Kirwan surjectivity for quiver varieties

TL;DR

This paper proves that the hyperkahler Kirwan map is surjective for Nakajima quiver varieties, confirming a long-standing conjecture using classical topological and geometric methods.

Contribution

It establishes the surjectivity of the hyperkahler Kirwan map for Nakajima quiver varieties, extending results to other cohomology theories and the derived category.

Findings

Hyperkahler Kirwan map is surjective for Nakajima quiver varieties

Results extend to other cohomology theories

Surjectivity proven using classical topological and geometric arguments

Abstract

For algebraic varieties defined by hyperkahler or, more generally, algebraic symplectic reduction, it is a long-standing question whether the "hyperkahler Kirwan map" on cohomology is surjective. We resolve this question in the affirmative for Nakajima quiver varieties. We also establish similar results for other cohomology theories and for the derived category. Our proofs use only classical topological and geometric arguments.