Degenerate Poisson algebras and derived Poisson degeneracy loci

Abstract

This paper originated as an attempt to answer a question: what are the natural derived structures on Poisson degeneracy loci? We argue that the question could be possibly answered via a construction of differential graded operads that ``naturally'' act on the degeneracy loci. For each $m \ge 0$, we suggest what looks like a reasonable condition for a Poisson structure on a commutative differential graded algebra to be $m$-degenerate, i.e. to ``have rank $\le 2m$''. That condition will turn out to be a universal property of the operad that controls such Poisson algebras; we denote that operad $\mathbb{P}_1^{\le m}$. We prove that the operad $\mathbb{P}_1^{\le m}$ does in fact exist, and we write an explicit simplicial resolution of it. The latter, in particular, will allow us to show that $\mathbb{P}_1^{\le m}$ sits in non-positive cohomological degrees and to compute $H^0(\mathbb{P}_1^{\le m})$.