Fibrations with constant scalar curvature Kähler metrics and the CM-line bundle (Q2459312)

The author gives a sufficient topological condition for the existence of a constant scalar curvature Kähler (=:cscK) metric on a compact Kähler manifold \(X\) for which there exists a holomorphic submersion \(\pi: X \rightarrow B\), where \(B\) is also a compact Kähler manifold, whose fibers and base have no non-zero holomorphic vectorfields and whose fibers admit cscK metrics. The condition is related to a certain natural line bundle on \(B\) for which \(c_{1} (B)<0\). Thus, one gets infinitely many cscK manifolds in every dimension, each having Kähler class arbitrarily far from the canonical class. Several other ways to produce cscK manifolds are also discussed.