The geometry of K\"ahler cones

TL;DR

This paper explores the geometric structure of K"ahler cones on compact manifolds, deriving their curvature tensor and examining properties like functorality and completeness, with a focus on their relative versions.

Contribution

It provides an explicit formula for the curvature tensor of the K"ahler cone metric and introduces relative versions of the cone and metric.

Findings

Curvature tensor of the K"ahler cone metric is explicitly computed.

The paper discusses functorality and completeness properties of the K"ahler cone.

Introduces relative versions of the K"ahler cone and the associated metric.

Abstract

The K\"ahler cone of a compact manifold carries a natural Riemannian metric, given by the intersection product of its cohomology ring. We write down the curvature tensor of this metric by embedding the K\"ahler cone in the space of hermitian metrics on the underlying manifold. After discussing weak functorality and completeness properties, we give a relative version of both the K\"ahler cone and the metric.