Dimension estimate and existence of holomorphic sections with polynomial growth on gradient K\"ahler Ricci shrinkers

TL;DR

This paper establishes bounds on the dimension of holomorphic functions with polynomial growth on gradient K"ahler Ricci shrinkers and proves the existence of certain holomorphic sections, with applications to embeddings.

Contribution

It provides new upper bounds for holomorphic functions and sections on Ricci shrinkers, and demonstrates the existence of sections leading to holomorphic embeddings.

Findings

Upper bounds for holomorphic functions with polynomial growth

Existence of holomorphic sections on asymptotically conical shrinkers

Holomorphic embedding into projective space via Kodaira map

Abstract

We prove an upper bound for the dimension of the linear space of holomorphic functions with polynomial growth on gradient K\"ahler Ricci shrinkers with bounded curvature. The upper bound is given as a power function of the growth rate. Similar results hold for holomorphic $(p, 0)-$forms, and holomorphic sections of the pluri-anticanonical line bundle $K_M^{-q}$. We also prove the existence of holomorphic sections of $K_M^{-q}$ with polynomial growth when the K\"ahler Ricci shrinker is asymptotically conical, provided $q$ is sufficiently large; as an application, we show that the Kodaira map constructed using such sections is a holomorphic embbedding into a complex projective space.