A virtual Kawasaki formula

TL;DR

This paper extends Kawasaki's formula to the virtual setting for orbifolds with perfect obstruction theories, enabling computations in virtual quantum K-theory of moduli spaces.

Contribution

It proves that Kawasaki's formula remains valid when using virtual structures on orbifolds with perfect obstruction theories.

Findings

Kawasaki's formula is compatible with virtual structures.

The virtual Kawasaki formula applies to moduli spaces of stable maps.

Facilitates computations in quantum K-theory.

Abstract

Kawasaki's formula is a tool to compute holomorphic Euler characteristics of vector bundles on a compact orbifold X. Let X be an orbispace with perfect obstruction theory which admits an embedding in a smooth orbifold. One can then construct the virtual structure sheaf and the virtual fundamental class of X. In this paper we prove that Kawasaki's formula behaves well " with working virtually" on X in the following sense: if we replace the structure sheaves, tangent and normal bundles in the formula by their virtual counterparts then Kawasaki's formula stays true. Our motivation comes from studying the quantum K-theory of a complex manifold X, with the formula applied to Kontsevich' moduli spaces of genus 0 stable maps to X.