12.2 Moment maps in algebraic geometry
5 min read•july 30, 2024
Moment maps bridge symplectic and algebraic geometry, capturing symmetries in Hamiltonian systems. They're key to understanding symplectic quotients and their algebraic counterparts, with applications in complex geometry and toric varieties.
The links moment maps to Geometric Invariant Theory, providing a symplectic interpretation of stability conditions. This connection is crucial for constructing and analyzing moduli spaces in algebraic geometry, especially for objects with symmetries.
Moment maps in algebraic geometry
Definition and fundamental properties
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- Moment maps capture symmetries of Hamiltonian systems in symplectic geometry extending to algebraic geometry through symplectic structure on complex projective varieties
- Smooth map μ: M → g* from M to dual of Lie algebra g of G acting on M satisfies equivariance and differential properties
- Zero set μ^(-1)(0) crucial for understanding geometry and topology of symplectic quotients and algebraic counterparts
- Exhibit natural equivariance property with respect to coadjoint action of Lie group G on dual of Lie algebra g*
- Formal definition: For symplectic manifold (M,ω) with action of Lie group G, moment map μ satisfies for all X in Lie algebra g (X_M vector field on M generated by X)
Connections to algebraic geometry
- Closely related to and Kempf-Ness theorem bridging symplectic and algebraic geometry
- Convexity properties () have implications for toric varieties and generalizations
- Example: For toric variety X with moment map μ, image μ(X) is convex polytope in ℝ^n corresponding to fan of X
- encodes combinatorial data of toric variety (face structure, lattice points)
Applications in complex geometry
- Used to study Kähler manifolds connecting symplectic and complex structures
- For (M,ω,J), moment map μ for U(1) action related to Kähler potential K by (θ angular coordinate of U(1) action)
- Example: On complex projective space ℂℙ^n with Fubini-Study metric moment map for U(1) action given by
Moment maps and GIT
Fundamental connections
- Geometric Invariant Theory (GIT) framework for constructing and studying quotients of algebraic varieties by group actions parallels
- Kempf-Ness theorem establishes link between GIT quotients and symplectic quotients showing homeomorphism under suitable conditions
- Moment maps provide symplectic interpretation of stability conditions in GIT (stable points correspond to zeros of moment map)
- Example: For linear action of reductive group G on projective space ℙ(V), GIT-stable points correspond to zeros of moment map μ: ℙ(V) → g*
Stability criteria and moment maps
- Hilbert-Mumford numerical criterion in GIT reformulated in terms of asymptotic behavior of moment map along one-parameter subgroups
- Gradient flow of norm-square of moment map relates to in GIT providing dynamical systems perspective on stability
- Example: For vector bundle E over curve X, Harder-Narasimhan filtration 0 ⊂ E_1 ⊂ ... ⊂ E_k = E corresponds to critical points of Yang-Mills functional ||F_A||^2 (F_A curvature of connection A)
Advanced concepts and generalizations
- GIT quotients interpreted as symplectic quotients "at infinity" (level set of moment map taken at non-zero value and scaled)
- Symplectic implosion (Guillemin, Jeffrey, Sjamaar) provides universal abelianization procedure connecting non-abelian moment maps to abelian ones
- Applications in both symplectic and algebraic geometry (moduli spaces, representation varieties)
- Example: Symplectic implosion of T*K (K compact Lie group) yields universal symplectic implosion space with rich combinatorial structure
Moment maps for moduli spaces
Construction and analysis of moduli spaces
- Moment maps crucial for constructing and analyzing moduli spaces in algebraic geometry (objects with symmetries like vector bundles or sheaves)
- Moduli space of vector bundles on Riemann surface described using moment maps ( relates stable bundles to irreducible unitary representations)
- Hyperkähler quotients generalize symplectic quotients constructed using moment maps for actions preserving hyperkähler structure
- Applications to moduli spaces of instantons and Higgs bundles
- Example: Moduli space M(n,d) of stable vector bundles of rank n and degree d on Riemann surface Σ constructed as symplectic quotient μ^(-1)(0)/G (G gauge group, μ curvature map)
Geometric structures on moduli spaces
- Atiyah-Bott-Goldman symplectic structure on moduli space of flat connections on Riemann surface understood through moment map techniques
- Moment maps provide tool for studying cohomology of moduli spaces (localization techniques, Duistermaat-Heckman formula)
- Geometry of toric varieties and generalizations () described using moment maps (combinatorial approach to studying properties)
- Example: For moduli space M_g of genus g curves moment map for SL(2,ℝ) action on Teichmüller space gives
Variations and transformations of moduli spaces
- Variation of Geometric Invariant Theory (VGIT) quotients interpreted as varying level set of moment map
- Leads to wall-crossing phenomena and birational transformations between moduli spaces
- Example: Variations of GIT quotients for Grassmannians G(k,n) correspond to different chambers in moment polytope leading to birational models of G(k,n)
Moment maps vs symplectic reduction
Fundamental concepts and relationships
- Symplectic () constructs reduced symplectic manifold from symplectic manifold with group action using moment map
- Reduced space M_red = μ^(-1)(0)/G inherits symplectic structure from original manifold M (G Lie group acting on M, μ moment map)
- Symplectic reduction theorem: Under regularity conditions reduced space symplectic manifold of dimension dim(M) - 2dim(G)
- Reduced space interpreted as space of orbits of group action on level set of moment map (geometric realization of quotient)
- Example: For S^2 × S^2 with diagonal SO(3) action moment map μ(x,y) = x + y ∈ ℝ^3 ≅ so(3)* reduced space μ^(-1)(0)/SO(3) diffeomorphic to S^2
Generalizations and extensions
- Symplectic reduction generalized to non-zero level sets μ^(-1)(ξ) (ξ coadjoint orbit in g*) leading to study of symplectic fibrations
- Principle of symmetric criticality relates critical points of invariant functions to critical points on fixed point sets (understood in terms of symplectic reduction)
- Quantization commutes with reduction principle relates geometric quantization of symplectic manifolds to representation theory (formulated using moment map techniques)
- Example: For cotangent bundle TG of Lie group G with natural G-action moment map μ: TG → g* given by μ(g,ξ) = Ad*(g)ξ reduced spaces μ^(-1)(O)/G symplectomorphic to coadjoint orbits O ⊂ g*
Applications in physics and geometry
- Symplectic reduction applied in to reduce symmetries and obtain simpler systems
- Used in gauge theory to construct moduli spaces of connections and study their properties
- Plays role in geometric quantization relating classical and quantum systems
- Example: In celestial mechanics reduction by rotational symmetry of n-body problem leads to study of reduced phase space describing relative motions of bodies
Key Terms to Review (27)
Algebraic variety: An algebraic variety is a fundamental concept in algebraic geometry, representing the solution set of a system of polynomial equations. It can be thought of as a geometric manifestation of algebraic equations, where each point in the variety corresponds to a solution of these equations. Algebraic varieties can be classified into two main types: affine varieties, which are defined by polynomial equations in affine space, and projective varieties, which are defined in projective space.
Atiyah-Guillemin-Sternberg Convexity Theorem: The Atiyah-Guillemin-Sternberg Convexity Theorem establishes a connection between symplectic geometry and convex geometry, asserting that the image of a moment map from a symplectic manifold under the action of a compact Lie group is a convex set. This theorem is crucial in understanding how symmetries in a physical system translate into geometric properties and offers deep insights into the structure of phase spaces in mechanics.
Classical Mechanics: Classical mechanics is a branch of physics that deals with the motion of objects and the forces acting on them, typically described by Newton's laws. It serves as the foundation for understanding physical systems, providing insight into energy conservation, the dynamics of motion, and the relationships between different physical quantities.
D. karshon: d. karshon is a mathematician known for significant contributions to symplectic geometry, particularly in the area of moment maps and their applications in algebraic geometry. His work focuses on the interplay between symplectic structures and algebraic varieties, emphasizing the role of moment maps in understanding geometric properties and the behavior of dynamical systems.
GIT quotient: The GIT quotient, or Geometric Invariant Theory quotient, is a construction that allows one to form a new space from a given space with group action by identifying points that are equivalent under this action. This process is crucial in both algebraic geometry and symplectic geometry, as it helps to simplify complex geometric structures by partitioning them based on symmetry. The GIT quotient creates a new space that reflects the orbits of the original space under the group action, enabling researchers to study properties of these orbits effectively.
Hamiltonian action: Hamiltonian action refers to a smooth action of a Lie group on a symplectic manifold that preserves the symplectic structure, allowing for the formulation of classical mechanics in a geometric framework. This concept connects deeply with the behavior of physical systems under transformations and leads to the definition of moment maps, which encapsulate important information about the dynamics and symmetry of the system.
Harder-Narasimhan Filtration: The Harder-Narasimhan filtration is a technique in algebraic geometry and representation theory that decomposes a given coherent sheaf or a vector bundle into a successive series of sub-sheaves or sub-bundles, each of which is semistable with respect to a fixed polarization. This filtration provides a way to analyze the geometric and algebraic structure of sheaves and is essential in understanding the behavior of moment maps, particularly when studying the stability of points in moduli spaces.
Hilbert-Mumford Numerical Criterion: The Hilbert-Mumford numerical criterion is a method used to determine the stability of a point in a projective variety under the action of a group. This criterion connects geometric properties of a variety with algebraic aspects by using moment maps, and it provides a way to analyze the stability of orbits in geometric invariant theory, particularly in the context of actions on projective spaces.
Image of the moment map: The image of the moment map is a concept in symplectic geometry that arises from the study of Hamiltonian systems and their interactions with symmetries. It refers to the range of values that a moment map can take, reflecting the relationship between a symplectic manifold and its associated Lie group action. Understanding this image is crucial for analyzing the geometric and algebraic structures involved in algebraic geometry, especially in the context of Hamiltonian actions.
Infinitesimal Generator: An infinitesimal generator is a mathematical operator associated with a one-parameter family of transformations, typically in the context of Lie groups and symmetries. It captures how a system evolves under continuous transformations, acting as the 'infinitesimal' version of a transformation that leads to the notion of flows in geometry. In relation to moment maps, infinitesimal generators help connect symmetries in algebraic geometry with physical concepts like conservation laws and stability.
Kähler manifold: A Kähler manifold is a special type of complex manifold that is equipped with a symplectic form that is also compatible with the complex structure. This means it has a rich geometric structure that combines features of both complex and symplectic geometry, allowing for unique insights in various mathematical contexts such as algebraic geometry and representation theory.
Kempf-ness Theorem: The Kempf-ness Theorem provides a powerful criterion for determining when a point in a variety is stable or semi-stable under the action of a reductive group, using the concept of moment maps. This theorem connects algebraic geometry and symplectic geometry by showing how the properties of moment maps can be used to classify points based on their stability, which is essential for understanding the geometric structure of quotients formed by these group actions.
Kirwan's Theorem: Kirwan's Theorem is a fundamental result in symplectic geometry and algebraic geometry that provides a way to describe the relationship between the geometry of a symplectic manifold and the behavior of moment maps under group actions. It states that the image of the moment map, which encodes information about the symplectic structure and symmetries of the manifold, can be decomposed into a simpler structure consisting of images of certain convex polytopes. This theorem connects geometric properties with algebraic structures, making it crucial for understanding the interplay between symmetry and geometry.
Lie Group: A Lie group is a mathematical structure that combines algebraic and geometric properties, specifically a group that is also a differentiable manifold. This dual nature allows for the study of continuous transformations, making Lie groups essential in understanding symmetries and conservation laws in various fields, including physics and geometry.
Marsden-Weinstein Quotient: The Marsden-Weinstein quotient is a mathematical construction used to describe the reduction of symplectic manifolds under the action of a Hamiltonian group. This process allows one to obtain a new symplectic manifold that captures the essential features of the original manifold while taking into account the symmetries induced by the group action. It provides a powerful framework for understanding how moment maps interact with symplectic geometry, particularly in algebraic geometry contexts.
Marsden-Weinstein Reduction: Marsden-Weinstein reduction is a process in symplectic geometry that allows for the simplification of symplectic manifolds by reducing the system with respect to group actions and associated moment maps. This technique connects geometric structures with physical systems, making it possible to analyze and understand complex dynamical systems by focusing on the behavior of trajectories and invariant properties under symmetry transformations.
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.
Morse Theory: Morse theory is a branch of differential topology that studies the topology of manifolds using smooth functions and their critical points. It connects the geometric properties of a manifold to the behavior of smooth functions defined on it, specifically focusing on how critical points correspond to the structure of the manifold. This approach can be applied to analyze Lagrangian submanifolds and plays a vital role in understanding moment maps in algebraic geometry.
Narasimhan-Seshadri Theorem: The Narasimhan-Seshadri theorem establishes a deep connection between the geometric properties of vector bundles and the theory of stable holomorphic structures. Specifically, it shows that a unitary representation of a compact group can be realized through the existence of stable vector bundles over a Riemann surface, linking algebraic geometry and symplectic geometry through moment maps.
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.
Reduction: Reduction is the process of simplifying a symplectic manifold by taking into account a symmetry that acts on it, often leading to a lower-dimensional manifold. This is typically achieved through a moment map, which identifies points in the manifold that correspond to particular orbits of a group action, allowing for a clearer understanding of the geometry and dynamics involved. Reduction can help to focus on essential features of the system by eliminating redundant degrees of freedom.
Spherical Varieties: Spherical varieties are a class of algebraic varieties that exhibit a certain symmetry, specifically having a dense orbit under the action of a connected algebraic group. This dense orbit condition leads to interesting geometric and representation-theoretic properties, making them essential in understanding various aspects of algebraic geometry and symplectic geometry, particularly in the study of moment maps and their applications.
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 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.
V. guillemin: In symplectic geometry, v. guillemin refers to the foundational work of Victor Guillemin, who introduced the concept of moment maps as a tool for studying symplectic manifolds and their symmetries. This concept plays a critical role in understanding how symplectic structures interact with group actions, particularly in areas such as mechanics and algebraic geometry.
Vgit quotients: Vgit quotients refer to the process of taking a geometric invariant theory (GIT) quotient of a projective variety under the action of a group, particularly in the context of vector fields. This concept connects algebraic geometry and symplectic geometry, especially when analyzing moment maps and their roles in establishing a correspondence between geometric structures and algebraic properties.
Weil-Petersson Symplectic Form: The Weil-Petersson symplectic form is a specific symplectic structure defined on the moduli space of complex structures on Riemann surfaces, particularly those of genus greater than one. This form arises from the study of complex geometry and algebraic geometry, providing crucial insights into deformation theory and the geometry of the moduli space through its association with moment maps.