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

TL;DR

This paper proves that under certain convexity conditions, the genus-zero Gromov-Witten invariants of a blow-up along a submanifold match those of the original manifold, and also establishes a vanishing theorem.

Contribution

It introduces conditions under which GW-invariants are preserved under blow-up along submanifolds with convex normal bundles, extending understanding of invariants in complex geometry.

Findings

GW-invariants of blow-up equal those of original manifold under convexity.

A vanishing theorem for GW-invariants is established under the same conditions.

Counter-examples show the theorems do not hold generally for all blow-ups.

Abstract

Given a submanifold Z inside X, let Y be the blow-up of X along Z. When the normal bundle of Z in X is convex with a minor assumption, we prove that genus-zero GW-invariants of Y 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 the normal bundle is generated by global sections. These two main theorems do not hold for arbitrary blow-ups, and counter-examples are included.