Determining biholomorphic type of a manifold using combinatorial and algebraic structures

TL;DR

This paper introduces methods to classify complex manifolds by leveraging combinatorial graph structures and algebraic properties of endomorphisms, advancing the understanding of biholomorphic equivalence.

Contribution

It presents novel approaches to determine biholomorphic types using graph-based and algebraic structures, addressing two previously unresolved problems.

Findings

Successfully reconstructs biholomorphic types from graph structures.

Uses semigroup properties of analytic endomorphisms for classification.

Provides new tools for complex manifold analysis.

Abstract

We settle two problems of reconstructing a biholomorphic type of a manifold. In the first problem we use graphs associated to Riemann surfaces of a particular class. In the second one we use the semigroup structure of analytic endomorphisms of domains in $\C^n$.