On a theorem of Narasimhan and Ramanan on deformations

TL;DR

This paper studies the deformation theory of moduli stacks of torsors over algebraic curves, using parahoric torsors to describe their cohomology, and recovers classical results of Narasimhan and Ramanan.

Contribution

It provides a new description of the cohomology groups of tangent and cotangent sheaves on moduli stacks of torsors using parahoric torsors, extending classical results.

Findings

Cohomology groups of moduli stacks are explicitly described in terms of the curve.

The approach generalizes classical results to more general group schemes.

Cohomology computations are achieved via parahoric-correspondences.

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 $\text{H}^i\left(\cM^{^s}, \Omega_{_{\cM}}\right), i = 0,1,2$ in terms of the curve $X$. The classical results of Narasimhan and Ramanan are derived as a consequence.