In this paper we consider a punctured Riemann surface endowed with a Hermitian metric that equals the Poincar{\'e} metric near the punctures, and a holomorphic line bundle that polarizes the metric. We show that the quotient of the Bergman kernel of high tensor powers of the line bundle and of the Bergman kernel of the Poincar{\'e} model near the singularity tends to one up to arbitrary negative powers of the tensor power.

We develop a general theory for the existence of extremal Kähler metrics of Poincaré type in the sense of Auvray (J Reine Angew Math 722:1–64, 2017), defined on the complement of a torus invariant divisor of a smooth compact toric variety. In the case when the divisor is smooth, we obtain a list of necessary conditions which must be satisfied for such a metric to exist. Using the explicit methods of Apostolov et al. (Ann Sci Ecole Norm Supp (4) 48:1075–1112, 2015; J Reine Angew Math 721:109–147, 2016, https://doi.org/10.1515/crelle-2014-0060 ) together with the computational approach of Sektnan (N Y J Math 24:317–354, 2018), we show that on a Hirzebruch complex surface the necessary conditions are also sufficient. In particular, on such a complex surface the complement of the infinity section admits an extremal Kähler metric of Poincaré type, whereas the complement of a fibre fixed by the torus action admits a complete ambitoric extremal Kähler metric which is not of Poincaré type.

We give an original analytic construction of hyperkahler ALF metrics on some ALE spaces of dihedral type, namely the spaces corresponding to minimal resolutions of Kleinian quotients relative to some binary dihedral group.

— A Poincaré type Kähler metric on the complement X\D of a simple normal crossing divisor D, in a compact Kähler manifold X, is a Kähler metric on X\D with cusp singularity along D. We relate the Futaki character for holomorphic vector fields parallel to the divisor, defined for any fixed Poincaré type Kähler class, to the classical Futaki character for the relative smooth class. As an application we express a numerical obstruction to the existence of extremal Poincaré type Kähler metrics, in terms of mean scalar curvatures and Futaki characters. Résumé. — On appelle métrique kählérienne de type Poincaré, sur le complémentaire X\D d’un diviseur à croisements normaux simples D dans une variété kählérienne compacte X, une métrique kählérienne sur X\D à singularités cusp le long de D. On relie le caractère de Futaki des champs de vecteurs holomorphes parallèles au diviseur, défini pour toute classe de Kähler de métriques de type Poincaré fixée, au caractère de Futaki classique de la classe lisse sous-jacente. On donne en application une obstruction numérique à l’existence de métriques extrémales de type Poincaré, exprimée en termes de courbures scalaires moyennes et de caractères de Futaki.

A Poincare type Kahler metric on the complement X\D of a simple normal crossing divisor D, in a compact Kahler manifold X, is a Kahler metric on X\D with cusp singularity along D. We relate the Futaki character for holomorphic vector fields parallel to the divisor, defined for any fixed Poincare type Kahler class, to the classical Futaki character for the relative smooth class. As an application we express a numerical obstruction to the existence of extremal Poincare type Kahler metrics, in terms of mean scalar curvatures and Futaki characters.

Consider a compact Kahler manifold X with a simple normal crossing divisor D, and define Poincare type metrics on X\D as Kahler metrics on X\D with cusp singularities along D. We prove that the existence of a constant scalar curvature (respectively an extremal) Poincare type Kahler metric on X\D implies the existence of a constant scalar curvature (respectively an extremal) Kahler metric, possibly of Poincare type, on every component of D. We also show that when the divisor is smooth, the constant scalar curvature/extremal metric on X\D is asymptotically a product near the divisor.

Consider a divisor D with simple normal crossings in a compact K\ahler manifold X. We show in this article that a K\ahler metric in an arbitrary class, with constant scalar curvature and cusp singularities along the divisor is unique in this class when K[D] is ample. This we do by generalizing Chen's construction of approximate geodesics in the space of K\ahler metrics, and proving an approximate version of the Calabi-Yau theorem, both independently of the ampleness of K[D].

A Poincaré type Kähler metric on the complement Xof a simple normal crossing divisor D, in a compact Kähler manifold X, is a Kähler metric on Xwith cusp singularity along D. We relate the Futaki character for holomorphic vector fields parallel to the divisor, defined for any fixed Poincaré type Kähler class, to the classical Futaki character for the relative smooth class. As an application we express a numerical obstruction to the existence of extremal Poincaré type Kähler metrics, in terms of mean scalar curvatures and Futaki characters.

Let D = Sigma(N)(j-1) D-j a divisor with simple normal crossings in a Kahler manifold (X, omega(0)) of complex dimension m >= 2. The purpose of this article was to show that the existence of a Poincare-type metric (omega) over bar with constant scalar curvature in PM[omega 0] on X backslash D implies for all j the inequality. (s) over bar < <(s)over bar>D-j. Here (s) over bar denotes the scalar curvature of (omega) over bar, whereas (s) over bar (Dj) denotes the mean scalar curvature associated to PM[omega 0 vertical bar Dj] or [omega(0)vertical bar(Dj)], depending on whether D-jj intersects with other components or not. We also explain how those results were already conjectured by G. Szekelyhidi when D is reduced to one component.

Ce travail de these s'interesse a la resolution d'equations de Monge-Ampere complexes et a ses applications sur certains types de varietes non compactes. Ce memoire decrit plus precisement deux situations distinctes dans lesquelles on resout des equations de Monge-Ampere, avant de tirer les consequences de ces resolutions. Dans une premiere partie, on travaille sur le complementaire d'un diviseur a croisements normaux dans une variete khalerienne compacte. On fixe sur le complementaire du diviseur une classe de metriques kahleriennes a singularites cusp le long du diviseur. %, classe que nous munissons elle-meme d'une metrique canonique. Pour construire des geodesiques entre metriques de cette classe, on resout une equation de Monge-Ampere homogene, sur le produit de notre ouvert de Zariski par une surface de Riemann a bord. On applique cette construction a un resultat d'unicite de metriques a courbure scalaire constante dans la classe consideree ; on resout encore pour cela une equation de Monge-Ampere avec second membre sur le complementaire du diviseur. On exhibe enfin des obstructions topologiques a l'existence de metriques a courbure scalaire constante au sein des classes de metriques kahleriennes singulieres envisagees. La seconde partie du memoire traite d'une construction analytique d'instantons gravitationnels ALF, ou varietes completes de dimension 4, hyperkahleriennes, a croissance cubique du volume. On donne la construction d'instantons diedraux ; on considere plus exactement des resolutions de singularites kleiniennes diedrales. Le traitement d'une equation de Monge-Ampere, donne pour des varietes kahleriennes ALF assez generales, nous permet sur nos exemples de corriger un prototype simple pour obtenir la metrique hyperkahlerienne recherchee.