Classical Poincaré metric pulled back off singularities using a Chow-type theorem and desingularization

Caroline Grant Melles; Pierre Milman

doi:10.5802/afst.1134

Caroline Grant Melles ¹; Pierre Milman ²

¹ Mathematics Department, United States Naval Academy, 572C Holloway Rd, Annapolis, Maryland 21402-5002, United States of America.
² Department of Mathematics, University of Toronto, 40 St George St, Toronto, Ontario M5S 2E4, Canada.

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 15 (2006) no. 4, pp. 689-771.

corrected-by Erratum: article Melles-Milman, Lemma 4.5, p. 719-720, fascicule 4, 2006

Abstract
Abstract

We construct complete Kähler metrics on the nonsingular set of a subvariety $X$ of a compact Kähler manifold. To that end, we develop (i) a constructive method for replacing a sequence of blow-ups along smooth centers, with a single blow-up along a product of coherent ideals corresponding to the centers and (ii) an explicit local formula for a Chern form associated to this ‘singular’ blow-up. Our metrics have a particularly simple local formula of a sum of the original metric and of the pull back of the classical Poincaré metric on the punctured disc by a ‘size-function’ $S_{I}$ of a coherent ideal $I$ used to resolve the singularities of $X$ by a ‘singular’ blow-up, where ${(S_{I})}^{2} : = \sum_{j = 1}^{r} {∣ f_{j} ∣}^{2}$ and the $f_{j}$ ’s are the local generators of the ideal $I$ . Our proof of (i) makes use of our generalization of Chow’s theorem for coherent ideals. We prove Saper type growth for our metric near the singular set and local boundedness of the gradient of a local generating function for our metric, motivated by results of Donnelly-Fefferman, Ohsawa, and Gromov on the vanishing of certain $L_{2}$ -cohomology groups.

Nous construisons des métriques complètes Kähleriennes sur le lieu non-singulier d’une sous-variété $X$ d’une variété compacte Kählerienne lisse. A cet effet, nous développons : (i) une méthode constructive pour le remplacement d’une suite d’éclatements le long des centres lisses par un seul éclatement le long d’un produit d’idéaux cohérents et (ii) une formule locale explicite pour une forme de Chern associée à cet éclatement. Nos métriques sont décrites par une formule locale particulièrement simple comme la somme de la métrique de départ et le tire-en-arrière de la métrique de Poincaré classique sur le disque épointé par une ‘fonction de grandeur’ $S_{I}$ de l’idéal cohérent $I$ utilisé pour la résolution des singularités de $X$ , ou ${(S_{I})}^{2} : = \sum_{j = 1}^{r} {∣ f_{j} ∣}^{2}$ et les $f_{j}$ sont des générateurs locaux de $I$ . Notre preuve de (i) utilise notre généralisation du théorème de Chow pour les idéaux cohérents. Nous montrons que la vitesse de croissance de notre métrique près du lieu singulier est de type Saper ainsi que le fait que le gradient d’une fonction génératrice locale de notre métrique est borné. Cela est motivé par les résultats de Donnelly-Fefferman, Ohsawa, et Gromov sur l’annulation de certains groupes de cohomologie $L_{2}$ .

EuDML MR Zbl

DOI: 10.5802/afst.1134

Author's affiliations:

Caroline Grant Melles ¹; Pierre Milman ²

¹ Mathematics Department, United States Naval Academy, 572C Holloway Rd, Annapolis, Maryland 21402-5002, United States of America.
² Department of Mathematics, University of Toronto, 40 St George St, Toronto, Ontario M5S 2E4, Canada.

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     author = {Caroline Grant Melles and Pierre Milman},
     title = {Classical {Poincar\'e} metric pulled back off singularities using a {Chow-type} theorem and desingularization},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {689--771},
     publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
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Caroline Grant Melles; Pierre Milman. Classical Poincaré metric pulled back off singularities using a Chow-type theorem and desingularization. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 15 (2006) no. 4, pp. 689-771. doi : 10.5802/afst.1134. https://afst.centre-mersenne.org/articles/10.5802/afst.1134/

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