METRIC GEOMETRY OF FINITE ENERGY CLASSES IN BIG COHOMOLOGY

Abstract

This thesis investigates the metric geometry of finite energy classes in big cohomology. These finite energy classes are made of functions that correspond to singular metrics on compact K\"ahler manifolds. These spaces of functions were introduced to find the canonical K\"ahler metrics. We extend their study to big cohomology classes.

On the space of finite energy potentials $\mathcal{E}^{p}(X,\theta)$ where $\theta$ represents a big cohomology class, we construct a complete geodesic metric $d_{p}$. We show that several metric properties of $(\mathcal{E}^{p}(X,\theta), d_{p})$ are the same as in the K\"ahler setting.

In the end, we study the space of geodesic rays in $\mathcal{E}^{p}(X,\theta)$, $\mathcal{R}^{p}_{\theta}$, and construct a chordal metric $d_{p}^{c}$ on it. We show that $(\mathcal{R}^{p}_{\theta}, d_{p}^{c})$ is a complete geodesic metric as well.