Soient une variété Kählerienne compacte et une forme fermée qui représente une classe de cohomologie grosse. On introduit une métrique sur l’espace d’énergie finie , 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 , 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.
Suppose is a compact Kähler manifold of dimension , and is closed -form representing a big cohomology class. We introduce a metric on the finite energy space , 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 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.
Mots clés : Variétés Kähleriénnes, théorie du pluripotentiel, classes d’énergie de Monge-Ampère, rayons géodésiques
@article{AIF_2018__68_7_3053_0, author = {Darvas, Tam\'as and Di Nezza, Eleonora and Lu, Chinh H.}, title = {$L^1$ metric geometry of big cohomology classes}, journal = {Annales de l'Institut Fourier}, pages = {3053--3086}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {68}, number = {7}, year = {2018}, doi = {10.5802/aif.3236}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.3236/} }
TY - JOUR AU - Darvas, Tamás AU - Di Nezza, Eleonora AU - Lu, Chinh H. TI - $L^1$ metric geometry of big cohomology classes JO - Annales de l'Institut Fourier PY - 2018 DA - 2018/// SP - 3053 EP - 3086 VL - 68 IS - 7 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.3236/ UR - https://doi.org/10.5802/aif.3236 DO - 10.5802/aif.3236 LA - en ID - AIF_2018__68_7_3053_0 ER -
Darvas, Tamás; Di Nezza, Eleonora; Lu, Chinh H. $L^1$ metric geometry of big cohomology classes. Annales de l'Institut Fourier, Tome 68 (2018) no. 7, pp. 3053-3086. doi : 10.5802/aif.3236. http://www.numdam.org/articles/10.5802/aif.3236/
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