The ADHM Variety and Perverse Coherent Sheaves

TL;DR

This paper explores the geometric structure of the ADHM variety, analyzing its subvarieties, singularities, and connections to perverse sheaves, advancing understanding of solutions to the ADHM equations.

Contribution

It provides a detailed geometric analysis of the ADHM variety, including dimension, singularity, reducibility, and links to perverse coherent sheaves.

Findings

Computed dimensions of ADHM subvarieties

Analyzed singularities and reducibility of the ADHM variety

Connected ADHM quiver representations to perverse sheaves on projective plane

Abstract

We study the full set of solutions to the ADHM equation as an affine algebraic set, the ADHM variety. We gather the points of the ADHM variety into subvarieties according to the dimension of the stabilizing subspace. We compute dimension, and analyze singularity and reducibility of all of these varieties. We also connect representations of the ADHM quiver to coherent perverse sheaves on $\mathbb{P}^2$ in the sense of Kashiwara.