Scalar curvature and Futaki invariant of Kähler metrics with cone singularities along a divisor  [ La courbure scalaire et l’invariant de Futaki des métriques kählériennes avec des singularités coniques le long d’un diviseur ]
Annales de l'Institut Fourier, à paraître, 62 p.

Nous étudions la courbure scalaire de certaines classes de métriques kählériennes qui ont des singularités coniques le long d’un diviseur. Notre résultat principal concerne le complété projectif du fibré pluricanonique sur un produit de variétés Fano Kähler–Einstein avec second nombre de Betti égal à 1. Sur cette variété, en utilisant une construction due à Hwang, nous démontrons qu’il existe une métrique kählérienne à courbure scalaire constante et singularités coniques si et seulement si l’invariant logarithmique de Futaki est nul pour l’action de ℂ * sur le fibré. Ce résultat conforte la version logarithmique de la conjecture de Yau–Tian–Donaldson pour des polarisations quelconques.

Nous démontrons aussi que la courbure scalaire peut être définie sur le lieu singulier comme un courant pour certaines classes de métriques à singularités coniques, ce qui nous permet de calculer l’invariant logarithmique de Futaki par rapport à des métriques singulières.

We study the scalar curvature of Kähler metrics that have cone singularities along a divisor, with a particular focus on certain specific classes of such metrics that enjoy some curvature estimates. Our main result is that, on the projective completion of a pluricanonical bundle over a product of Kähler–Einstein Fano manifolds with the second Betti number 1, momentum-constructed constant scalar curvature Kähler metrics with cone singularities along the ∞-section exist if and only if the log Futaki invariant vanishes on the fibrewise ℂ * -action, giving a supporting evidence to the log version of the Yau–Tian–Donaldson conjecture for general polarisations.

We also show that, for these classes of conically singular metrics, the scalar curvature can be defined on the whole manifold as a current, so that we can compute the log Futaki invariant with respect to them. Finally, we prove some partial invariance results for them.

Reçu le : 2017-04-27
Révisé le : 2017-09-19
Accepté le : 2017-11-14
Publié le : 2019-03-08
Classification:  53C55,  53C25
Mots clés: métriques kählériennes de courbure scalaire constante, l’invariant de Futaki, métriques kählériennes avec les singularités coniques 
@unpublished{AIF_0__0_0_A16_0,
     author = {Hashimoto, Yoshinori},
     title = {Scalar curvature and Futaki invariant of K\"ahler metrics with cone singularities along a divisor},
     note = {to appear in \emph{Annales de l'Institut Fourier}},
}

Hashimoto, Yoshinori. Scalar curvature and Futaki invariant of Kähler metrics with cone singularities along a divisor. Annales de l'Institut Fourier, à paraître, 62 p.

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