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Conformal blocks, parahoric torsors and Borel-Weil-Bott

V. Balaji, Y. Pandey

TL;DR

The paper develops a unified parahoric framework for conformal blocks by leveraging Hecke correspondences to relate cohomology of line bundles on moduli stacks of G-torsors. It proves the Pappas–Rapoport conjecture 3.7 in characteristic zero, establishes propagation of vacua, and shows independence of central charge from base points, with projective flatness following from Faltings’ Hitchin connection. By connecting to Teleman’s BWB results and Hong–Kumar’s twisted vacua, it derives vanishing theorems for higher cohomology and a Verlinde-type theory in the parahoric setting. The work also clarifies the role of equvariant/vacua formalisms, clarifies descent through uniformization, and situates the parahoric approach as a broad generalization that includes and extends several prior frameworks ( bs, pr, Hong–Kumar, dh).

Abstract

Let $X$ be a smooth projective curve over an algebraically closed field $k$. Let $\mathcal{G}$ be a parahoric group scheme on $X$ as in \cite{pr}. Via the principle of Hecke correspondences, we set-up relationships between the cohomology of lines bundles on various moduli stacks of torsors. This approach gives a proof of \cite[Conjecture 3.7]{pr} for group schemes $\mathcal G$ as above in characteristic zero. This further gives as a consequence, the principle of propagation of vacua. We give a direct proof of the independence of central charge on base points. Projective flatness is recovered as a corollary of Faltings construction of the Hitchin connection. Using C.Teleman's basic results (\cite{bwb}), we deduce the analogous result that cohomology of line bundles on the stack of principal $G$-bundles vanish in all degrees except possibly one. Results on twisted vacua \cite{hongkumar} are obtained as immediate consequences.

Conformal blocks, parahoric torsors and Borel-Weil-Bott

TL;DR

The paper develops a unified parahoric framework for conformal blocks by leveraging Hecke correspondences to relate cohomology of line bundles on moduli stacks of G-torsors. It proves the Pappas–Rapoport conjecture 3.7 in characteristic zero, establishes propagation of vacua, and shows independence of central charge from base points, with projective flatness following from Faltings’ Hitchin connection. By connecting to Teleman’s BWB results and Hong–Kumar’s twisted vacua, it derives vanishing theorems for higher cohomology and a Verlinde-type theory in the parahoric setting. The work also clarifies the role of equvariant/vacua formalisms, clarifies descent through uniformization, and situates the parahoric approach as a broad generalization that includes and extends several prior frameworks ( bs, pr, Hong–Kumar, dh).

Abstract

Let be a smooth projective curve over an algebraically closed field . Let be a parahoric group scheme on as in \cite{pr}. Via the principle of Hecke correspondences, we set-up relationships between the cohomology of lines bundles on various moduli stacks of torsors. This approach gives a proof of \cite[Conjecture 3.7]{pr} for group schemes as above in characteristic zero. This further gives as a consequence, the principle of propagation of vacua. We give a direct proof of the independence of central charge on base points. Projective flatness is recovered as a corollary of Faltings construction of the Hitchin connection. Using C.Teleman's basic results (\cite{bwb}), we deduce the analogous result that cohomology of line bundles on the stack of principal -bundles vanish in all degrees except possibly one. Results on twisted vacua \cite{hongkumar} are obtained as immediate consequences.

Paper Structure

This paper contains 42 sections, 24 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Loop groups and Bruhat-Tits group schemes
  3. Notations and Conventions
  4. Some attributes of facets
  5. From Bruhat-Tits bt
  6. The moduli stacks $\mathcal{M}(\mathcal{G})$ and their uniformization
  7. The group scheme
  8. Uniformization of $\mathcal{M}(\mathcal{G})$
  9. The Hecke Correspondence
  10. Construction of compatible atlases $\pi: \mathcal{M}^{\tt I} \rightarrow \mathcal{M}(\mathcal{G})$ of \ref{['IwahoritoGstack']}
  11. Picard functor and central charge
  12. The notion of central charge pradv
  13. Central charge of line bundles on moduli stack of $\mathcal{G}$-torsors
  14. Line bundles on $\mathcal{M}(\mathcal{G})$
  15. Interpreting pullback of line bundles under $\pi_{_{\small\text{\cursive s}}}: \mathcal{M}^{\tt I} \rightarrow \mathcal{M}(\mathcal{G})$ in terms of $h_{_{\small\text{\cursive s}}}: \mathcal{M}^{\tt I} \rightarrow \mathcal{M}(\mathfrak{H})$ (see \ref{['heckecorresp']})
  16. ...and 27 more sections

Key Result

Proposition 5.1

pradv Let $\mathcal{G}$ be a parahoric group scheme that splits over a tamely ramified extension. There is an isomorphism $\mathrm{Pic}(\mathcal{F}\ell_{\mathbf{a}}) \simeq \mathbb{Z}^{\mathbb{S}}$ defined by the degrees of the restrictions to $\mathbb{P}^1_\alpha= L^+\mathcal{G}_{{\tt F}^\alpha}/L^

Theorems & Definitions (60)

  • Remark 4.1
  • Proposition 5.1
  • Proposition 5.2
  • Theorem 5.3
  • Remark 5.4
  • Remark 5.5
  • Corollary 5.6
  • proof
  • Proposition 5.7
  • proof
  • ...and 50 more