Poisson Traces and D-Modules on Poisson Varieties |
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Authors: | Pavel Etingof Travis Schedler |
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Institution: | 1. MIT/Department of Mathematics, Room 2-176, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA, 02139, USA 2. AIM and MIT/Department of Mathematics, Room 2-179, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA, 02139, USA
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Abstract: | To every Poisson algebraic variety X over an algebraically closed field of characteristic zero, we canonically attach a right D-module M(X) on X. If X is affine, solutions of M(X) in the space of algebraic distributions on X are Poisson traces on X, i.e. distributions invariant under Hamiltonian flow. When X has finitely many symplectic leaves, we prove that M(X) is holonomic. Thus, when X is affine and has finitely many symplectic leaves, the space of Poisson traces on X is finite-dimensional. More generally, to any morphism ${\phi : X \to Y}To every Poisson algebraic variety X over an algebraically closed field of characteristic zero, we canonically attach a right D-module M(X) on X. If X is affine, solutions of M(X) in the space of algebraic distributions on X are Poisson traces on X, i.e. distributions invariant under Hamiltonian flow. When X has finitely many symplectic leaves, we prove that M(X) is holonomic. Thus, when X is affine and has finitely many symplectic leaves, the space of Poisson traces on X is finite-dimensional. More generally, to any morphism f: X ? Y{\phi : X \to Y} and any quasicoherent sheaf of Poisson modules N on X, we attach a right D-module Mf(X,N){M_\phi(X,N)} on X, and prove that it is holonomic if X has finitely many symplectic leaves, f{\phi} is finite, and N is coherent. |
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