Quot scheme and deformation quantization

TL;DR

This paper demonstrates that the cotangent bundle of a Zariski open subset of a quot scheme over a Riemann surface admits a natural deformation quantization, extending previous results to more general cases.

Contribution

It introduces a canonical deformation quantization for the cotangent bundle of a subset of the quot scheme on a Riemann surface, generalizing earlier work for the case r=1=d.

Findings

Cotangent bundle admits a canonical deformation quantization.

Deformation quantization constructed on a Zariski open subset.

Extension of previous results to more general quot schemes.

Abstract

Let $X$ be a compact connected Riemann surface, and let ${\mathcal Q}(r,d)$ denote the quot scheme parametrizing the torsion quotients of ${\mathcal O}^{\oplus r}_X$ of degree $d$. Given a projective structure $P$ on $X$, we show that the cotangent bundle $T^*{\mathcal U}$ of a certain nonempty Zariski open subset ${\mathcal U}\, \subset\, {\mathcal Q}(r,d)$, equipped with the natural Liouville symplectic form, admits a canonical deformation quantization. When $r\,=\,1\,=\, d$, then ${\mathcal Q}(r,d)\,=\, X$; this case was addressed earlier in \cite{BB}.