Holomorphic self-maps of singular rational surfaces

TL;DR

This paper provides a new proof for classifying singular surface germs with non-invertible holomorphic self-maps and introduces the concept of minimal holomorphic models, drawing an analogy with foliation classification.

Contribution

It offers a novel proof of Wahl's classification and proposes the minimal holomorphic model concept with conditions for its uniqueness.

Findings

New proof of Wahl's classification of singular surface germs

Introduction of minimal holomorphic models for holomorphic maps

Conditions ensuring the uniqueness of these models

Abstract

We give a new proof of the classification of normal singular surface germs admitting non-invertible holomorphic self-maps and due to J. Wahl. We then draw an analogy between the birational classification of singular holomorphic foliations on surfaces, and the dynamics of holomorphic maps. Following this analogy, we introduce the notion of minimal holomorphic model for holomorphic maps. We give sufficient conditions which ensure the uniqueness of such a model.