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

Abstract

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