L^1 metric geometry of big cohomology classes

TL;DR

This paper develops a new L^1 metric on the space of finite energy classes of big cohomology classes on compact Kähler manifolds, establishing its completeness and geodesic structure using pluripotential theory.

Contribution

It introduces a novel L^1 metric on the finite energy space for big classes, proving its completeness and geodesic properties without relying on infinite dimensional Finsler geometry.

Findings

The space becomes a complete geodesic metric space under the new metric.

Construction of geodesic rays in the space is made flexible by adapting existing results.

The approach relies solely on pluripotential theory, enhancing rigidity and applicability.

Abstract

Suppose $(X,\omega)$ is a compact K\"ahler manifold of dimension $n$, and $\theta$ is closed $(1,1)$-form representing a big cohomology class. We introduce a metric $d_1$ on the finite energy space $\mathcal{E}^1(X,\theta)$, making it a complete geodesic metric space. This construction is potentially more rigid compared to its analog from the K\"ahler case, as it only relies on pluripotential theory, with no reference to infinite dimensional $L^1$ Finsler geometry. Lastly, by adapting the results of Ross and Witt Nystr\"om to the big case, we show that one can construct geodesic rays in this space in a flexible manner.