Resumen de Fibrations with constant scalar curvature K\"ahler metrics and the CM-line bundle

Joel Fine

• 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.