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

Let $\pi \colon X \to B$ be a holomorphic submersion between compact Kähler manifolds of any dimensions, whose fibres and base have no non-zero holomorphic vector fields and whose fibres admit constant scalar curvature Kähler metrics. This article gives a sufficient topological condition for the existence of a constant scalar curvature Kähler metric on $X$. 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 Kähler manifolds of constant scalar curvature in every dimension, each with Kähler class arbitrarily far from the canonical class.

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Published 1 January 2007