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Fibrations with constant scalar curvature Kahler metrics and the CM-line bundle

Joel Fine
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Abstract

Let X --> B be a holomorphic submersion between compact Kahler manifolds of any dimension, whose fibres and base have no non-zero holomorphic vector fields and whose fibres all admit constant scalar curvature Kahler metrics. This article gives a sufficient topological condition for the existence of a constant scalar curvature Kahler metric on the total space X. The condition involves the CM-line bundle--a certain natural line bundle on B--which is proved to be nef. Knowing this, the condition is then implied by c_1(B)<0. This provides infinitely many Kahler manifolds of constant scalar curvature in every dimension, each with Kahler class arbitrarily far from the canonical class.

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Joel Fine. 2006-03-04. Fibrations with constant scalar curvature Kahler metrics and the CM-line bundle. https://arxiv.org/abs/math/0510075

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