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.