Criteria for existence of stable parahoric $\SO_n$, $\Sp_n$ and $\Spin$ bundles on $\PP^1$

TL;DR

This paper establishes criteria for the existence of stable parahoric orthogonal, symplectic, and spin bundles on the projective line, linking their properties to intrinsic objects on the base curve and analyzing their moduli spaces.

Contribution

It provides necessary and sufficient conditions for the non-emptiness of moduli spaces of stable parahoric bundles on ^1, introducing an intrinsic description in terms of objects on the base curve.

Findings

Criteria for existence of stable parahoric bundles on ^1

Intrinsic description of parahoric bundles via base curve objects

Conditions for non-emptiness of moduli spaces

Abstract

Let $p: Y \ra X$ be a Galois cover of smooth projective curves over $\CC$ with Galois group $\Gamma$. This paper is devoted to the study of principal orthogonal and symplectic bundles $E$ on $Y$ to which the action of $\Gamma$ on $Y$ lifts. We notably describe them intrinsically in terms of objects defined on $X$ and call these objects parahoric bundles. We give necessary and sufficient conditions for the non-emptiness of the moduli of stable (and semi-stable) parahoric special orthogonal, symplectic and spin bundles on the projective line $\PP^1$.