Detailed view  PhD thesis 
Ecole Polytechnique X (09/01/2001), Olivier Biquard (Dir.) 
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Métriques kählériennes de volume fini, uniformisation des surfaces complexes réglées et équations de SeibergWitten 
Yann Rollin^{1} 
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 Burnsde 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 SeibergWitten 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 SeibergWitten solutions for these smooth metrics. 
1:  CMLSEcolePolytechnique  Centre de Mathématiques Laurent Schwartz 
Geometrie differentielle – fibres paraboliques – SeibergWitten 
Kaehler metrics of finite volume, unifomization of ruled complex surfaces and SeibergWitten 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 Burnsde 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 SeibergWitten 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 SeibergWitten solutions for these smooth metrics. 
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From: Yann Rollin  
Submitted on: Monday, 21 May 2007 15:07:35  
Updated on: Thursday, 9 June 2011 20:59:58 