Finite generation of the ring of holomorphic functions with polynomial growth on the Kähler-Ricci shrinker
Jiangtao Li
TL;DR
The paper addresses whether holomorphic functions on non-compact gradient shrinking Kähler-Ricci solitons with polynomial growth form a finitely generated algebra. It leverages the polarized Fano fibration structure on shrinkers to reduce the problem to the affine base $Y$ and proves that $\mathcal{O}_P(X)$ coincides with $R(Y)$, hence is finitely generated, under the mild condition $\gamma<1$ on scalar curvature growth. The approach combines gradient estimates for the soliton potential with algebro-geometric structure to connect $X$ and its affine base, culminating in a finite generation result that partially confirms Munteanu-Wang's conjecture. This work highlights a deep link between differential-geometric solitons and affine algebraic geometry and broadens the scope of uniformization-type results for complete Kähler manifolds.
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]).