Kähler-Einstein metrics with mixed Poincaré and cone singularities along a normal crossing divisor

Guenancia, Henri

doi:10.5802/aif.2881

Kähler-Einstein metrics with mixed Poincaré and cone singularities along a normal crossing divisor
[MÃ©triques de Kähler-Einstein Ã singularités mixtes Poincaré et coniques le long d’un diviseur à croisements normaux]

Guenancia, Henri ¹

¹ Université Pierre et Marie Curie Institut de Mathématiques de Jussieu, Paris & École Normale Supérieure Département de Mathématiques et Applications Paris (France)

Annales de l'Institut Fourier, Tome 64 (2014) no. 3, pp. 1291-1330.

Résumé
Abstract

Soit $X$ une variété compacte kählerienne et $Δ$ un $ℝ$ -diviseur dont le support est à croisements normaux simples et à coefficients entre $1 / 2$ et $1$ . En supposant $K_{X} + Δ$ ample, on prouve l’existence et l’unicité d’une métrique de Kähler-Einstein à courbure négative sur $X ∖ Supp (Δ)$ ayant des singularités mixtes Poincaré et coniques suivant les coefficients de $Δ$ . Nous appliquons ensuite ce résultat pour prouver un théorème d’annulation concernant certains champs de tenseurs holomorphes naturellement attachés à la paire $(X, Δ)$ .

Let $X$ be a compact Kähler manifold and $Δ$ be a $ℝ$ -divisor with simple normal crossing support and coefficients between $1 / 2$ and $1$ . Assuming that $K_{X} + Δ$ is ample, we prove the existence and uniqueness of a negatively curved Kahler-Einstein metric on $X ∖ Supp (Δ)$ having mixed Poincaré and cone singularities according to the coefficients of $Δ$ . As an application we prove a vanishing theorem for certain holomorphic tensor fields attached to the pair $(X, Δ)$ .

MR Zbl

DOI : 10.5802/aif.2881

Classification : 32Q05, 32Q10, 32Q15, 32Q20, 32U05, 32U15
Keywords: Kähler-Einstein metrics, cone singularities, Poincaré singularities, cusps, orbifold tensors, complex Monge-Ampère equation
Mot clés : métriques de Kähler-Einstein, singularités coniques, singularités Poincaré, cusps, tenseurs orbifoldes, équation de Monge-Ampère complexe

Affiliations des auteurs :

Guenancia, Henri ¹

¹ Université Pierre et Marie Curie Institut de Mathématiques de Jussieu, Paris & École Normale Supérieure Département de Mathématiques et Applications Paris (France)

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     title = {K\"ahler-Einstein metrics with mixed {Poincar\'e} and cone singularities along a normal crossing divisor},
     journal = {Annales de l'Institut Fourier},
     pages = {1291--1330},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {64},
     number = {3},
     year = {2014},
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Guenancia, Henri. Kähler-Einstein metrics with mixed Poincaré and cone singularities along a normal crossing divisor. Annales de l'Institut Fourier, Tome 64 (2014) no. 3, pp. 1291-1330. doi : 10.5802/aif.2881. https://aif.centre-mersenne.org/articles/10.5802/aif.2881/

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