Background Cohomology and Symplectic Reduction

TL;DR

This paper introduces a background cohomology for Hamiltonian G-actions on Kähler manifolds and proves it aligns with the classical cohomology after symplectic reduction, bridging geometric quantization and reduction.

Contribution

It establishes that the background cohomology of a prequantum line bundle commutes with symplectic reduction, extending previous definitions to a new setting.

Findings

Background cohomology is defined for G-equivariant holomorphic bundles.

The background cohomology of a prequantum line bundle matches the Dolbeault cohomology of the reduced space.

The invariant part of background cohomology is isomorphic to the reduced space's cohomology.

Abstract

We consider a Hamiltonian action of a compact Lie group $G$ on a complete \ka manifold $M$ with a proper moment map. In a previous paper, we defined a regularized version of the Dolbeault cohomology of a $G$-equivariant holomorphic vector bundle, called the background cohomology. In this paper, we show that the background cohomology of a prequantum line bundle over $M$ `commutes with reduction', i.e. the invariant part of the background cohomology is isomorphic to the usual Dolbeault cohomology of the symplectic reduction.