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Abstract

Abstract. We study Ka¨hler surfaces with harmonic anti-self-dual Weyl tensor. We provide an explicit local description, which we use to obtain the complete classification in the compact case. We give new examples of extremal Ka¨hler metrics, including Ka¨hler-Einstein metrics and conformally Einstein Ka¨hler metrics. We also extend some of our results to almost Ka¨hler 4-manifolds, providing new examples of Ricci-flat almost Ka¨hler metrics which are not Ka¨hler. Mathematics Subject Classifications (2000). 53B35, 53C55.

On the uniqueness of supersymmetric AdS(5) black holes with toric symmetry

“…In this section we will focus on a particular subclass of toric Kähler bases that are of Calabi type [30] (we will define this in section 3.1). Our motivation for considering this class of solutions is that the CCLP black hole, which is the most general known black hole in this theory, has a Kähler base of Calabi type [27].…”

“…In this subsection, we will introduce toric Kähler manifolds of Calabi type. A general definition of Kähler surfaces of Calabi type was given by Apostolov, Calderbank and Gauduchon in their study of Kähler surfaces admitting a hamiltonian two-form, the latter a concept which they also introduce [30] 10 . They find that Kähler surfaces admitting a hamiltonian two-form necessarily possess two commuting hamiltonian Killing vectors, and if these vectors are independent the Kähler surface is orthotoric, whereas if they are dependent it is of Calabi type.…”

“…An orthotoric Kähler surface is one that admits two linearly independent hamiltonian Killing fields with moment maps ξ + η and ξη such that dξ and dη are orthogonal. We will give our own analogous definition of Calabi type in the context of toric Kähler surfaces (which is a subclass of Calabi type according to [30]). Definition 1.…”

“…Curiously, it turns out that the supersymmetric CCLP black hole has a Kähler base of Calabi type [27]. Kähler surfaces of Calabi type naturally appear in the classification of Kähler surfaces that admit a hamiltonian two-form [30]. We will give a self-contained definition of Calabi type in the context of toric Kähler surfaces in terms of a certain orthogonality property of the associated moment maps (analogous to orthotoric Kähler, see Definition 1).…”

Bihermitian Gray surfaces

“…For other examples of manifolds of the type studied here, see [Madsen et al 1997]; their Ricci tensor has two eigenvalues of multiplicities 1 and 3, whereas ours have Ricci tensor with two eigenvalues of the same multiplicity 2. Noncompact examples of bihermitian Kähler ᏭC ⊥ -surfaces (also of cohomogeneity 1) were first given [Derdziński 1981] and recently the general explicit expression of such Kähler surfaces was discovered by Apostolov, Calderbank and Gauduchon [Apostolov et al 2003]. …”