Constant scalar curvature Kahler metrics on fibred complex surfaces

Details Constant scalar curvature Kahler metrics on fibred complex surfaces Constant scalar curvature Kahler metrics on fibred complex surfaces

2004-01-21

arxiv.org/abs/math/0401275v2

Journal of Differential Geometry, 2004, 68(3), 397--432.

Mathematics

Differential Geometry

32 pages, no figures (with minor corrections and a larger font for tired eyes)

Scientific paper

This article finds constant scalar curvature Kahler metrics on certain compact complex surfaces. The surfaces considered are those admitting a holomorphic submersion to a curve, with fibres of genus at least 2. The proof is via an adiabatic limit. An approximate solution is constructed out of the hyperbolic metrics on the fibres and a large multiple of a certain metric on the base. A parameter-dependent inverse function theorem is then used to perturb the approximate solution to a genuine solution in the same cohomology class. The arguments also apply to certain higher dimensional fibred Kahler manifolds.

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