Donaldson-Thomas theory of $[\mathbb{C}^2/\mathbb{Z}_{n+1}]\times \mathbb{P}^1$

TL;DR

This paper explores the orbifold Donaldson-Thomas theory of a specific threefold, establishing a correspondence with quantum cohomology and linking it to crepant resolution, advancing understanding in enumerative geometry.

Contribution

It introduces a new correspondence between relative orbifold DT theory and quantum multiplication, and relates it to crepant resolutions for the studied threefold.

Findings

Established a DT-quantum multiplication correspondence.

Determined the entire theory under a nondegeneracy condition.

Connected the DT theory to crepant resolution of the orbifold.

Abstract

We study the relative orbifold Donaldson-Thomas theory of $[\mathbb{C}^2/\mathbb{Z}_{n+1}]\times \mathbb{P}^1$. We establish a correspondence between the DT theory relative to 3 fibers to quantum multiplication by divisors in the Hilbert scheme of points on $[\mathbb{C}^2/\mathbb{Z}_{n+1}]$. This determines the whole theory if a further nondegeneracy condition is assumed. The result can also be viewed as a crepant resolution correspondence to the DT theory of $\mathcal{A}_n\times \mathbb{P}^1$.