Springer correspondence, hyperelliptic curves, and cohomology of Fano varieties

TL;DR

This paper advances Springer theory for symmetric pairs, explicitly determines the correspondence for specific nilpotent orbits, and applies these results to compute cohomology of Fano varieties related to hyperelliptic curves.

Contribution

It explicitly describes the Springer correspondence for IC sheaves on order 2 nilpotent orbits within a symmetric pair setting, linking to hyperelliptic curves and Fano variety cohomology.

Findings

Explicit Springer correspondence for certain nilpotent orbits

Connection between hyperelliptic curves and cohomology calculations

Cohomology of Fano varieties of k-planes in intersections of quadrics

Abstract

In \cite{CVX3}, we have established a Springer theory for the symmetric pair $(\operatorname{SL}(N),\operatorname{SO}(N))$. In this setting we obtain representations of (the Tits extension) of the braid group rather than just Weyl group representations. These representations arise from cohomology of families of certain (Hessenberg) varieties. In this paper we determine the Springer correspondence explicitly for IC sheaves supported on order 2 nilpotent orbits. In this process we encounter universal families of hyperelliptic curves. As an application we calculate the cohomolgy of Fano varieties of $k$-planes in the smooth intersection of two quadrics in an even dimensional projective space.