Arnol'd's type theorem on a neighborhood of a cycle of rational curves

TL;DR

This paper extends Arnol'd's theorem, demonstrating the uniqueness of complex analytic structures in neighborhoods of cycles of rational curves, analogous to the original result for elliptic curves.

Contribution

It provides a new analogue of Arnol'd's theorem for neighborhoods of cycles of rational curves, broadening the scope of complex structure uniqueness results.

Findings

Establishes the uniqueness of complex analytic structures near cycles of rational curves.

Generalizes Arnol'd's theorem from elliptic curves to rational curve cycles.

Provides foundational results for understanding neighborhoods of rational curve cycles.

Abstract

Arnol'd showed the uniqueness of the complex analytic structure of a small neighborhood of a non-singular elliptic curve embedded in a non-singular surface whose normal bundle satisfies Diophantine condition in the Picard variety. We show an analogue of this theorem for a neighborhood of a cycle of rational curves.