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

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 ${\mathbb{P}}^2$.