one often encounters degenerations, where the variety becomes singular. The theory of limit mixed Hodge structures describes how the Hodge structure behaves in these situations, providing insights into the geometry of singularities.
Hodge Theory on Non-Compact Manifolds
The extension of Hodge theory to non-compact manifolds involves additional challenges, such as the need to consider L^2 cohomology and the use of Hodge modules. These extensions are crucial for understanding the geometry of non-compact moduli spaces and their applications in both mathematics and physics.