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

Abstract

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.

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

“…Here is a result which encompasses the previous results of [11], Kobayashi ([23]) and ). It is the technical result expressing in terms of Monge-Ampère equations the content of Theorem A given in the introduction (cf.…”

“…The proof of this results follows closely the one of its analogue in [11]: we use a Bochner formula applied to the truncated holomorphic tensors, and the key point is to control the error term. However, a new difficulty pops up here, namely we have to deal with an additional term coming from the curvature of the line bundle O X ( ∆ ); fortunately, it has the right sign.…”

“…We claim (see also [11,Lemma 2.2]) that there exist constants C 1 , C 2 depending only B, inf ∆F and n such that…”

“…when the coefficients of ∆ are strictly less than 1), under the assumption that K X + ∆ is positive or zero, has been studied by R. Mazzeo [26], T. Jeffres [21] and recently solved independently by S. Brendle [8] and R. Mazzeo, T. Jeffres, Y. Rubinstein [22] in the case of an (irreducible) smooth divisor, and by Campana-Guenancia-Păun [11] in the general case of a simple normal crossing divisor (having though all its coefficients greater than 1/2). In the conic case where K X + ∆ < 0, some interesting existence results were obtained by R. Berman in [5] and T. Jeffres, R. Mazzeo and Y. Rubinstein in [22].…”

“…In the last part of the paper, and as in [11], we try to use the Kähler-Einstein metric constructed in the previous sections to obtain the vanishing of some particular holomorphic tensors attached to a pair (X, ∆), ∆ being still a R-divisor with simple normal crossing support and having coefficients in [0,1]. This specific class consists in the holomorphic tensors which are the global sections of the locally free sheaf T r s (X|∆) introduced by Campana in [10]: they are holomorphic tensors on X 0 with prescribed zeros or poles along ∆.…”