The DT/PT correspondence for smooth curves

TL;DR

This paper establishes a version of the DT/PT correspondence for smooth curves in Calabi-Yau threefolds, linking local curve counting invariants with global Donaldson-Thomas theory using Quot schemes.

Contribution

It introduces a new approach to relate local and global invariants for smooth curves via the Hilbert-Chow morphism and Quot schemes, specifically for genus 3 Abel-Jacobi curves.

Findings

Derived local curve counting invariants using Hilbert-Chow morphism.

Determined the global Donaldson-Thomas theory for genus 3 Abel-Jacobi curves.

Established a version of the DT/PT correspondence for smooth curves.

Abstract

We show a version of the DT/PT correspondence relating local curve counting invariants, encoding the contribution of a fixed smooth curve in a Calabi-Yau threefold. We exploit a local study of the Hilbert-Chow morphism about the cycle of a smooth curve. We determine, via Quot schemes, the global Donaldson-Thomas theory of a general Abel-Jacobi curve of genus $3$.