Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields

Campana, Frédéric; Guenancia, Henri; Păun, Mihai

Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 46 (2013) no. 6, pp. 879-916.

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.

Dans cet article, nous prouvons l'existence de métriques de Kähler-Einstein à courbure négative ayant des singularités coniques le long d'un diviseur à croisements normaux simples sur une variété kählérienne compacte, sous une hypothèse technique sur les angles des cones. Nous discutons également du cas des métriques de Kähler-Einstein à 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'étendent à notre cadre.

Classification: 32Q05, 32Q10, 32Q15, 32Q20, 32U05, 32U15
Keywords: kähler-Einstein metrics, cone singularities, orbifold tensors, Monge-ampère equations
Mot clés : métriques de kähler-Einstein, singularités coniques, tenseurs orbifoldes, équation de Monge-ampère

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