On the Existence of a Global Neighbourhood

TL;DR

This paper establishes conditions under which local complex manifold embeddings can be combined into a global neighborhood, with applications to quantum cohomology and TEP structures.

Contribution

It proves a global neighborhood existence theorem for complex manifolds and extends Hertling--Manin's unfolding theorem to a global setting.

Findings

Global neighborhood of a complex manifold can be constructed from compatible local germs.

A global version of Hertling--Manin's unfolding theorem is established.

Applications to quantum cohomology and TEP structures are demonstrated.

Abstract

Suppose that a complex manifold M is locally embedded into a higher-dimensional neighbourhood as a submanifold. We show that, if the local neighbourhood germs are compatible in a suitable sense, then they glue together to give a global neighbourhood of M. As an application, we prove a global version of Hertling--Manin's unfolding theorem for germs of TEP structures; this has applications in the study of quantum cohomology.