On sharp rates and analytic compactifications of asymptotically conical K\"ahler metrics

TL;DR

This paper develops precise estimates for asymptotically conical Calabi-Yau metrics on complex manifolds, relating embedding properties to deformation rates, and proves an analytic compactification theorem for cone deformations.

Contribution

It introduces a method to obtain optimal asymptotic rates for Calabi-Yau metrics using deformation to the normal cone and relates embedding conditions to deformation weights.

Findings

Constructs a diffeomorphism with high-order holomorphic approximation

Provides optimal decay estimates for Calabi-Yau metrics

Establishes an analytic compactification theorem for cone deformations

Abstract

Let $X$ be a complex manifold and $S\hookrightarrow X$ be an embedding of complex submanifold. Assuming that the embedding is $(k-1)$-linearizable or $(k-1)$-comfortably embedded, we construct via the deformation to the normal cone a diffeomorphism $F$ from a small neighborhood of the zero section in the normal bundle $N_{S}$ to a small neighborhood of $S$ in $X$ such that $F$ is in a precise sense holomorphic to the $(k-1)$-th order. Using this $F$ we obtain optimal estimates on asymptotical rates for asymptotically conical Calabi-Yau metrics constructed by Tian-Yau. Furthermore, when $S$ is an ample divisor satisfying an appropriate cohomological condition, we relate the order of comfortable embedding to the weight of the deformation of the normal isolated cone singularity arising from the deformation to the normal cone. We also give an example showing that the condition of comfortable embedding depends on the splitting liftings. We then prove an analytic compactification result for the deformation of the complex structure on a complex cone that decays to any positive order at infinity. This can be seen as an analytic counterpart of Pinkham's result on deformations of cone singularities with negative weights.