Derived Varieties of Complexes and Kostant’s Theorem for $\mathfrak{gl}(\mathrm{m}|\mathrm{n})$

Abstract

Given a graded vector space V, the variety of complexes Com(V) consists of all differentials making V into a cochain complex. This variety was first introduced by Buchsbaum and Eisenbud and later studied by Kempf, De Concini, Strickland and many other people. It is highly singular and can be seen as a proto-typical singular moduli space in algebraic geometry. We introduce a natural derived analog of Com(V) which is a smooth derived scheme RCom(V). It can be seen as the derived scheme classifying twisted complexes. We study the cohomology of the dg-algebra of regular functions on RCom(V). It turns out that the natural action of the group GL(V) (automorphisms of V as a graded space) on the cohomology has simple spectrum. This generalizes the known properties of Com(V) and the classical theorem of Kostant on the Lie algebra cohomology of upper triangular matrices.

To Maxim Kontsevich for his 50th birthday