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

Zhiyuan Wang; Chenglang Yang

Home/Archive/arXiv

arXiv · 2610.04483

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

Zhiyuan Wang·Chenglang Yang

View original source ↗Read source document ↗Export metadata ↓

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 $\mathbb P^2$.

Explore related subjects

math.AG math-ph math.MP math.SG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Zhiyuan Wang, Chenglang Yang. 2026-10-03. On leading Gopakumar--Vafa invariants of local $\mathbb{P}^2$ and leading Betti numbers of moduli spaces of one-dimensional sheaves on $\mathbb{P}^2$. https://arxiv.org/abs/2610.04483

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the Kodaira dimension of the moduli space of curves of genus 16

We show that the moduli space of curves of genus 16 is NOT of general type.

math.AG↗

arXiv2026

On the minimal model of semi-isogenous mixed surfaces

The aim of this paper is to determine minimal models of the semi-isogenous mixed surfaces with $χ=1$ and $K^2>0$ constructed by Cancian and Frapporti (Math. Nachr., 291(2-3): 264-283, 2018). In order to do this, we further develop the idea of orbit divisors introduced by Frapporti and Lee (Pac. J. Math., 318(2): 233-247, 2022), to construct effective divisors on surfaces isogenous to a product of mixed type, extending it to the semi-isogenous mixed surfaces.

math.AG↗

arXiv2026

The cohomology of framed moduli spaces and the coordinate ring of torus fixed points of quotient singularities

If two conical symplectic resolutions $X\to X_0$ and $X^!\to X_0^!$ are symplectic dual, the cohomology ring $H^*(X)$ and the coordinate ring of $\mathbb{C}^*$-fixed points in $X_0^!$ are expected to be isomorphic as graded algebras. This statement is called Hikita conjecture and it is known that the conjecture holds for some cases. In this paper, we deal with the cohomology of framed moduli spaces over the projective plane and the coordinate ring of $\mathbb{C}^*$- fixed points of $\mathbb{C}^{2n}/((\mathbb{Z}/r\mathbb{Z})\wr S_n) $ and show that these are isomorphic as graded vector spaces. Building on subsequent work of Krylov and Shlykov on the Hikita--Nakajima conjecture for the Gieseker variety, the appendix establishes an isomorphism of graded algebras by a different approach and gives an explicit presentation by generators and relations. This revised version also corrects an error in the proof in the previous version.

math.AG↗