Framed symplectic sheaves on surfaces

TL;DR

This paper constructs and analyzes a moduli space of framed symplectic sheaves on surfaces, providing an explicit description and studying its properties, especially on the complex projective plane.

Contribution

It introduces the first construction of a moduli space for framed symplectic sheaves on surfaces, including an ADHM-type description and birational map for the projective plane.

Findings

The moduli space is irreducible.

It admits an ADHM-type description.

There is a birational proper map to the space of framed symplectic ideal instantons.

Abstract

A framed symplectic sheaf on a smooth projective surface $X$ is a torsion-free sheaf $E$ together with a trivialization on a divisor $D\subseteq X$ and a morphism $\Lambda^{2}E\rightarrow\mathcal{O}_{X}$ satisfying some additional conditions. We construct a moduli space for framed symplectic sheaves on a surface, and present a detailed study for $X=\mathbb{P}_{\mathbb{C}}^{2}$. In this case, the moduli space is irreducible and admits an ADHM-type description and a birational proper map into the space of framed symplectic ideal instantons.