Gromov–Witten invariants of blow-ups along submanifolds with convex normal bundles

Abstract

When the normal bundle $N_{Z∕X}$ is convex with a minor assumption, we prove that genus$-0$ GW–invariants of the blow-up ${Bl}_{Z}X$ of $X$ along a submanifold $Z$, with cohomology insertions from $X$, are identical to GW–invariants of $X$. Under the same hypothesis, a vanishing theorem is also proved. An example to which these two theorems apply is when $N_{Z∕X}$ is generated by its global sections. These two main theorems do not hold for arbitrary blow-ups, and counterexamples are included.