Note on Poincaré type Futaki characters
Annales de l'Institut Fourier, Volume 68 (2018) no. 1, pp. 319-344.

A Poincaré type Kähler metric on the complement XD of a simple normal crossing divisor D, in a compact Kähler manifold X, is a Kähler metric on XD with cusp singularity along D. We relate the Futaki character for holomorphic vector fields parallel to the divisor, defined for any fixed Poincaré type Kähler class, to the classical Futaki character for the relative smooth class. As an application we express a numerical obstruction to the existence of extremal Poincaré type Kähler metrics, in terms of mean scalar curvatures and Futaki characters.

On appelle métrique kählérienne de type Poincaré, sur le complémentaire XD d’un diviseur à croisements normaux simples D dans une variété kählérienne compacte X, une métrique kählérienne sur XD à singularités cusp le long de D. On relie le caractère de Futaki des champs de vecteurs holomorphes parallèles au diviseur, défini pour toute classe de Kähler de métriques de type Poincaré fixée, au caractère de Futaki classique de la classe lisse sous-jacente. On donne en application une obstruction numérique à l’existence de métriques extrémales de type Poincaré, exprimée en termes de courbures scalaires moyennes et de caractères de Futaki.

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DOI: 10.5802/aif.3162
Classification: 53C55,  32Q15
Keywords: Extremal Kähler metrics, Poincaré type Kähler metrics, Futaki character/invariant, Yau–Tian–Donaldson conjecture.
Auvray, Hugues 1

1 Univ. Paris-Sud, Laboratoire de Mathématiques d’Orsay, CNRS, Université Paris-Saclay, F-91405 Orsay cedex (France)
License: CC-BY-ND 4.0
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Auvray, Hugues. Note on Poincaré type Futaki characters. Annales de l'Institut Fourier, Volume 68 (2018) no. 1, pp. 319-344. doi : 10.5802/aif.3162. https://aif.centre-mersenne.org/articles/10.5802/aif.3162/

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