Higher canonical asymptotics of Kähler-Einstein metrics on quasi-projective manifolds

In this thesis, we derive the asymptotic expansion of the Kiihler-Einstein metrics on certain quasi-projective varieties, which can be compactified by adding a divisor with simple normal crossings. The weighted Cheng-Yau Hilder spaces and the log-filtrations based on the bounded geometry are introduced to characterize the asymptotics. We first develop the analysis of the Monge-Ampere operators on these weighted spaces. We construct a family of linear elliptic operators which can be viewed as certain conjugacies of the specially linearized Monge-Ampbre operators. We derive a theorem of Fredholm alternative for such elliptic operators by the Schauder theory and Yau's generalized maximum principle. Together these results derive the isomorphism theorems of the Monge-Ampbre operators, which imply that the Monge-Ampere operators preserve the log-filtration of the Cheng-Yau Holder ring. Next, by choosing a canonical metric on the submanifold, we construct an initial Kidhler metric on the quasi-projective manifold such that the unique solution of the Monge-Ampere equation belongs to the weighted -1 Cheng-Yau Hölder ring. Moreover, we generalize the Fefferman's operator to act on the volume forms and obtain an iteration formula.

Department dc:contributor.department

Massachusetts Institute of Technology. Dept. of Mathematics.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2005

Author dc:creator

• Wu, Damin, Ph. D. Massachusetts Institute of Technology

Advisor dc:contributor.advisor

• Shing-Tung Yau.

• Mathematics.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/33600

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/33600