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

Let $\tilde{X}$ 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 $\tilde{X}$. 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 $\tilde{X}$ 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 $(\tilde{X},E)$ and $X$, where $E$ is the exceptional divisor.