Coinvariants of Lie algebras of vector fields on algebraic varieties
File(s)
Accepted version
Author(s)
Schedler, TJ
Etingof, PI
Type
Journal Article
Abstract
We prove that the space of coinvariants of functions on an affine variety by a Lie algebra
of vector fields whose flow generates finitely many leaves is finite-dimensional. Cases of the theorem
include Poisson (or more generally Jacobi) varieties with finitely many symplectic leaves under
Hamiltonian flow, complete intersections in Calabi-Yau varieties with isolated singularities under
the flow of incompressible vector fields, quotients of Calabi-Yau varieties by finite volume-preserving
groups under the incompressible vector fields, and arbitrary varieties with isolated singularities
under the flow of all vector fields. We compute this quotient explicitly in many of these cases. The
proofs involve constructing a natural D-module representing the invariants under the flow of the
vector fields, which we prove is holonomic if it has finitely many leaves (and whose holonomicity
we study in more detail). We give many counterexamples to naive generalizations of our results.
These examples have been a source of motivation for us.
of vector fields whose flow generates finitely many leaves is finite-dimensional. Cases of the theorem
include Poisson (or more generally Jacobi) varieties with finitely many symplectic leaves under
Hamiltonian flow, complete intersections in Calabi-Yau varieties with isolated singularities under
the flow of incompressible vector fields, quotients of Calabi-Yau varieties by finite volume-preserving
groups under the incompressible vector fields, and arbitrary varieties with isolated singularities
under the flow of all vector fields. We compute this quotient explicitly in many of these cases. The
proofs involve constructing a natural D-module representing the invariants under the flow of the
vector fields, which we prove is holonomic if it has finitely many leaves (and whose holonomicity
we study in more detail). We give many counterexamples to naive generalizations of our results.
These examples have been a source of motivation for us.
Date Issued
2017-02-22
Date Acceptance
2015-04-24
Citation
Asian Journal of Mathematics, 2017, 20 (5), pp.795-868
ISSN
1093-6106
Publisher
International Press
Start Page
795
End Page
868
Journal / Book Title
Asian Journal of Mathematics
Volume
20
Issue
5
Copyright Statement
© 2017 International Press
Sponsor
National Science Foundation
Grant Number
DMS-1406553
Subjects
General Mathematics
0101 Pure Mathematics