Non-Affine Stein Manifolds and Normal Crossing Divisors

TL;DR

This paper demonstrates the existence of Stein manifolds with normal crossing divisor compactifications that are neither affine nor quasi-projective, using contact geometry and plumbing calculus to analyze their boundaries.

Contribution

It introduces conditions under which neighborhoods of symplectic normal crossing divisors are uniquely determined by their boundary contact structures.

Findings

Existence of Stein manifolds with non-affine, non-quasi-projective compactifications

Application of contact-geometric plumbing calculus to boundary analysis

Conditions for boundary contact structures to determine neighborhoods

Abstract

We show that there are Stein manifolds that admit normal crossing divisor compactifications despite being neither affine nor quasi-projective. To achieve this, we study the contact boundaries of neighborhoods of symplectic normal crossing divisors via a contact-geometric analog of W. Neumann's plumbing calculus. In particular, we give conditions under which the neighborhood is determined by the contact structure on its boundary.