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1 June 2020 On sharp rates and analytic compactifications of asymptotically conical Kähler metrics
Chi Li
Duke Math. J. 169(8): 1397-1483 (1 June 2020). DOI: 10.1215/00127094-2019-0073


Let X be a complex manifold, and let SX be an embedding of a complex submanifold. Assuming that the embedding is (k1)-linearizable or (k1)-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 NS to a small neighborhood of S in X such that F is in a precise sense holomorphic up to the (k1)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.


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Chi Li. "On sharp rates and analytic compactifications of asymptotically conical Kähler metrics." Duke Math. J. 169 (8) 1397 - 1483, 1 June 2020.


Received: 26 July 2014; Revised: 15 September 2019; Published: 1 June 2020
First available in Project Euclid: 27 March 2020

zbMATH: 07226644
MathSciNet: MR4101736
Digital Object Identifier: 10.1215/00127094-2019-0073

Primary: 32G05
Secondary: 53C55

Keywords: analytic compactifications , asymptotic rate , asymptotically conical Kähler metrics , comfortably embedded , deformation of affine cones , embeddings of submanifolds , linearizable , splitting

Rights: Copyright © 2020 Duke University Press


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Vol.169 • No. 8 • 1 June 2020
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