L^1 metric geometry of big cohomology classes

Abstract

Suppose $(X,\omega )$ is a compact Kähler 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 Kähler 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 Nyström to the big case, we show that one can construct geodesic rays in this space in a flexible manner.

v2. published version

References in corpus (5)

• Tian's properness conjectures and Finsler geometry of the space of Kahler metrics
• A variational approach to the Yau-Tian-Donaldson conjecture
• On the singularity type of full mass currents in big cohomology classes
• On the constant scalar curvature Kähler metrics, existence results
• Geometry and Topology of the space of Kähler metrics on singular varieties