Geometry and Cohomology of Schubert varieties

General information:

Lecture: Fr, 10-12, V2-105/115 

Discussion/Exercise session: Fr, 12-14, V3-204 (at this new time starting May 8)

Current version of the notes (version: 27.5.)

No classes June 12 and June 19

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)

Exercise sheets:

• Sheet 1 (updated 17.4.)

• Sheet 2 (updated 29.4.)

• Sheet 3

• Sheet 4

Literature:

For the classical part of the story there are two text books as well as sets of lecture notes introducing the affine part of the story:

• 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

• Zhu: An introduction to affine Grassmannians and the geometric Satake equivalence, In: Geometry of moduli spaces and representation theory. Lecture notes from the 2015 IAS/Park City Mathematics Institute (PCMI) Summer School, 2017

• Richarz: Basics on affine Grassmannians, 2020. URL: https://timo-richarz.com/wp-content/uploads/2020/02/BoAG_02.pdf.

• Görtz: Affine Springer fibers and affine Deligne-Lusztig varieties. In: Affine flag manifolds and principal bundles, Trends Math., pages 1-50. Birkhäuser/Springer Basel AG, Basel, 2010

• Görtz: Lecture notes "Algebraic Geometry 3", WS 2023/2024, URL: https://math.ug/alggeom3-ws2324/sect0004.html

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