Métriques kählériennes de volume fini, uniformisation des surfaces complexes réglées et équations de Seiberg-Witten

Abstract : 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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Mathématiques [math]. Ecole Polytechnique X, 2001. Français
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• HAL Id : tel-00148005, version 1

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Yann Rollin. Métriques kählériennes de volume fini, uniformisation des surfaces complexes réglées et équations de Seiberg-Witten. Mathématiques [math]. Ecole Polytechnique X, 2001. Français. <tel-00148005>

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