Frobenius splitting of Schubert varieties of semi-infinite flag manifolds

Abstract

We exhibit basic algebro-geometric results on the formal model of semi-infinite flag varieties and its Schubert varieties over an algebraically closed field $\mathbb{K}$ of characteristic $\ne 2$ from scratch. We show that the formal model of a semi-infinite flag variety admits a unique nice (ind)scheme structure, its projective coordinate ring has a $\mathbb{Z}$-model, and it admits a Frobenius splitting compatible with the boundaries and opposite cells in positive characteristic. This establishes the normality of the Schubert varieties of the quasi-map space with a fixed degree (instead of their limits proved in [K, Math. Ann. {\bf 371} no.2 (2018)]) when $\mathsf{char}\mathbb{K}=0$ or $\gg 0$, and the higher cohomology vanishing of their nef line bundles in arbitrary characteristic $\ne 2$. Some particular cases of these results play crucial roles in our proof [K, arXiv:1805.01718] of a conjecture by Lam-Li-Mihalcea-Shimozono [J. Algebra {\bf 513} (2018)] that describes an isomorphism between affine and quantum $K$-groups of a flag manifold.

64pages, v8: post publication update, cited [69] (as an original reference), corrected a typo in Lemma 3.28 and an argument in the second paragraph of the proof of Theorem 3.33 is corrected

References in corpus (4)

• Loop structure on equivariant $K$-theory of semi-infinite flag manifolds
• Demazure character formula for semi-infinite flag varieties
• Equivariant $K$-theory of semi-infinite flag manifolds and Pieri-Chevalley formula
• Vertex algebras and coordinate rings of semi-infinite flags

Cited by in corpus (1)

• Beilinson-Drinfeld Schubert varieties and global Demazure modules