Corks, covers, and complex curves

TL;DR

This paper constructs pairs of complex curves in ^2 that are topologically isotopic but not smoothly isotopic, using corks as branched covers of holomorphic disks and exotic braid factorizations.

Contribution

It introduces a novel method to distinguish smooth structures on complex curves via corks and braid factorizations in ^2.

Findings

Existence of topologically isotopic but not smoothly isotopic complex curves in ^2.

Construction of corks as branched covers of holomorphic disks.

Connection between exotic braid factorizations and complex curve embeddings.

Abstract

We show that $\mathbb{C}^2$ contains pairs of properly embedded, smooth complex curves that are isotopic through homeomorphisms but not diffeomorphisms of $\mathbb{C}^2$. The construction is based on realizing corks as branched covers of holomorphic disks in the 4-ball. These disks can also be described using exotic factorizations of quasipositive braids.