$L^1$ metric geometry of big cohomology classes

Darvas, Tamás; Di Nezza, Eleonora; Lu, Chinh H.

Annales de l'Institut Fourier, Volume 68 (2018) no. 7, pp. 3053-3086.

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 ${ℰ}^1(X,\theta )$, 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.

Soient $X$ une variété Kählerienne compacte et $\theta$ une forme fermée qui représente une classe de cohomologie grosse. On introduit une métrique $d_1$ sur l’espace d’énergie finie ${ℰ}^1(X,\theta )$, ce qui en fait un espace métrique géodésique complet. Cette construction s’appuie seulement sur la théorie du pluripotentiel et ne se réfère pas à la géométrie finsleriénne $L^1$, et donc a priori elle est plus rigide par rapport à la construction analogue dans le cas Kählerien. Enfin, on adapte des résultats de Ross et Witt Nyström au cas d’une classe grosse pour montrer que l’on peut construire des rayons géodésiques dans cet espace de façon très flexible.

Keywords: Kähler manifolds, pluripotential theory, Monge-Ampère energy classes, geodesic rays
Mot clés : Variétés Kähleriénnes, théorie du pluripotentiel, classes d’énergie de Monge-Ampère, rayons géodésiques

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