Métriques kählériennes de volume fini, uniformisation des surfaces complexes réglées et équations de Seiberg-Witten
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...
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| Language: | FRE |
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Ecole Polytechnique X
2001
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| Online Access: | http://tel.archives-ouvertes.fr/tel-00148005 http://tel.archives-ouvertes.fr/docs/00/14/80/05/PDF/these_rollin.pdf |
| Summary: | 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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