Mathematical Research Letters

Volume 27 (2020)

Number 5

Springer correspondence, hyperelliptic curves, and cohomology of Fano varieties

Pages: 1281 – 1323

DOI: https://dx.doi.org/10.4310/MRL.2020.v27.n5.a2

Authors

Tsao-Hsien Chen (Department of Mathematics, University of Chicago, Illinois, U.SA.)

Kari Vilonen (School of Mathematics and Statistics, University of Melbourne, Victoria, Australia; and Department of Mathematics and Statistics, University of Helsinki, Finland)

Ting Xue (School of Mathematics and Statistics, University of Melbourne, Victoria, Australia; and Department of Mathematics and Statistics, University of Helsinki, Finland)

Abstract

In [CVX3] (T. H. Chen, K. Vilonen, and T. Xue, “Springer correspondence for the split symmetric pair in type A”, Compos. Math. 154 (2018), no. 11, 2403–2425), 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.

Tsao-Hsien Chen was partially supported by NSF grants DMS-1702337 and DMS-2001257.

Kari Vilonen was supported in part by the ARC grants DP150103525 and DP180101445, the Academy of Finland, the Humboldt Foundation, the Simons Foundation, and the NSF grant DMS-1402928.

Ting Xue was supported in part by the ARC grants DP150103525 and DE160100975 and the Academy of Finland.

Received 27 January 2019

Accepted 28 May 2019

Published 12 January 2021