Curve counting on $\mathcal{A}_n \times \mathbb{C}^2$

Yalong Cao (Kavli Institute for the Physics and Mathematics of the Universe (WPI), and the University of Tokyo Institutes for Advanced Study, Kashiwa, Chiba, Japan)

Abstract

Let $\mathcal{A}_n \to \mathbb{C}^2 / \mathbb{Z}_{n+1}$ be the minimal resolution of $\mathcal{A}_n$-singularity and $X = \mathcal{A}_n \times \mathbb{C}^2$ be the associated toric Calabi–Yau $4$-fold. In this note, we study curve counting on $X$ from both Donaldson–Thomas and Gromov–Witten perspectives. In particular, we verify conjectural formulae relating them proposed by the author, Maulik and Toda.

Keywords

curve counting, $\mathcal{A}_n$-surfaces, Calabi–Yau $4$-folds

2010 Mathematics Subject Classification

14J32, 14N35

Full Text (PDF format)

The author is partially supported by the World Premier International Research Center Initiative (WPI), MEXT, Japan, the JSPS KAKENHI Grant Number JP19K23397 and Royal Society Newton International Fellowships Alumni 2019.

Received 30 January 2019

Accepted 2 April 2020

Published 11 November 2020