11.3 Mixed Hodge structures and variations

Hodge structures are a powerful tool in algebraic geometry, helping us understand complex varieties. Mixed Hodge structures extend this to singular or non-compact varieties, revealing key geometric and topological info.

Variations of Hodge structures let us study families of algebraic varieties, like moduli spaces. They encode how Hodge structures change continuously, giving insights into geometry and arithmetic of whole families.

Mixed Hodge Structures in Algebraic Geometry

Definition and Properties

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• A mixed Hodge structure is a finite-dimensional vector space over the complex numbers equipped with a weight filtration and a Hodge filtration satisfying certain compatibility conditions
• The weight filtration is an increasing filtration, while the Hodge filtration is a decreasing filtration
  • The filtrations induce a pure Hodge structure of weight $k$ on each graded piece of the weight filtration
• Mixed Hodge structures arise naturally in the study of the cohomology of algebraic varieties, particularly when considering singular or non-compact varieties (singular curves, open varieties)
• The mixed Hodge structure on the cohomology of a complex algebraic variety encodes important geometric and topological information about the variety (Hodge numbers, Betti numbers)

Fundamental Results and Applications

• The existence of mixed Hodge structures on the cohomology of algebraic varieties is a fundamental result in Hodge theory, proven by Deligne
• Mixed Hodge structures provide a powerful tool for studying the geometry and arithmetic of algebraic varieties, as well as their moduli spaces and degeneration behavior
• They can be used to extract information about the intermediate Jacobians, cycle classes, and period integrals of algebraic varieties
• Mixed Hodge structures play a crucial role in the study of the cohomology of singular varieties and their compactifications (resolution of singularities, divisor complements)

Variations of Hodge Structures

Definition and Properties

• A variation of Hodge structure is a family of Hodge structures parametrized by a complex manifold or algebraic variety, satisfying certain compatibility and holomorphicity conditions
• The notion of a variation of Hodge structure generalizes the concept of a single Hodge structure to a family of Hodge structures that vary continuously or algebraically over a base space
• Variations of Hodge structures arise naturally in the study of families of algebraic varieties, such as in the theory of moduli spaces (moduli of curves, moduli of abelian varieties)
• The period map associated to a variation of Hodge structure encodes the variation of the Hodge structures in the family and provides important geometric and arithmetic information

Key Concepts and Generalizations

• Griffiths transversality is a key property of variations of Hodge structures, which relates the derivatives of the period map to the Hodge filtration
• Admissible variations of mixed Hodge structures are a generalization of variations of Hodge structures that allow for singular fibers and incorporate mixed Hodge structures
• The study of variations of Hodge structures and their period maps is a central topic in Hodge theory and has applications to moduli theory, arithmetic geometry, and mirror symmetry (Hodge conjecture, Torelli theorem)

Hodge Structures under Degeneration

Degeneration and Limiting Mixed Hodge Structures

• Degeneration of Hodge structures refers to the study of how Hodge structures behave when the underlying algebraic variety degenerates to a singular or non-reduced variety
• The limiting mixed Hodge structure is a key concept in the study of degeneration, which describes the mixed Hodge structure that arises in the limit of a family of Hodge structures
• The Clemens-Schmid exact sequence relates the limiting mixed Hodge structure to the mixed Hodge structures of the special and generic fibers in a degeneration (semistable reduction, monodromy)

Specialization and the Specialization Theorem

• Specialization of Hodge structures refers to the process of restricting a variation of Hodge structure to a point or a subvariety of the base space
• The specialization theorem, proven by Deligne, states that the specialization of a variation of Hodge structure at a point is a mixed Hodge structure
• The study of degeneration and specialization of Hodge structures is crucial for understanding the behavior of Hodge structures in families and their relationship to the geometry of the underlying varieties (degeneration of abelian varieties, Neron models)

Mixed Hodge Structures for Cohomology

Cohomology of Algebraic Varieties

• Mixed Hodge structures provide a powerful framework for studying the cohomology of algebraic varieties, particularly in the presence of singularities or non-compact components
• The mixed Hodge structure on the cohomology of an algebraic variety can be used to extract important geometric and topological information, such as Hodge numbers, Betti numbers, and the structure of the intermediate Jacobians
• The study of mixed Hodge structures on the cohomology of algebraic varieties has applications to the topology of complex algebraic varieties, the theory of motives, and the study of algebraic cycles (Hodge conjecture, Bloch-Beilinson filtrations)

Compactifications and Mixed Hodge Modules

• Compactification is a process of embedding a non-compact algebraic variety into a compact variety, often by adding a boundary or completing the variety in a suitable manner (smooth compactifications, toroidal compactifications)
• The mixed Hodge structure on the cohomology of a compactification of an algebraic variety is closely related to the mixed Hodge structure on the cohomology of the original variety, often through a long exact sequence or a spectral sequence
• The study of mixed Hodge structures on compactifications can provide insights into the geometry and arithmetic of the original variety, such as its intersection theory, cycle classes, and period integrals
• The theory of mixed Hodge modules, developed by Saito, provides a sheaf-theoretic approach to mixed Hodge structures and allows for the study of mixed Hodge structures on the cohomology of algebraic varieties in a more general and functorial setting (perverse sheaves, $\mathcal{D}$-modules)