Torsors over moduli spaces of vector bundles over curves of fixed determinant

TL;DR

This paper studies the structure of torsors over moduli spaces of stable vector bundles with fixed determinant on curves, revealing a natural holomorphic projective connection and identifying torsors with sheaves of connections.

Contribution

It establishes a natural holomorphic projective connection on certain moduli spaces and identifies associated torsors with sheaves of connections on line bundles.

Findings

Existence of a natural holomorphic projective connection on the moduli space.

Identification of the torsor with the sheaf of connections on an ample line bundle.

Characterization of the moduli space of pairs (E, ∇) as an algebraic T*𝓜-torsor.

Abstract

Let ${\mathcal M}$ be a moduli space of stable vector bundles of rank $r$ and determinant $\xi$ on a compact Riemann surface $X$. Fix a semistable holomorphic vector bundle $F$ on $X$ such that $\chi(E\otimes F)= 0$ for $E \in \mathcal M$. Then any $E\in \mathcal M$ with $H^0(X, E\otimes F) = 0 = H^1(X, E\otimes F)$ has a natural holomorphic projective connection. The moduli space of pairs $(E,\, \nabla)$, where $E\, \in\, \mathcal M$ and $\nabla$ is a holomorphic projective connection on $E$, is an algebraic $T^*{\mathcal M}$--torsor on $\mathcal M$. We identify this $T^*{\mathcal M}$--torsor on $\mathcal M$ with the $T^*{\mathcal M}$--torsor given by the sheaf of connections on an ample line bundle over $\mathcal M$.