Non-algebraizable neighborhoods of curves

TL;DR

This paper constructs examples of complex curves in smooth surfaces whose neighborhoods cannot be embedded into algebraic surfaces, expanding understanding of non-algebraizable neighborhoods using various geometric techniques.

Contribution

It introduces new families of non-algebraizable neighborhoods of curves, generalizing previous examples and employing novel geometric and extension theorems.

Findings

Constructed families of non-algebraizable neighborhoods

Generalized previous examples by Lvovski

Used extension theorem of Ivashkovich in proofs

Abstract

We provide several families of compact complex curves embedded in smooth complex surfaces such that no neighborhood of the curve can be embedded in an algebraic surface. Different constructions are proposed, by patching neighborhoods of curves in projective surfaces, and blowing down exceptional curves. These constructions generalize examples recently given by S. Lvovski. One of our non algebraic argument is based on an extension theorem of S. Ivashkovich.