Microlocal sheaves and quiver varieties

TL;DR

This paper establishes a connection between microlocal sheaves on nodal curves and Nakajima quiver varieties, generalizing the concept of perverse sheaves and exploring their algebraic and geometric structures.

Contribution

It introduces microlocal sheaves on nodal curves, relates them to quiver varieties, and generalizes preprojective algebras for higher genus components.

Findings

DM(X) is Calabi-Yau of dimension 2 for compact X

M(X) is equivalent to representations of multiplicative pre-projective algebra when components are rational

Quiver varieties are realized as moduli spaces of microlocal sheaves with framing

Abstract

We relate Nakajima Quiver Varieties (or, rather, their multiplicative version) with moduli spaces of perverse sheaves. More precisely, we consider a generalization of the concept of perverse sheaves: microlocal sheaves on a nodal curve X. They are defined as perverse sheaves on normalization of X with a Fourier transform condition near each node and form an abelian category M(X). One has a similar triangulated category DM(X) of microlocal complexes. For a compact X we show that DM(X) is Calabi-Yau of dimension 2. In the case when all components of X are rational, M(X) is equivalent to the category of representations of the multiplicative pre-projective algebra associated to the intersection graph of X. Quiver varieties in the proper sense are obtained as moduli spaces of microlocal sheaves with a framing of vanishing cycles at singular points. The case when components of X have higher genus, leads to interesting generalizations of preprojective algebras and quiver varieties. We analyze them from the point of view of pseudo-Hamiltonian reduction and group-valued moment maps.