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We prove that the space of coinvariants of functions on an affine variety bya Lie algebra of vector fields whose flow generates finitely many leaves isfinite-dimensional. Cases of the theorem include Poisson (or more generallyJacobi) varieties with finitely many symplectic leaves under Hamiltonian flow,complete intersections in Calabi-Yau varieties with isolated singularitiesunder the flow of incompressible vector fields, quotients of Calabi-Yauvarieties by finite volume-preserving groups under the incompressible vectorfields, and arbitrary varieties with isolated singularities under the flow ofall vector fields. We compute this quotient explicitly in many of these cases.The proofs involve constructing a natural D-module representing the invariantsunder the flow of the vector fields, which we prove is holonomic if it hasfinitely many leaves (and whose holonomicity we study in more detail). We givemany counterexamples to naive generalizations of our results. These exampleshave been a source of motivation for us.

Coinvariants of Lie algebras of vector fields on algebraic varieties