Equivariant Jeffrey-Kirwan localization theorem in non-compact setting

TL;DR

This paper extends the Jeffrey-Kirwan localization theorem to non-compact symplectic and hyperKahler quotients, introducing a formal equivariant residue approach and applying it to compute the cohomology of Hilbert schemes.

Contribution

It introduces a generalized localization formula for non-compact quotients using equivariant residues, broadening the theorem's applicability.

Findings

Defined equivariant integrals on non-compact manifolds via localization

Introduced an equivariant Jeffrey-Kirwan residue with similar properties to the classical one

Computed the equivariant cohomology ring of the Hilbert scheme of points on the plane

Abstract

We generalize the Jeffrey-Kirwan localization theorem for non-compact symplectic and hyperKahler quotients. Similarly to the circle compact integration of Hausel and Proudfoot we define equivariant integrals on non-compact manifolds using the Atiyah-Bott-Berline-Vergne localization formula as formal definition. We introduce a so called equivariant Jeffrey-Kirwan residue and we show that it shares similar properties as the usual one. Our localization formula has the same structure as the usual Jeffrey-Kirwan formula, but it uses formal integration and equivariant residue. We also give a version for hyperKahler quotients. Finally, we apply our formula to compute the equivariant cohomology ring of Hilbert scheme of points on the plane constructed as a hyperKahler quotient.