关于局部$\mathbb{P}^2$的领先Gopakumar--Vafa不变式及$\mathbb{P}^2$上一维层模空间的领先Betti数
On leading Gopakumar--Vafa invariants of local $\mathbb{P}^2$ and leading Betti numbers of moduli spaces of one-dimensional sheaves on $\mathbb{P}^2$
- School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院）
- Institute for Math and AI, Wuhan University（武汉大学数学与人工智能中心）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文用拓扑顶点形式推导局部$\mathbb{P}^2$的Gromov--Witten不变式领先项公式，并应用它推广Guo--Zhou公式及证明关于$\mathbb{P}^2$上一维层模空间领先Betti数的猜想。
AI中文摘要：
本文利用拓扑顶点形式推导了局部$\mathbb P^2$的Gromov--Witten不变式生成级数领先项的结构公式。然后我们给出该公式的两个应用。首先，我们利用该结果推导了局部$\mathbb P^2$的变换后Gopakumar--Vafa不变式领先项的显式公式。这是Guo--Zhou \cite{gz}公式从次数$2d-5$到$3d-1$的推广，并与著名的Göttsche--Yau--Zaslow公式具有一些相似特征。作为第二个应用，我们证明了Guo--Wu--Moreira关于$\mathbb P^2$上一维Gieseker半稳定层模空间领先Betti数的猜想\cite{gw}。
英文摘要：
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 $\mathbb P^2$.
