Moduli spaces of framed sheaves and quiver varieties

TL;DR

This paper surveys moduli spaces of framed sheaves on surfaces, describing their structure via quiver varieties, and provides explicit geometric descriptions for certain cases on Hirzebruch surfaces, including isomorphisms to Grassmannians.

Contribution

It offers a comprehensive survey of moduli spaces of framed sheaves, characterizes them as quiver varieties, and explicitly describes their structure on Hirzebruch surfaces in the minimal case.

Findings

Moduli spaces on $P^2$ and blowups are described via ADHM data.

On $F_n$, moduli spaces in the minimal case are isomorphic to Grassmannians or their cotangent bundles.

These moduli spaces can be realized as quiver varieties through a generalized Nakajima construction.

Abstract

In the first part of this paper we provide a survey of some fundamental results about moduli spaces of framed sheaves on smooth projective surfaces. In particular, we outline a result by Bruzzo and Markushevich, and discuss a few significant examples. The moduli spaces of framed sheaves on $\mathbb{P}^2$, on multiple blowup of $\mathbb{P}^2$ are described in terms of ADHM data and, when this characterization is available, as quiver varieties. The second part is devoted to a detailed study of framed sheaves on the Hirzebruch surface $\Sigma_n$ in the case when the invariant expressing the necessary and sufficient condition for the nonemptiness of moduli spaces attains its minimum (what we call the "minimal case"). Our main result is that, under this assumption, the corresponding moduli space is isomorphic to a Grassmannian (when $n=1$), or to the direct sum of $n-1$ copies of the cotangent bundle of a Grassmannian (when $n\geq 2$). Finally, by slightly generalizing a construction due to Nakajima, we prove that these moduli spaces admit a description as quiver varieties.