12.1 Symplectic geometry and complex algebraic varieties
4 min read•july 30, 2024
Symplectic geometry and complex algebraic varieties are deeply intertwined, with symplectic structures naturally arising on smooth complex varieties. The on complex manifolds induces a symplectic structure, establishing a fundamental connection between these two mathematical realms.
This connection enables powerful tools from both fields to be applied in tandem. parallels quotients, while moment maps provide a bridge between symplectic geometry and algebraic geometry in the context of group actions on varieties.
Symplectic Geometry and Algebraic Varieties
Fundamental Connections
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- Symplectic geometry studies manifolds equipped with a , 2-form called a
- Complex algebraic varieties constitute geometric objects defined by polynomial equations in complex projective space
- Kähler form on a complex manifold induces a natural symplectic structure establishes fundamental connection between complex algebraic geometry and symplectic geometry
- Holomorphic maps between Kähler manifolds automatically preserve symplectic structure induced by Kähler form
- of a complex manifold has canonical symplectic structure plays crucial role in relating symplectic and complex algebraic geometry
Parallel Constructions and Correspondences
- Symplectic reduction and geometric invariant theory (GIT) quotients provide parallel constructions in symplectic and algebraic geometry often yield equivalent results
- in symplectic geometry corresponds to notion of linearized group action on line bundles in algebraic geometry
- Symplectic subvarieties within smooth locus correspond to certain algebraic subvarieties provide geometric interpretation of algebraic properties
- of symplectic structures on smooth locus relates to deformations of underlying algebraic variety connects symplectic and algebraic deformation theories
Invariants and Computations
- on smooth locus defined using symplectic form encodes important algebraic and geometric information about variety
- of smooth locus computed using intersection theory and characteristic classes provides bridge between symplectic and algebraic invariants
- of coherent sheaves on closely related to of resolved symplectic variety connects algebraic and symplectic invariants
Symplectic Structure on Smooth Varieties
Induced Structures and Constructions
- Smooth locus of inherits natural symplectic structure from ambient projective space induced by
- Symplectic forms on smooth locus constructed using and Kähler identities relate to variety's cohomology structure
- Symplectic structure on smooth locus interacts with variety's singularities influences local and global properties of algebraic variety
- Process of used to construct symplectic resolutions provides geometric interpretation of resolution process
Applications in Algebraic Geometry
- Symplectic subvarieties within smooth locus correspond to certain algebraic subvarieties provide geometric interpretation of algebraic properties
- Deformation theory of symplectic structures on smooth locus relates to deformations of underlying algebraic variety connects symplectic and algebraic deformation theories
- Symplectic volume of smooth locus computed using intersection theory and characteristic classes provides bridge between symplectic and algebraic invariants
Symplectic Resolution of Singularities
Definitions and Properties
- Symplectic resolution constitutes proper birational morphism from symplectic variety to singular variety isomorphic over smooth locus
- Existence of symplectic resolutions closely related to concept of in algebraic geometry
- Symplectic resolutions preserve associated with symplectic form extend it from smooth locus to entire resolved variety
- Symplectic resolutions play crucial role in study of singularities and their deformations
Applications in Representation Theory
- Symplectic resolutions essential in study of and geometric representation theory link symplectic geometry to representation theory
- Derived category of coherent sheaves on symplectic resolution closely related to Fukaya category of resolved symplectic variety connects algebraic and symplectic invariants
- Symplectic resolutions provide geometric realization of certain representation-theoretic constructions (crystal bases, canonical bases)
Moment Maps in Algebraic Geometry
Fundamental Concepts and Properties
- Moment maps provide fundamental tool for relating symplectic geometry to algebraic geometry particularly in context of group actions on varieties
- Moment map for Hamiltonian torus action on projective variety encodes combinatorial data about variety leads to and its applications
- Image of moment map for Hamiltonian group action on projective variety constitutes convex polytope known as captures important geometric and algebraic information
- Symplectic quotients obtained via moment maps often have natural interpretations as GIT quotients in algebraic geometry establish key link between symplectic and algebraic reduction
Advanced Applications and Techniques
- relates critical points of norm-square of moment map to important algebraic subvarieties (semistable points, orbit closures)
- Moment maps play crucial role in study of and symplectic cutting techniques allow for construction of new symplectic varieties from given ones
- associated with moment map provides dynamical system approach to studying geometry of complex algebraic varieties connects symplectic techniques to algebraic geometry
- relates symplectic volume of reduced spaces to combinatorial properties of moment polytope (volume formula, localization)
Key Terms to Review (31)
Closed: In the context of symplectic geometry, 'closed' refers to a differential form that has zero exterior derivative. This means that when the symplectic form, typically denoted as $$\\omega$$, is closed, it satisfies the condition $$d\\omega = 0$$. This property is crucial because it establishes a foundational aspect of symplectic structures, ensuring that the form can be integrated over surfaces without changing its characteristics, and it plays a significant role in the conservation laws in both physics and mathematics.
Complex algebraic variety: A complex algebraic variety is a fundamental object in algebraic geometry that represents the solution set of a system of polynomial equations in several complex variables. These varieties can be either affine, defined by polynomial equations without any restrictions, or projective, which are defined in a projective space and can include points at infinity. They play a crucial role in connecting algebraic structures to geometric properties and have deep implications in symplectic geometry.
Cotangent Bundle: The cotangent bundle of a manifold is the vector bundle that consists of all the cotangent spaces at each point of the manifold, effectively capturing the linear functionals on the tangent spaces. This construction plays a crucial role in symplectic geometry as it provides a natural setting for defining symplectic structures and studying Hamiltonian dynamics.
Deformation Theory: Deformation theory is a mathematical framework that studies how geometric structures can be varied or 'deformed' while retaining their essential properties. This concept plays a critical role in understanding the stability and classification of geometric objects, particularly in symplectic geometry and the study of complex algebraic varieties, where the deformation of structures such as Lagrangian submanifolds can reveal important information about their topology and interactions.
Derived category: A derived category is a construction in homological algebra that allows for the systematic study of chain complexes and their morphisms, providing a framework for understanding derived functors and other aspects of category theory. This concept plays a crucial role in connecting algebraic geometry, particularly in the context of complex algebraic varieties and symplectic geometry, where it helps to analyze the relationships between various geometric structures and their associated cohomological properties.
Duistermaat-Heckman Theorem: The Duistermaat-Heckman theorem is a fundamental result in symplectic geometry that establishes a deep connection between the symplectic structure of a Hamiltonian manifold and the geometry of its moment map. Specifically, it describes how the integral of a certain function over the preimage of a regular value of the moment map relates to the volume of the image under this map, thus bridging symplectic geometry and algebraic geometry.
Fubini-Study Form: The Fubini-Study form is a symplectic form on complex projective space, which provides a natural way to study the geometry of complex projective varieties. It arises from the metric structure induced by the Fubini-Study metric, capturing important features of both complex geometry and symplectic geometry. This form allows one to analyze the relationships between complex manifolds and their associated symplectic structures.
Fukaya Category: The Fukaya category is a mathematical structure that arises in the study of symplectic geometry, particularly in relation to Lagrangian submanifolds. It organizes these submanifolds and their associated morphisms into a category, allowing for the exploration of the geometric and topological properties of symplectic manifolds. The Fukaya category has deep connections to both algebraic geometry and mirror symmetry, revealing profound links between seemingly disparate areas of mathematics.
Geometric Invariant Theory: Geometric Invariant Theory (GIT) is a framework that studies the action of groups on algebraic varieties and classifies objects up to equivalence by analyzing their invariant properties. It plays a vital role in understanding how symmetries can affect the structure of geometric objects, especially within the context of symplectic geometry and the interplay with complex algebraic varieties. This theory enables us to create quotients that reflect the geometric structures influenced by group actions.
Gradient-hamiltonian flow: Gradient-hamiltonian flow refers to the evolution of a system governed by a Hamiltonian function, where the flow is determined by the gradient of that function. This concept connects symplectic geometry and dynamics, as it describes how points in a symplectic manifold move over time under the influence of energy gradients. Understanding this flow is crucial when studying how complex algebraic varieties can exhibit symplectic structures, as it reveals the deep relationship between geometry and dynamics in these mathematical settings.
Hamiltonian group action: A Hamiltonian group action is a smooth action of a Lie group on a symplectic manifold that preserves the symplectic structure and is generated by a Hamiltonian function. This concept connects the dynamics of the system with geometric properties, allowing for the analysis of symplectic manifolds in the context of group actions. It plays a crucial role in understanding how symmetries influence the geometry of complex algebraic varieties and in defining symplectic quotients.
Hodge Theory: Hodge Theory is a mathematical framework that studies the relationship between differential forms, topology, and algebraic geometry. It provides tools to analyze the decomposition of cohomology groups and their connections with complex geometry, particularly in the context of Kähler manifolds and symplectic geometry, which are important for understanding complex algebraic varieties.
Holomorphic map: A holomorphic map is a function between complex manifolds that is complex differentiable at every point in its domain. This property of complex differentiability ensures that holomorphic maps preserve the structure of complex algebraic varieties and are crucial in the study of symplectic geometry, where the preservation of certain structures under mappings is essential for understanding geometric properties.
Kähler Form: The Kähler form is a specific type of closed differential 2-form that arises in the study of Kähler manifolds, which are complex manifolds with a symplectic structure. This form is crucial because it allows for the integration of symplectic geometry and complex algebraic geometry, highlighting the deep connections between these areas. A Kähler manifold has a compatible metric that is both Hermitian and symplectic, meaning the Kähler form not only encodes geometric information but also plays a role in defining the symplectic structure of the manifold.
Kirwan-ness Theorem: The Kirwan-ness theorem provides a powerful connection between symplectic geometry and the geometry of complex algebraic varieties. It essentially states that the symplectic reduction of a symplectic manifold at a certain level can be described in terms of the GIT (Geometric Invariant Theory) quotient of the corresponding algebraic variety, showing how both realms interact and provide insights into the structure of these spaces.
Moment Polytope: A moment polytope is a geometric object associated with a symplectic manifold and a Hamiltonian action of a compact Lie group. It captures the image of the moment map, which relates symplectic geometry to algebraic geometry, providing a way to visualize how symplectic structures interact with complex varieties. Moment polytopes are important in understanding the geometric properties of these actions, as they reflect crucial information about the symplectic manifold's topology and the group action.
Nakajima Quiver Varieties: Nakajima quiver varieties are a class of complex algebraic varieties that arise from representations of quivers, which are directed graphs used in representation theory. These varieties have rich geometric structures and are significant in the study of symplectic geometry, as they provide examples of symplectic manifolds with deep connections to mathematical physics, particularly in the context of gauge theory and moduli spaces.
Non-degenerate: In symplectic geometry, a non-degenerate structure refers to a bilinear form that does not have any non-zero vectors that are annihilated by it. This concept is crucial because it ensures the existence of a unique symplectic orthogonal complement for every subspace and allows for the establishment of a well-defined symplectic manifold. A non-degenerate symplectic form guarantees that the dynamics of a system can be properly described and facilitates the transition from geometric to analytical perspectives in various mathematical and physical contexts.
Poisson bracket: The Poisson bracket is a binary operation defined on the algebra of smooth functions over a symplectic manifold, capturing the structure of Hamiltonian mechanics. It quantifies the rate of change of one observable with respect to another, linking dynamics with the underlying symplectic geometry and establishing essential relationships among various physical quantities.
Poisson Structure: A Poisson structure is a mathematical framework that defines a way to describe the dynamics of a system through a bilinear operation on functions, allowing for the formulation of Hamiltonian mechanics. It serves as a bridge between symplectic geometry and the algebra of functions, enabling the study of systems with inherent symmetries and conservation laws. This structure provides a natural setting for analyzing the relationships between coordinates and momenta in phase space, as well as for understanding the geometric properties of complex algebraic varieties and representation theory.
Rational Singularities: Rational singularities are a type of singularity in algebraic geometry where the local cohomology of a space is 'rational' in the sense that it can be computed using a rational function. This concept is particularly important because it provides insights into the geometry of complex algebraic varieties and their behavior under various transformations, linking ideas from both algebraic geometry and symplectic geometry.
Symplectic Blowup: A symplectic blowup is a process used in symplectic geometry to modify a symplectic manifold by replacing a symplectic submanifold with a new space, typically a projective space. This operation allows for the resolution of singularities and the construction of new symplectic manifolds while preserving their symplectic structure. Symplectic blowups are particularly important when studying complex algebraic varieties, as they help to analyze their geometric properties in relation to the underlying symplectic structure.
Symplectic Form: A symplectic form is a closed, non-degenerate 2-form defined on a differentiable manifold, which provides a geometric framework for the study of Hamiltonian mechanics and symplectic geometry. It plays a crucial role in defining the structure of symplectic manifolds, facilitating the formulation of Hamiltonian dynamics, and providing insights into the conservation laws in integrable systems.
Symplectic Implosion: Symplectic implosion is a process that relates to symplectic geometry, particularly in the context of complex algebraic varieties, where one can 'implode' a symplectic manifold to produce a new manifold that retains some of the original structure. This technique often involves taking a Hamiltonian action of a Lie group and modifying the phase space in a way that emphasizes certain geometric features, effectively reducing dimensionality while preserving symplectic properties. It provides a powerful method to study the geometry and topology of symplectic manifolds by simplifying them into more manageable forms.
Symplectic Manifold: A symplectic manifold is a smooth, even-dimensional differentiable manifold equipped with a closed, non-degenerate differential 2-form called the symplectic form. This structure allows for a rich interplay between geometry and physics, especially in the formulation of Hamiltonian mechanics and the study of dynamical systems.
Symplectic quotient: A symplectic quotient is a construction in symplectic geometry that arises from the process of taking a symplectic manifold and applying a group action to it, leading to a reduction of dimensions while preserving symplectic structure. This concept is closely related to notions in algebraic geometry and involves the use of moment maps, which encode the way a symplectic manifold interacts with symmetry. By forming the quotient with respect to a group action, one can analyze the geometric and topological properties of the resulting space, often yielding insights into both symplectic and algebraic structures.
Symplectic Reduction: Symplectic reduction is a process in symplectic geometry that simplifies a symplectic manifold by factoring out symmetries, typically associated with a group action, leading to a new manifold that retains essential features of the original. This process is crucial for understanding the structure of phase spaces in mechanics and connects to various mathematical concepts and applications.
Symplectic Resolution: A symplectic resolution is a symplectic manifold that serves as a smooth resolution of singularities for a given complex algebraic variety. This concept plays a crucial role in connecting symplectic geometry with algebraic geometry, particularly in the study of varieties with singular points. By resolving these singularities, one can better understand the underlying geometric structure and properties of the algebraic variety.
Symplectic Subvariety: A symplectic subvariety is a subset of a symplectic manifold that inherits a symplectic structure from the ambient manifold, typically defined by the restriction of the symplectic form to the subvariety. These subvarieties can be viewed as solutions to certain geometric problems, revealing how complex algebraic structures interact with symplectic geometry. The understanding of these subvarieties is crucial for studying phenomena like Lagrangian intersections and Hamiltonian dynamics.
Symplectic Volume: Symplectic volume is a measure of the 'size' of a symplectic manifold, defined using a symplectic form that provides a natural volume element. This concept is crucial in symplectic geometry as it relates to the behavior and properties of Hamiltonian systems and the geometric structures of complex algebraic varieties. Understanding symplectic volume helps in studying how these varieties can be embedded or realized within symplectic manifolds.
Toric Geometry: Toric geometry is a branch of algebraic geometry that studies algebraic varieties defined by combinatorial data, specifically using torus actions. It connects the geometric properties of varieties to their combinatorial structures, making it particularly useful in understanding symplectic geometry and complex algebraic varieties through polyhedral techniques and fan constructions.