The Monge-Ampère operator and geodesics in the space of Kähler potentials

Abstract

√−1, where s(z) is a local, nowhere vanishing holomorphic section. In particular, R(h k ) = kR(h). Let

Curvature of vector bundles associated to holomorphic fibrations

“…plays a crucial role in the recent work on variations of Kähler metrics on compact manifolds; see [27], [23], [11], [12], [26] and [8], to cite just a few. Fixing a line bundle L on Z, these papers consider the space K(L) of all Kähler metrics whose Kähler form is cohomologous to the Chern class of L. This means precisely that the Kähler form can be written…”

Complex Monge-Ampère equations

“…It has also been shown by Berndtsson [Be2] that the convergence described in [PS06] can actually be strengthened to uniform convergence.…”

“…This translates precisely into whether the solutions of the homogeneous complex Monge-Ampère equations can be approximated by one-parameter subgroups of Bergman kernels [PS06]. We shall see below that the answer is affirmative, see [PS06,PS07,PS09b]. Some refinements of these approximations and their rate of convergence can be found in Berndtsson [Be1,Be2], and in [SZ07,SZ10] in the case of toric varieties.…”

“…A prime example is whether geodesics in K can be approximated by geodesics in K k . This translates precisely into whether the solutions of the homogeneous complex Monge-Ampère equations can be approximated by one-parameter subgroups of Bergman kernels [PS06]. We shall see below that the answer is affirmative, see [PS06,PS07,PS09b].…”

“…We can now state the main theorems of this section and complete their proofs. For geodesic segments, we have [PS06]: Proof. We compute explicitly the right hand side of (13.30).…”