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Frobenius splitting of Schubert varieties of semi-infinite flag manifolds

Syu Kato

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.

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 of characteristic 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 -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 or , and the higher cohomology vanishing of their nef line bundles in arbitrary characteristic . 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 -groups of a flag manifold.

Paper Structure

This paper contains 21 sections, 107 theorems, 232 equations.

Table of Contents

  1. Preliminaries
  2. Groups, root systems, and Weyl groups
  3. Representations of affine and current algebras
  4. Semi-infinite flag manifolds
  5. Quasi-map spaces and Zastava spaces
  6. Semi-infinite flag manifolds over $\mathbb{Z} [\frac{1}{2}]$
  7. Frobenius splittings
  8. Representations of affine Lie algebras over $\mathbb{Z}$
  9. Projectivity of the module $\mathbb{W} ( \lambda )_{\Bbbk}$
  10. Frobenius splitting of $\mathbf{Q}_{G,\mathtt J}$
  11. Frobenius splitting of quasi-map spaces
  12. The scheme $\mathscr{Q}'_\mathtt J ( v, w )$ and its Frobenius splitting
  13. Modular interpretation of $\mathbf{Q}_{G,\mathtt J}^{\mathrm{rat}}$
  14. Coarse representability of the scheme $\mathbf{Q}_G^{\mathrm{rat}}$
  15. The properties of the schemes $\mathscr{Q}'_{\mathtt J} ( v, w )$
  16. ...and 6 more sections

Key Result

Theorem A

There is an indscheme $\mathbf{Q}_G ^{\mathrm{rat}}$ with the following properties:

Theorems & Definitions (219)

  • Theorem A: $\doteq$ Theorem \ref{['bQ-int']} and Proposition \ref{['bQcoarse']}
  • Theorem B
  • Corollary C: $\doteq$ Theorem \ref{['coh']}
  • Corollary D: $\doteq$ Corollary \ref{['Bconn+']}
  • Theorem 1.1: Lusztig Lus80 cf. LS10 Proposition 4.4
  • Theorem 1.2: LNSSS LNSSS2, Chari-Ion CI15, cf. Kat18 Theorem 1.6
  • Theorem 1.3: Kat18 Theorem 3.3 and Corollary 3.4
  • proof : Sketch of proof
  • Theorem 1.4: FM99FFKMKNS17Lus80
  • Theorem 1.5: KNS17 Theorem 4.26 and Corollary 4.27
  • ...and 209 more