Pure and Applied Mathematics Quarterly

Volume 16 (2020)

Number 3

Special Issue: In Honor of Prof. Kyoji Saito’s 75th Birthday

Guest Editors: Stanislaw Janeczko, Si Li, Jie Xiao, Stephen S.T. Yau, and Huaiqing Zuo

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

Pages: 659 – 674

DOI: https://dx.doi.org/10.4310/PAMQ.2020.v16.n3.a10


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)


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.


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

2010 Mathematics Subject Classification

14J32, 14N35

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