  Given a vertex operator algebra V , one can attach a graded Poisson algebra
called the C2-algebra R(V). The associate Poisson scheme provides an important
invariant for V and has been studied by Arakawa as the associated variety. In
this article, we define and examine the cohomological variety of a vertex
algebra, a notion cohomologically dual to that of the associated variety, which
measures the smoothness of the associated scheme at the vertex point. We study
its basic properties and then construct a closed subvariety of the
cohomological variety for rational affine vertex operator algebras constructed
from finite dimensional simple Lie algebras. We also determine the
cohomological varieties of the simple Virasoro vertex operator algebras. These
examples indicate that, although the associated variety for a rational
C2-cofinite vertex operator algebra is always a simple point, the cohomological
variety can have as large a dimension as possible. In this paper, we study R(V)
as a commutative algebra only and do not use the property of its Poisson
structure, which is expected to provide more refined invariants. The goal of
this work is to study the cohomological supports of modules for vertex algebras
as the cohomological support varieties for finite groups and restricted Lie
algebras.
