Poisson--Kähler fibration I: curvature of the base manifold

Abstract

We start from a finite dimensional Higgs bundle description of a result of Burns on negative curvature property of the space of complex structures, then we apply the corresponding infinite dimensional Higgs bundle picture and obtain a precise curvature formula of a Weil-Petersson type metric for general relative Kähler fibrations. In particular, our curvature formula implies a Burns type negative curvature property of the base manifold for a special class of maximal variation Kähler fibrations (named Poisson-Kähler fibrations).

Long geodesics in the space of Kähler metrics

“…This last part of the paper depends heavily on a result of Dan Burns on Monge-Ampère foliations. It also follows from recent work of Wan and Wang, [17], who, among many other things, give a direct proof of a curvature estimate from which our Theorem 4.2 follows. It seems to be an interesting problem if Theorem 4.2 also holds for generalized geodesics (in which case it is known that the K-energy is still convex).…”