Coinvariants of Lie algebras of vector fields on algebraic varieties - Schedler,TJ; Etingof,PI (Asian Journal of Mathematics, 2017)
We prove that the space of coinvariants of functions on an affine variety by a Lie algebraof vector fields whose flow generates finitely many leaves is finite-dimensional. Cases of the theoreminclude Poisson (or more generally Jacobi) varieties with finitely many symplectic leaves underHamiltonian flow, complete intersections in Calabi-Yau varieties with isolated singularities underthe flow of incompressible vector fields, quotients of Calabi-Yau varieties by finite volume-preservinggroups under the incompressible vector fields, and arbitrary varieties with isolated singularitiesunder the flow of all vector fields. We compute this quotient explicitly in many of these cases. Theproofs involve constructing a natural D-module representing the invariants under the flow of thevector fields, which we prove is holonomic if it has finitely many leaves (and whose holonomicitywe study in more detail). We give many counterexamples to naive generalizations of our results.These examples have been a source of motivation for us.