Let X be a non-singular compact Kähler manifold, endowed with an effective divisor D = (1 − β k )Y k having simple normal crossing support, and satisfying β k ∈ (0, 1). The natural objects one has to consider in order to explore the differentialgeometric properties of the pair (X, D) are the so-called metrics with conic singularities. In this article, we complete our earlier work [CGP13] concerning the Monge-Ampère equations on (X, D) by establishing Laplacian and C 2,α,β estimates for the solution of this equations regardless to the size of the coefficients 0 < β k < 1. In particular, we obtain a general theorem concerning the existence and regularity of Kähler-Einstein metrics with conic singularities along a normal crossing divisor.

We prove the existence of non-positively curved Kähler-Einstein metrics with cone singularities along a given simple normal crossing divisor of a compact Kähler manifold, under a technical condition on the cone angles, and we also discuss the case of positively-curved Kähler-Einstein metrics with cone singularities. As an application we extend to this setting classical results of Lichnerowicz and Kobayashi on the parallelism and vanishing of appropriate holomorphic tensor fields.Résumé. -Dans cet article, nous prouvons l'existence de métriques de Kähler-Einsteinà courbure négative ayant des singularités coniques le long d'un diviseurà croisements normaux simples sur une variété kählerienne compacte, sous une hypothèse technique sur les angles des cones. Nous discutonségalement du cas des métriques de Kähler-Einsteinà courbure strictement positive avec des singularités coniques. Nous en déduisons que les résultats classiques de Lichnerowicz et Kobayashi sur le parallélisme et l'annulation des champs de tenseurs holomorphes s'étendentà notre cadre.It is a pleasure to thank S. Boucksom and V. Tosatti: their remarks and comments helped us to improve the content and the presentation of the present article.

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