正性条件下的Gromov-Witten不变量与虚拟基本类的Blow-up公式
Blow-up formulas of Gromov-Witten invariants and virtual cycles under positivity conditions
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中文总结 AI 辅助
本文在正性条件下将Chen-Du的数值blow-up公式提升为虚拟基本类的推出恒等式，证明主空间局部化修正项消失，推广Gathmann公式，给出高亏格点blow-up猜想反例，并建立所有亏格的绝对-相对对应。
中文摘要 AI 辅助
设$\widetilde{X}$是光滑射影簇$X$沿满足数值正性条件的光滑子簇$Z \subset X$的blow-up。我们研究到$\widetilde{X}$的稳定映射模空间的虚拟基本类。我们将Chen和Du的数值blow-up公式提升为到$\widetilde{X}$和$X$的亏格零稳定映射模空间虚拟基本类的推出恒等式。利用不动点轨迹的虚拟维数计算和自由叶收缩态射，我们证明主空间局部化公式中的所有修正项消失。该方法的变体产生进一步的blow-up公式，包括Gathmann公式的推广。我们还给出了高亏格点blow-up猜想（由Hu提出）的一个反例，并在正性和维数假设下建立了所有亏格的绝对-相对对应，比较了$(\widetilde{X},E)$与$X$的Gromov-Witten理论，其中$E$是例外因子。
英文摘要
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.
发表机构
- Ajou University（延世大学）
- Seoul National University（首尔大学）
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