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On leading Gopakumar--Vafa invariants of local ℙ2 and leading Betti numbers of moduli spaces of one-dimensional sheaves on ℙ2

  • Zhiyuan Wang
  • Chenglang Yang
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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 ℙ2. 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 ℙ2. 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 ℙ2.

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Cite

BibTeXRIS
BibTeX
@misc{wang2026leading,
  title = {{On leading Gopakumar--Vafa invariants of local $\mathbbP^2$ and leading Betti numbers of moduli spaces of one-dimensional sheaves on $\mathbbP^2$}},
  author = {Zhiyuan Wang and Chenglang Yang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.04483},
  eprint = {2610.04483},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.04483},
}
RIS
TY  - UNPB
TI  - On leading Gopakumar--Vafa invariants of local ℙ^2 and leading Betti numbers of moduli spaces of one-dimensional sheaves on ℙ^2
AU  - Zhiyuan Wang
AU  - Chenglang Yang
PY  - 2026
DA  - 2026/10/03
PB  - arXiv
DO  - 10.48550/arxiv.2610.04483
UR  - https://arxiv.org/abs/2610.04483
AB  - 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 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 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 ℙ^2.
ER  -