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arXiv math/0106077math.DGmath.CV
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有限体积的Kahler曲面与Seiberg-Witten方程

Kahler surfaces of finite volume and Seiberg-Witten equations

Yann Rollin

备注：
42 pages, 1 figure
发表信息：
Bull. Soc. Math. Fr. 130 (3), 2002, p. 409-456
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摘要：

设M=P（E）是直纹曲面。我们在M上引入了有限体积度量，它的奇性由E上的抛物结构参数化。然后推广了Burns-de Bartolomeis和LeBrun的结果，证明了M上有限体积和常数非正数量曲率的奇异Kahler度量的存在等价于E的抛物多稳定性，而且这些度量都来自于$H^2 \times CP^1$的有限体积导数.为了证明这个定理，我们必须给出Seiberg-Witten方程对于奇异度量g的解。我们使用orbifold紧化$\overline M$上，我们近似g的一个序列的光滑度量，所需的解决方案g获得的限制序列的Seiberg-Witten解决方案，这些光滑度量。

英文摘要：

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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