Kaehler metrics of finite volume, unifomization of ruled complex surfaces and Seiberg-Witten equations

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

Yann Rollin (1)

1 CMLS - Centre de Mathématiques Laurent Schwartz

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.

Keywords: Geometrie differentielle; fibres paraboliques; Seiberg-Witten

Domains: Mathematics [math]

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