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arXiv · 2609.14553

Blow-up formulas of Gromov-Witten invariants and virtual cycles under positivity conditions

Sanghyeon Lee·Seungjae Yun
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Abstract

Let $\widetilde{X}$ be the blow-up of a smooth projective variety $X$ along a smooth subvariety $Z \subset X$ satisfying a numerical positivity condition. We study virtual cycles of moduli spaces of stable maps to $\widetilde{X}$. We lift the numerical blow-up formula of Chen and Du to a pushforward identity for virtual cycles of moduli spaces of genus-zero stable maps to $\widetilde{X}$ and $X$. Using virtual dimension calculations for the fixed loci and a free-leaf contraction morphism, we show that all correction terms in the master space localization formula vanish. Variants of this method yields further blow-up formulas, including a generalization of Gathmann's formula. We also give a counterexample to the point-blow-up conjecture in higher genus (proposed by Hu), and establish an absolute--relative correspondence in all genera under positivity and dimension assumptions, comparing the Gromov--Witten theories of $(\widetilde{X},E)$ and $X$, where $E$ is the exceptional divisor.

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Sanghyeon Lee, Seungjae Yun. 2026-09-13. Blow-up formulas of Gromov-Witten invariants and virtual cycles under positivity conditions. https://arxiv.org/abs/2609.14553

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