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arXiv · 2503.17713

Higher genus Gromov-Witten invariants from projective bundles on smooth log Calabi-Yau pairs

Benjamin Zhou
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Abstract

Let $(X,E)$ be a smooth log Calabi-Yau pair consisting of a smooth Fano surface $X$ and a smooth anticanonical divisor $E$. We obtain certain higher genus local Gromov-Witten invariants from the projectivization of the canonical bundle $Z := \mathbb{P}(K_X \oplus \mathcal{O}_X)$, using the degeneration formula for stable log maps [KLR]. We evaluate an invariant in the degeneration using the relationship between $q$-refined tropical curve counting and logarithmic Gromov-Witten theory with $\lambda_g$-insertion [Bou]. As a corollary, we use flops to prove a blow up formula for higher genus invariants of $Z$. Additionally assuming $X$ is toric, we prove an all-genus correspondence between open invariants of an outer Aganagic-Vafa brane $L \subset K_X$ and closed invariants of $Z$ that generalizes a genus-0 open-closed equality of [Cha] to all-genus, by using an argument in [GRZZ].

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Benjamin Zhou. 2025-03-22. Higher genus Gromov-Witten invariants from projective bundles on smooth log Calabi-Yau pairs. https://arxiv.org/abs/2503.17713

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