Geometry and Cohomology of Schubert varieties

General information:

Lecture: Fr, 10-12, V2-105/115 

Discussion/Exercise session: Mo, 16-18, V4-116 (starting April 13)

Content:

• Classical and affine Grassmannians and (partial) flag varieties

• Schubert varieties and their Demazure resolutions

• Singularities in characteristic p, Frobenius splittings, (derived) splinters

• Normality (and global F/+-regularity) of affine Schubert varieties (following Cass and Lourenco)

Literature:

I plan to write lecture notes that will be uploaded here.

For the classical part of the story there are two text books:

• Brion, Kumar: Frobenius Splitting Methods in Geometry and Representation Theory, Progr. Math., 231, 2005

• Kumar: Kac-Moody Groups, their Flag Varieties and Representation Theory, Progr. Math., 204, 2002

Further references are the following papers:

• G. Pappas, M. Rapoport: Twisted loop groups and their affine flag varieties (with an appendix by T. Haines and M. Rapoport), Adv. Math. 219 (2008), [journal],  arxiv.org/abs/math/0607130 

• R. Cass, J. Lourenco: Mod p sheaves on Witt flags, arxiv.org/abs/2503.01796 

• R. Cass: Perverse Fp-sheaves on the affine Grassmannian, J.Reine Angew. Math. 785 (2022), [journal] , arxiv.org/abs/1910.03377 

• J. Lourenco: Distributions and normality theorems, arxiv.org/abs/2312.17121 

• B. Bhatt: Derived splinters in positive characteristic, Compositio Math. 148 (2012), [journal], [arxiv] 

• B. Bhatt, L. Ma, Z. Patakfalvi, K. Schwede, K. Tucker, J. Waldron, J. Witaszek: Globally +-regular varieties and the minimal model program for threefolds in mixed characteristic , arxiv.org/abs/2012.15801