Glossary

11.3 Mixed Hodge structures and variations

4 min read•august 14, 2024

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 is a finite-dimensional vector space over the complex numbers equipped with a and a satisfying certain compatibility conditions
  • The weight filtration is an increasing filtration, while the Hodge filtration is a decreasing filtration
    • The filtrations induce a of weight kk on each graded piece of the weight filtration
  • Mixed Hodge structures arise naturally in the study of the , 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 (, 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 (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

  • is a key property of variations of Hodge structures, which relates the derivatives of the period map to the Hodge filtration
  • 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 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 , 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 (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
  • 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 , 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, D\mathcal{D}-modules)

Key Terms to Review (25)

Admissible Variations of Mixed Hodge Structures: Admissible variations of mixed Hodge structures are families of mixed Hodge structures that vary continuously in a way that is compatible with the underlying topology of the space they inhabit. They serve as a bridge between algebraic geometry and topology, allowing for the analysis of how these structures behave under deformation, particularly in relation to complex geometry and algebraic cycles.
Algebraic Cycles: Algebraic cycles are formal sums of subvarieties of an algebraic variety, defined over a field, that serve as important geometric and topological constructs in algebraic geometry. They help in understanding the structure of varieties and provide a framework to study the intersection theory, which is crucial for deeper concepts like Hodge theory and mixed Hodge structures.
Cohomology of algebraic varieties: Cohomology of algebraic varieties is a mathematical framework that studies the global properties of algebraic varieties using tools from algebraic topology and sheaf theory. This approach allows mathematicians to understand the relationships between different geometric objects and provides deep insights into their structure, especially in terms of classes and cycles. It connects with mixed Hodge structures, where cohomology groups can be interpreted in terms of Hodge decomposition and variations, enriching the understanding of the geometry of algebraic varieties.
Compactifications: Compactifications refer to the process of extending a given space by adding 'points at infinity' or 'boundary points' to make it compact, which means that it is both closed and bounded. This process is essential in various fields, including algebraic geometry and complex analysis, as it allows for better control and understanding of the properties of spaces, particularly in relation to Hodge structures and their variations.
De Rham cohomology: de Rham cohomology is a mathematical tool used in algebraic geometry that studies the topology of smooth manifolds through differential forms. It connects analysis and topology by associating differential forms on a manifold to algebraic invariants, which can reveal important geometric information. This concept is significant for understanding the relationships between various cohomology theories, such as Čech cohomology, and provides insights into the structure of Hodge and mixed Hodge structures.
Deligne's Theorem: Deligne's Theorem states that the mixed Hodge structure on the cohomology of a smooth projective variety is not only a purely algebraic object but also carries deep topological and geometric information. This theorem is pivotal in connecting the realms of algebraic geometry, topology, and number theory, particularly through the concepts of variations of Hodge structures. It provides a powerful framework for understanding how the geometry of algebraic varieties relates to their topological properties.
Griffiths Transversality: Griffiths transversality refers to a concept in algebraic geometry that deals with the behavior of variations of Hodge structures. It focuses on the idea that certain mappings between complex manifolds preserve the structure of these Hodge structures in a specific way, ensuring that the fibers of variations behave nicely in relation to the base space. This notion is vital for understanding the relationships between different geometric objects and their deformations.
Hodge Decomposition: Hodge decomposition is a fundamental concept in algebraic geometry that describes how differential forms on a smooth manifold can be decomposed into orthogonal components. It shows that any differential form can be expressed as the sum of an exact form, a co-exact form, and a harmonic form, providing insight into the relationship between topology and analysis on manifolds.
Hodge Filtration: Hodge filtration is a key concept in the study of mixed Hodge structures, providing a way to organize the cohomology groups of algebraic varieties by separating their contributions from different degrees. It essentially breaks down the structure into a filtered series of subspaces, allowing mathematicians to analyze complex geometrical and topological properties of varieties through their Hodge decompositions. This filtration is crucial in understanding how variations in parameters affect the geometry of the underlying space.
Hodge numbers: Hodge numbers are integer values that arise in the study of the Hodge decomposition of cohomology groups of a Kähler manifold. They provide important information about the structure of the manifold and are denoted as $h^{p,q}$, representing the dimensions of the spaces of harmonic forms. These numbers are intimately connected to the geometry and topology of the manifold, helping to classify complex structures and understand variations in mixed Hodge structures.
Kähler Manifolds: Kähler manifolds are a special class of complex manifolds that possess a Riemannian metric compatible with the complex structure and a symplectic form. These manifolds play a significant role in bridging complex geometry with differential geometry, allowing the study of geometric properties through both complex and symplectic perspectives.
Limiting Mixed Hodge Structure: A limiting mixed Hodge structure is a specific kind of mixed Hodge structure that arises in the study of algebraic varieties and their degenerations, particularly in the context of variations of Hodge structures. It encodes the asymptotic behavior of Hodge structures as one approaches a limit point in a moduli space, allowing for the analysis of degenerating families of complex structures. This concept plays a critical role in understanding how the topology and geometry of algebraic varieties behave under deformation.
M. Saito: M. Saito is a mathematician known for his contributions to the theory of mixed Hodge structures and variations. His work focuses on the interplay between algebraic geometry and Hodge theory, particularly in understanding how variations of Hodge structures can arise in families of algebraic varieties. Saito's insights have led to significant advancements in the study of how these structures behave under different conditions, deepening our understanding of their geometric and topological properties.
Mixed hodge modules: Mixed Hodge modules are a mathematical framework that integrates the concepts of Hodge theory and algebraic geometry, allowing for a richer understanding of the interplay between complex geometry and homological algebra. They extend the notion of Hodge structures to include singular spaces and have applications in areas such as deformation theory and the study of perverse sheaves.
Mixed Hodge Structure: A mixed Hodge structure is a mathematical framework that combines both Hodge structures and the concept of weight to study the cohomology of algebraic varieties. It allows for the handling of more complex spaces by integrating aspects of both algebraic and topological features, particularly in the context of variations. This concept plays a significant role in understanding the behavior of families of algebraic varieties and their singularities.
Mixed Tate Motives: Mixed Tate motives are a class of algebraic objects that generalize Tate motives, allowing for a richer structure in the study of algebraic geometry. They play a crucial role in the connection between Hodge theory and motivic cohomology, providing a framework to understand the relationship between various cohomology theories and their applications in arithmetic geometry.
Pierre Deligne: Pierre Deligne is a renowned Belgian mathematician known for his contributions to algebraic geometry and number theory, particularly in the realm of mixed Hodge structures. His work has significantly influenced the understanding of the relationship between topology and algebraic geometry, and he was awarded the Fields Medal in 1978 for his groundbreaking achievements in these areas.
Polarizable mixed Hodge structure: A polarizable mixed Hodge structure is a mathematical framework that combines aspects of both Hodge theory and algebraic geometry, allowing for the study of mixed Hodge structures that have a polarization. This means that there exists a non-degenerate bilinear form on the underlying vector space which satisfies certain conditions, linking the geometry of complex varieties to their topology through the theory of mixed Hodge structures.
Pure Hodge Structure: A pure Hodge structure is a special type of mathematical structure that arises in the study of algebraic geometry and complex manifolds, characterized by a decomposition of the cohomology groups into parts that reflect both algebraic and geometric properties. This structure allows one to understand how different cohomology classes relate to each other through their Hodge decomposition, which splits them into components that can be analyzed independently. Pure Hodge structures serve as the foundational concept in understanding mixed Hodge structures, particularly in how they combine and interact with variations in the context of algebraic varieties.
Singular cohomology: Singular cohomology is a powerful mathematical tool in algebraic topology that assigns a sequence of abelian groups or vector spaces to a topological space, capturing its shape and structure. This concept plays a crucial role in understanding topological spaces by providing a way to classify them based on their 'holes' and 'voids', linking closely to Hodge structures and the interplay between topology and algebraic geometry. The relationship between singular cohomology and these structures provides insights into how geometric properties translate into algebraic forms.
Specialization Theorem: The specialization theorem states that if you have a family of algebraic varieties over a base scheme, the geometry of the fibers can be understood in terms of the geometry of the generic fiber and its specializations. This concept connects the properties of the varieties in the family to those of their limits, revealing how they behave as one approaches certain points in the base space. It plays a crucial role in understanding variations and their mixed Hodge structures.
Spectral sequence: A spectral sequence is a mathematical tool used in homological algebra and algebraic topology to compute homology groups and other invariants through a series of approximations. It organizes data in a multi-layered fashion, allowing for the systematic extraction of information from complex structures like mixed Hodge structures and variations, ultimately simplifying the computations involved.
Theorems on the Topology of Variations: Theorems on the topology of variations refer to a collection of results in algebraic geometry and Hodge theory that investigate how variations of complex structures behave under continuous deformations. These theorems are crucial for understanding the interplay between algebraic geometry, complex differential geometry, and the broader implications of Hodge theory, particularly in relation to mixed Hodge structures.
Variation of Mixed Hodge Structures: A variation of mixed Hodge structures is a mathematical framework that captures the changes in the mixed Hodge structure as one moves through a family of algebraic varieties. This concept is significant in understanding how the topological and algebraic properties of a family of complex manifolds interact and vary continuously, reflecting how the Hodge decomposition alters as parameters change.
Weight Filtration: Weight filtration is a concept in algebraic geometry and Hodge theory that organizes the structure of a mixed Hodge structure according to the weights of the elements involved. This filtration allows mathematicians to study how the topology of a space interacts with complex geometry by categorizing cohomology groups based on their 'weight', thus revealing intricate relationships between different levels of geometry and algebra.
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