[Paper Review] Poisson traces and D-modules on Poisson varieties

Pavel Etingof, Travis Schedler+1 more|arXiv (Cornell University)|Aug 26, 2009

Advanced Algebra and Geometry15 references11 citations

TL;DR

This paper introduces a canonical right D-module M(X) on any Poisson variety X, linking Poisson traces—functionals invariant under Hamiltonian flow—to the zeroth cohomology of M(X). It proves that if X has finitely many symplectic leaves, M(X) is holonomic, implying the space of Poisson traces on X is finite-dimensional. This result extends to morphisms and Poisson modules, yielding finiteness theorems for noncommutative filtered algebras with Poisson centers having finitely many symplectic leaves.

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 flows. 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. As an application, we deduce that noncommutative filtered algebras whose associated graded algebras are coordinate rings of Poisson varieties with finitely many symplectic leaves have finitely many irreducible finite-dimensional representations. The appendix, by Ivan Losev, strengthens this to show that in such algebras, there are finitely many prime ideals, and they are all primitive. More generally, to any morphism phi: X -> Y and any quasicoherent sheaf of Poisson modules N on X, we attach a right D-module M_phi(X, N) on X, and prove that it is holonomic if X has finitely many symplectic leaves, phi is finite, and N is coherent. As an application, the finiteness result for irreducible representations of noncommutative filtered algebras extends to the case where the associated graded algebra is not necessarily commutative, but is finitely generated as a module over its center, which is the coordinate ring of a Poisson variety with finitely many symplectic leaves. We also describe explicitly (in the settings of affine varieties and compact smooth manifolds) the space of Poisson traces on X when X=V/G, where V is symplectic and G is a finite group acting faithfully on V. In particular, we show that this space is finite-dimensional.

Motivation & Objective

• To resolve the non-local nature of Poisson traces by constructing a geometric D-module framework on Poisson varieties.
• To establish a canonical D-module M(X) on any Poisson variety X whose cohomology computes Poisson homology HP₀(O_X).
• To prove that M(X) is holonomic when X has finitely many symplectic leaves, ensuring finiteness of Poisson traces.
• To extend the theory to morphisms φ:X→Y and Poisson modules N, constructing D-modules M_φ(X,N) with similar finiteness properties.
• To apply the results to noncommutative algebras, showing finite-dimensionality of HH₀(A) and finitely many finite-dimensional representations under Poisson center conditions.

Proposed method

• Construct a right D-module M(X) on a Poisson variety X using the Lie algebra action of Hamiltonian vector fields.
• Define M_φ(X,N) for a morphism φ:X→Y and a quasicoherent sheaf of Poisson modules N on X, generalizing M(X).
• Use the underived direct image of M_φ(X,N) under φ to realize N / {O_Y, N} as the space of Poisson traces with respect to Y.
• Prove holonomicity of M_φ(X) and M_φ(X,N) when X has finitely many symplectic leaves, φ is finite, and N is coherent.
• Apply holonomicity to deduce finite-dimensionality of HP₀(O_X) and related spaces via properties of D-module direct images.
• Leverage deformation-theoretic techniques and Rees algebras to extend results to filtered quantizations and their representations.

Experimental results

Research questions

• RQ1 Can Poisson traces on a Poisson variety be described via a local D-module construction?
• RQ2 Under what geometric conditions on a Poisson variety X is the space of Poisson traces finite-dimensional?
• RQ3 How does the D-module M(X) relate to the Poisson homology HP₀(O_X)?
• RQ4 What finiteness conditions on the center of a filtered noncommutative algebra imply finitely many finite-dimensional representations?
• RQ5 Are the prime and primitive ideals in such algebras finite in number and well-behaved?

Key findings

• The D-module M(X) on a Poisson variety X is holonomic if X has finitely many symplectic leaves, implying the space of Poisson traces is finite-dimensional.
• For any finite morphism φ:X→Y and coherent Poisson module N on X, the D-module M_φ(X,N) is holonomic under the same symplectic leaf condition.
• The space O_X / {O_Y, O_X} is finite-dimensional when X has finitely many symplectic leaves and φ is finite, generalizing the case of quotient varieties.
• When X = V/G for a symplectic vector space V and finite group G acting faithfully, the space of Poisson traces is explicitly described and finite-dimensional.
• For filtered noncommutative algebras whose associated graded is finite over its center with finitely many symplectic leaves, HH₀(A) = A/[A,A] is finite-dimensional.
• In such algebras, Ivan Losev's appendix proves there are finitely many prime ideals, all of which are primitive, including symplectic reflection algebras.