Kahler surfaces of finite volume and Seiberg-Witten equations

Details Kahler surfaces of finite volume and Seiberg-Witten equations Kahler surfaces of finite volume and Seiberg-Witten equations

2001-06-11

arxiv.org/abs/math/0106077v1

Bull. Soc. Math. Fr. 130 (3), 2002, p. 409-456

Mathematics

Differential Geometry

42 pages, 1 figure

Scientific paper

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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