Derived Resolution Property for Stacks, Euler Classes and Applications

TL;DR

This paper introduces a method to define Euler classes for perfect derived objects over stacks and applies it to compute Euler numbers for Calabi-Yau threefolds, linking to Gromov-Witten invariants.

Contribution

It develops a new approach to define Euler classes for derived objects on stacks and applies this to Calabi-Yau threefolds, connecting to Gromov-Witten theory.

Findings

Defined Euler numbers for Calabi-Yau threefolds in projective space

Conjectured these Euler numbers as reduced Gromov-Witten invariants

Linked the Euler numbers to Gromov-Witten numbers of the quintic

Abstract

By resolving an arbitrary perfect derived object over a Deligne-Mumford stack, we define its Euler class. We then apply it to define the Euler numbers for a smooth Calabi-Yau threefold in the 4-dimensional projective space. These numbers are conjectured to be the reduced Gromov-Witten invariants and to determine the usual Gromov-Witten numbers of the smooth quintic as speculated by J. Li and A. Zinger.