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# Mathematical Research Letters

## Volume 14 (2007)

### Number 2

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

Pages: 239 – 247

DOI: http://dx.doi.org/10.4310/MRL.2007.v14.n2.a7

#### Author

#### Abstract

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.