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Constant scalar curvature Kahler metrics on fibred complex surfaces

Joel Fine
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Abstract

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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Joel Fine. 2004-03-05. Constant scalar curvature Kahler metrics on fibred complex surfaces. https://arxiv.org/abs/math/0401275

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