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.
 Publication:

arXiv eprints
 Pub Date:
 January 2018
 arXiv:
 arXiv:1802.00087
 Bibcode:
 2018arXiv180200087D
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Complex Variables
 EPrint:
 v2. published version