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

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 (2006)

- Volume: 15, Issue: 4, page 689-771
- ISSN: 0240-2963

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topGrant Melles, Caroline, and Milman, Pierre. "Classical Poincaré metric pulled back off singularities using a Chow-type theorem and desingularization." Annales de la faculté des sciences de Toulouse Mathématiques 15.4 (2006): 689-771. <http://eudml.org/doc/10020>.

@article{GrantMelles2006,

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 $(S_I)^2 := \sum _\{j=1\}^r \{\mid f_j \mid \}^2$ 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.},

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

author = {Grant Melles, Caroline, Milman, Pierre},

journal = {Annales de la faculté des sciences de Toulouse Mathématiques},

language = {eng},

number = {4},

pages = {689-771},

publisher = {Université Paul Sabatier, Toulouse},

title = {Classical Poincaré metric pulled back off singularities using a Chow-type theorem and desingularization},

url = {http://eudml.org/doc/10020},

volume = {15},

year = {2006},

}

TY - JOUR

AU - Grant Melles, Caroline

AU - Milman, Pierre

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

JO - Annales de la faculté des sciences de Toulouse Mathématiques

PY - 2006

PB - Université Paul Sabatier, Toulouse

VL - 15

IS - 4

SP - 689

EP - 771

AB - 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 $(S_I)^2 := \sum _{j=1}^r {\mid f_j \mid }^2$ 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.

LA - eng

UR - http://eudml.org/doc/10020

ER -

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