Kirwan surjectivity for quiver varieties

For algebraic varieties defined by hyperk¨ahler or, more generally, algebraic symplectic reduction, it is a long-standing question whether the “hyperk¨ahler 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.