Symplectic structures on moduli spaces of framed sheaves on surfaces

TL;DR

This paper generalizes classical geometric notions to framed sheaves, constructs symplectic structures on their moduli spaces, and applies these to specific surfaces, advancing the understanding of their geometric properties.

Contribution

It introduces generalized Atiyah classes and Kodaira-Spencer maps for framed sheaves and constructs symplectic forms on their moduli spaces on certain surfaces.

Findings

Defined symplectic structures on moduli spaces of framed sheaves.

Extended classical notions to the framed sheaves context.

Applied constructions to birationally ruled surfaces.

Abstract

We provide generalizations of the notions of Atiyah class and Kodaira-Spencer map to the case of framed sheaves. Moreover, we construct closed two-forms on the moduli spaces of framed sheaves on surfaces. As an application, we define a symplectic structure on the moduli spaces of framed sheaves on some birationally ruled surfaces.