On sharp rates and analytic compactifications of asymptotically conical Kähler metrics

Abstract

Let $X$ be a complex manifold, and let $S\hookrightarrow X$ be an embedding of a 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 up to the $(k-1)$th order. Using this $F$, we obtain optimal estimates on asymptotic rates for asymptotically conical (AC) Calabi–Yau (CY) metrics constructed by Tian and 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 an affine 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.