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

Abstract

Let X̃ be the blow-up of a smooth projective variety X along a smooth subvariety Z ⊂ X satisfying a numerical positivity condition. We study virtual cycles of moduli spaces of stable maps to X̃. 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 X̃ 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 (X̃,E) and X, where E is the exceptional divisor.

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BibTeX RIS

BibTeX

@misc{lee2026blowup,
  title = {{Blow-up formulas of Gromov-Witten invariants and virtual cycles under positivity conditions}},
  author = {Sanghyeon Lee and Seungjae Yun},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.14553},
  eprint = {2609.14553},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14553},
}

RIS

TY  - UNPB
TI  - Blow-up formulas of Gromov-Witten invariants and virtual cycles under positivity conditions
AU  - Sanghyeon Lee
AU  - Seungjae Yun
PY  - 2026
DA  - 2026/09/13
PB  - arXiv
DO  - 10.48550/arxiv.2609.14553
UR  - https://arxiv.org/abs/2609.14553
AB  - Let X̃ be the blow-up of a smooth projective variety X along a smooth subvariety Z ⊂ X satisfying a numerical positivity condition. We study virtual cycles of moduli spaces of stable maps to X̃. 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 X̃ 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 (X̃,E) and X, where E is the exceptional divisor.
ER  -

Abstract

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