We give a criterion for the coercivity of the Mabuchi functional for general Kähler classes on Fano manifolds in terms of Tian’s alpha invariant. This generalises a result of Tian in the anti-canonical case implying the existence of a Kähler-Einstein metric. We also prove the alpha invariant is a continuous function on the Kähler cone. As an application, we provide new Kähler classes on a general degree one del Pezzo surface for which the Mabuchi functional is coercive.
On donne un critère pour la coercivité de la fonctionnelle de Mabuchi pour des classes de Kähler générales sur les variétés de Fano en termes d’invariant alpha de Tian. Cela généralise un théorème de Tian dans le cas anticanonique, ce qui implique l’existence d’une métrique Kähler-Einstein. On montre également que l’invariant alpha est une fonction continue sur le cône de Kähler. On en déduit de nouvelles classes de Kähler sur des surfaces de Del Pezzo pour lesquelles la fonctionnelle de Mabuchi est coercive.
@article{AFST_2016_6_25_4_919_0,
author = {Ruadha{\'\i} Dervan},
title = {Alpha invariants and coercivity of the {Mabuchi} functional on {Fano} manifolds},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {919--934},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 25},
number = {4},
year = {2016},
doi = {10.5802/afst.1515},
zbl = {1357.32018},
mrnumber = {3564131},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1515/}
}
TY - JOUR AU - Ruadhaí Dervan TI - Alpha invariants and coercivity of the Mabuchi functional on Fano manifolds JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2016 SP - 919 EP - 934 VL - 25 IS - 4 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1515/ DO - 10.5802/afst.1515 LA - en ID - AFST_2016_6_25_4_919_0 ER -
%0 Journal Article %A Ruadhaí Dervan %T Alpha invariants and coercivity of the Mabuchi functional on Fano manifolds %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2016 %P 919-934 %V 25 %N 4 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1515/ %R 10.5802/afst.1515 %G en %F AFST_2016_6_25_4_919_0
Ruadhaí Dervan. Alpha invariants and coercivity of the Mabuchi functional on Fano manifolds. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 25 (2016) no. 4, pp. 919-934. doi : 10.5802/afst.1515. https://afst.centre-mersenne.org/articles/10.5802/afst.1515/
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