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On a theorem of Narasimhan and Ramanan on deformations

V. Balaji, Y. Pandey

Abstract

Let $X$ be a smooth projective curve genus $G$ (as elaborated in \ref{main1}), over an algebraically closed field $k$ of arbitrary characteristics. Let $\cH$ {\em be a tamely ramified absolutely simple, simply connected connected group scheme (see \eqref{quasisplitcase})}. Let $\cM$ denote the moduli stack $\cM_X(\cH)$ of $\cH$-torsors on $X$ and $\cM^{^s}$ be the open substack of {\em stable torsors}. Using the theory of parahoric torsors and Parahoric-correspondences, we describe the cohomology groups $\text{H}^i\left(\cM^{^s}, \cT_{_{\cM}}\right), i = 0,1,2$ and $\text{H}^i\left(\cM^{^s}, Ω_{_{\cM}}\right), i = 0,1,2$ in terms of the curve $X$. The classical results of Narasimhan and Ramanan are derived as a consequence.

On a theorem of Narasimhan and Ramanan on deformations

Abstract

Let be a smooth projective curve genus (as elaborated in \ref{main1}), over an algebraically closed field of arbitrary characteristics. Let {\em be a tamely ramified absolutely simple, simply connected connected group scheme (see \eqref{quasisplitcase})}. Let denote the moduli stack of -torsors on and be the open substack of {\em stable torsors}. Using the theory of parahoric torsors and Parahoric-correspondences, we describe the cohomology groups and in terms of the curve . The classical results of Narasimhan and Ramanan are derived as a consequence.

Paper Structure

This paper contains 23 sections, 12 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Group theoretical data
  3. Parahoric groups
  4. Parahoric subgroups
  5. Standard parahorics
  6. Maximal parahorics
  7. Hyperspecial Parahorics
  8. Bruhat-Tits group schemes
  9. Some group-theoretical computations in the split case
  10. For exceptional groups with trivial centre
  11. Parahoric-correspondence and some geometry
  12. Relations between universal torsors
  13. The Kodaira-Spencer map
  14. Closer look at Parahoric-correspondences
  15. Some general considerations on flag bundles
  16. ...and 8 more sections

Key Result

Lemma 5.1

Let $\{V_s\}_{_{s \in S}}$ be a family of vector bundles on $X$ parametrized by $S$ and let $\small\text{\cursive q}:\mathbb P(V^*_x) \to S$ be the projection. Let $\tau_x$ denote the torsion sheaf on $X$ of height $1$ at $x \in X$. Consider the vector bundle $\mathcal{K}^*$ on $X \times \mathbb P \ Then the Kodaira-Spencer map for the family $\left\{\mathcal{K}_y \right\}_{_{y \in X}}$ of vector

Theorems & Definitions (26)

  • Definition 4.1
  • Lemma 5.1
  • proof
  • Theorem 5.2
  • proof
  • Theorem 6.1
  • proof
  • Remark 6.2
  • Theorem 6.3
  • proof
  • ...and 16 more