Note on Poincaré type Futaki characters

Auvray, Hugues

Annales de l'Institut Fourier, Tome 68 (2018) no. 1, pp. 319-344.

On appelle métrique kählérienne de type Poincaré, sur le complémentaire $X\setminus D$ d’un diviseur à croisements normaux simples $D$ dans une variété kählérienne compacte $X$, une métrique kählérienne sur $X\setminus D$ 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.

A Poincaré type Kähler metric on the complement $X\setminus D$ of a simple normal crossing divisor $D$, in a compact Kähler manifold $X$, is a Kähler metric on $X\setminus D$ 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.

Classification : 53C55, 32Q15
Keywords: Extremal Kähler metrics, Poincaré type Kähler metrics, Futaki character/invariant, Yau–Tian–Donaldson conjecture.
Mot clés : Métriques kählériennes extrémales, métriques kählériennes de type Poincaré, caractère/invariant de Futaki, conjecture de Yau–Tian–Donaldson.

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