Work

On leading Gopakumar--Vafa invariants of local $\mathbb{P}^2$ and leading Betti numbers of moduli spaces of one-dimensional sheaves on $\mathbb{P}^2$

Year: 2026

Type: preprint

Abstract: In this paper we use the topological vertex formalism to derive a structural formula for the leading terms of the generating series of Gromov--Witten invariants of local $\mathbb P^2$. Then we give two applications of this formula. First we use this result to derive an explicit formula for the leading terms of the transformed Gopakumar--Vafa invariants of local $\mathbb P^2$. This is an extension of a formula of Guo--Zhou \cite{gz} from degree $2d-5$ to $3d-1$, and shares some similar features with the famous Göttsche--Yau--Zaslow formula. And as the second application, we prove Guo--Wu--Moreira's Conjecture \cite{gw} about the leading Betti numbers of the moduli space of one-dimensional Gieseker semistable sheaves on $\mathbb P^2$.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.04483

Authors: Wang, Zhiyuan, Yang, Chenglang

Institutions: Huazhong University of Science and Technology, Wuhan University

Language: English

Topic: Algebraic Geometry and Number Theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: National Natural Science Foundation of China, Fundamental Research Funds for the Central Universities

Award: G335815763

OpenAlex ID: W7220745862

Cites: 8

Cited by:

Related to: 10