Dimension Estimate and Existence of Holomorphic Sections With Polynomial Growth on Gradient Kähler Ricci Shrinkers

Abstract In this paper, we show that the dimension for the space of holomorphic $(p, 0)$-forms and holomorphic sections of the pluri-anticanonical line bundle $K_{M}^{-q}$ with polynomial growth on gradient Kähler–Ricci shrinkers with bounded curvature is bounded from above by a power function of the growth rate. We also prove the existence of holomorphic sections of $K_{M}^{-q}$ with polynomial growth when the Kähler–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 embedding into a complex projective space.