Finite generation of the ring of holomorphic functions with polynomial growth on the K\"{a}hler-Ricci shrinker

TL;DR

This paper proves that under certain conditions, the ring of polynomial growth holomorphic functions on a non-compact Kähler-Ricci shrinker is finitely generated, supporting a conjecture by Munteanu and Wang.

Contribution

It establishes finite generation of the polynomial growth holomorphic functions ring on Kähler-Ricci shrinkers under mild scalar curvature assumptions, partially confirming a conjecture.

Findings

Ring OP(X) is finitely generated under mild scalar curvature conditions.

Supports the conjecture of Munteanu and Wang regarding polynomial growth functions.

Provides new insights into the structure of holomorphic functions on Kähler-Ricci shrinkers.

Abstract

Let (X, g, J, f ) be a non-compact gradient shrinking Kahler-Ricci soliton. We prove that if the scalar curvature of X satisfies a mild assumption, then OP (X), the ring of holomorphic functions with polynomial growth on X, is finitely generated. This gives a partial confirmation to a conjecture of Munteanu and Wang (cf.[MW14]).