Frobenius splitting of Schubert varieties of semi-infinite flag manifolds

TL;DR

This paper establishes foundational algebro-geometric properties of semi-infinite flag varieties and their Schubert varieties, including Frobenius splitting and normality, with implications for affine and quantum K-theory.

Contribution

It provides the first comprehensive algebro-geometric framework for semi-infinite flag varieties, including Frobenius splitting and normality results, crucial for related conjectures in K-theory.

Findings

Unique ind-scheme structure of semi-infinite flag varieties

Frobenius splitting compatible with boundaries and cells

Normality and cohomology vanishing of Schubert varieties

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 $\neq 2$ 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 $\neq 2$. 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.