Guenancia, Henri
Kähler-Einstein metrics with mixed Poincaré and cone singularities along a normal crossing divisor  [ Métriques de Kähler-Einstein à singularités mixtes Poincaré et coniques le long d’un diviseur à croisements normaux ]
Annales de l'institut Fourier, Tome 64 (2014) no. 3 , p. 1291-1330
MR 3330171 | Zbl 06387308
doi : 10.5802/aif.2881
URL stable : http://www.numdam.org/item?id=AIF_2014__64_3_1291_0

Classification:  32Q05,  32Q10,  32Q15,  32Q20,  32U05,  32U15
Mots clés: métriques de Kähler-Einstein, singularités coniques, singularités Poincaré, cusps, tenseurs orbifoldes, équation de Monge-Ampère complexe
Soit X une variété compacte kählerienne et Δ un -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 XSupp(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 -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 XSupp(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).

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