The Poisson Cohomology of k-step Nilmanifolds
Nilmanifolds that admit invariant abelian complex structures are a particular class of compact complex manifold. We can consider their deformations, which are tied to a holomorphic Poisson bivector. Together with an adaptation of the classic d bar operator, the adjoint action of such a bivector generates a bi-grading, and thus a double complex. The associated hypercohomology then provides a spectral sequence. Under constraints on both the choice of bivector and the nature of the nilmanifold, we find a condition which determines whether the spectral sequence will degenerate on the first page, and provide examples and counterexamples. This same condition determines when the holomorphic Poisson hypercohomology admits a Hodge-type decomposition, and also determines whether such a decomposition is one of vector spaces, or one of Schouten algebras.