Cohomology of Flag Superschemes and Syzygies of Compositional Varieties

Abstract

We study the coherent cohomology of partial flag supervarieties and calculate the sheaf cohomology of the structure sheaves of four new infinite families of flag supervarieties. We introduce compositional varieties, a generalization of determinantal varieties obtained by imposing rank conditions on compositions of a pair of maps. We study geometric properties of compositional varieties and, in cases of interest, compute their $\mathrm{Tor}$-groups. For each of the four families of flag supervarieties, we show that the graded sheaf cohomology is isomorphic to the tensor product of the singular cohomology ring of an ordinary partial flag variety and the graded $\mathrm{Tor}$-groups of a compositional variety.