Moduli of ADHM Sheaves and Local Donaldson-Thomas Theory

TL;DR

This paper explores the geometry of moduli spaces of ADHM sheaves on curves, establishing their virtual smoothness and connecting them to local Donaldson-Thomas theory and curve counting via stable pairs.

Contribution

It introduces a new geometric framework for moduli spaces of ADHM sheaves on curves and relates them to local Donaldson-Thomas invariants, providing a higher rank generalization.

Findings

Moduli space of ADHM sheaves is virtually smooth.

Established a spectral sequence relating sheaves to curve counting.

Proposed a new conjectural approach to local Donaldson-Thomas theory.

Abstract

The ADHM construction establishes a one-to-one correspondence between framed torsion free sheaves on the projective plane and stable framed representations of a quiver with relations in the category of complex vector spaces. This paper studies the geometry of moduli spaces of representations of the same quiver with relations in the abelian category of coherent sheaves on a smooth complex projective curve $X$. In particular it is proven that this moduli space is virtually smooth and related byrelative Beilinson spectral sequence to the curve counting construction via stable pairs of Pandharipande and Thomas. This yields a new conjectural construction for the local Donaldson-Thomas theory of curves as well as a natural higher rank generalization.