Curvature of the base manifold of a Monge-Ampère fibration and its existence

Abstract

In this paper, we consider a special relative Kähler fibration that satisfies a homogenous Monge-Ampère equation, which is called a Monge-Ampère fibration. There exist two canonical types of generalized Weil-Petersson metrics on the base complex manifold of the fibration. For the second generalized Weil-Petersson metric, we obtain an explicit curvature formula and prove that the holomorphic bisectional curvature is non-positive, the holomorphic sectional curvature, the Ricci curvature, and the scalar curvature are all bounded from above by a negative constant. For a holomorphic vector bundle over a compact Kähler manifold, we prove that it admits a projectively flat Hermitian structure if and only if the associated projective bundle fibration is a Monge-Ampère fibration. In general, we can prove that a relative Kähler fibration is Monge-Ampère if and only if an associated infinite rank Higgs bundle is Higgs-flat. We also discuss some typical examples of Monge-Ampère fibrations.

arXiv commentfinal version, to appear in Math. Ann