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| Detailed view | PhD thesis |
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| Ecole Polytechnique X (2001-01-09), Olivier Biquard (Dir.) |
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| Métriques kählériennes de volume fini, uniformisation des surfaces complexes réglées et équations de Seiberg-Witten |
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| Yann Rollin1 |
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| Let M=P(E) be a ruled surface. We introduce metrics of finite volume on M whose singularities are parametrized by a parabolic structure over E. Then, we generalise results of Burns--de Bartolomeis and LeBrun, by showing that the existence of a singular Kahler metric of finite volume and constant non positive scalar curvature on M is equivalent to the parabolic polystability of E; moreover these metrics all come from finite volume quotients of $H^2 \times CP^1$. In order to prove the theorem, we must produce a solution of Seiberg-Witten equations for a singular metric g. We use orbifold compactifications $\overline M$ on which we approximate g by a sequence of smooth metrics; the desired solution for g is obtained as the limit of a sequence of Seiberg-Witten solutions for these smooth metrics. |
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| 1: | CMLS-EcolePolytechnique - Centre de Mathématiques Laurent Schwartz |
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| Geometrie differentielle – fibres paraboliques – Seiberg-Witten |
| Kaehler metrics of finite volume, unifomization of ruled complex surfaces and Seiberg-Witten equations |
| Let M=P(E) be a ruled surface. We introduce metrics of finite volume on M whose singularities are parametrized by a parabolic structure over E. Then, we generalise results of Burns--de Bartolomeis and LeBrun, by showing that the existence of a singular Kahler metric of finite volume and constant non positive scalar curvature on M is equivalent to the parabolic polystability of E; moreover these metrics all come from finite volume quotients of $H^2 \times CP^1$. In order to prove the theorem, we must produce a solution of Seiberg-Witten equations for a singular metric g. We use orbifold compactifications $\overline M$ on which we approximate g by a sequence of smooth metrics; the desired solution for g is obtained as the limit of a sequence of Seiberg-Witten solutions for these smooth metrics. |
| tel-00148005, version 1 | |
| http://tel.archives-ouvertes.fr/tel-00148005 | |
| oai:tel.archives-ouvertes.fr:tel-00148005 | |
| From: Yann Rollin | |
| Submitted on: Monday, 21 May 2007 15:07:35 | |
| Updated on: Thursday, 9 June 2011 20:59:58 | |
