2022/12/31 by Konovalov, Grigorii
#Algebraic Geometry (math.AG) #FOS: Mathematics #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.2301.00244
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 ≥ 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 ≤ 2m''. That condition will turn out to be a universal property of the operad that controls such Poisson algebras; we denote that operad ℙ1≤ m. We prove that the operad ℙ1≤ m does in fact exist, and we write an explicit simplicial resolution of it. The latter, in particular, will allow us to show that ℙ1≤ m sits in non-positive cohomological degrees and to compute H0(ℙ1≤ m).