Classiflcation of indeflnite hyper-Kahler symmetric spaces

Journal of Geometry and …, 2007

For each non-compact quaternion-Kähler symmetric space of dimension 8, all of its descriptions as a homogeneous Riemannian space (obtained through the Witte's refined Langlands decomposition) and the associated homogeneous quaternionic Kähler structures are studied.

2009

We consider hyper-Kähler manifolds of complex dimension 4 which are fibrations. It is known that the fibers are abelian varieties and the base is P. We assume that the general fiber is isomorphic to a product of two elliptic curves. We prove that such a hyper-Kähler manifold is deformation equivalent to a Hilbert scheme of two points on a K3 surface. 1. Preliminaries First we define our main objects of study, irreducible symplectic manifolds or hyperKähler manifolds. Definition 1.1. A compact complex Kähler manifold X is called irreducible symplectic if it is simply connected and if H(X, ΩX) is spanned by an everywhere non-degenerate 2-form ω. Any holomorphic two-form σ induces a homomorphism TX → ΩX . The two-form is everywhere non-degenerate if and only if TX → ΩX is bijective. The last condition in the definition implies that h(X) = h(X) = 1 and KX ∼= OX , i.e., c1(X) = 0. Definition 1.2. A compact connected 4n-dimensional Riemannian manifold (M, g) is called irreducible hyper-Kä...

Transactions of the London Mathematical Society, 2021

We study symmetry properties of quaternionic Kähler manifolds obtained by the HK/QK correspondence. To any Lie algebra g of infinitesimal automorphisms of the initial hyper-Kähler data we associate a central extension of g, acting by infinitesimal automorphisms of the resulting quaternionic Kähler manifold. More specifically, we study the metrics obtained by the one-loop deformation of the c-map construction, proving that the Lie algebra of infinitesimal automorphisms of the initial projective special Kähler manifold gives rise to a Lie algebra of Killing fields of the corresponding one-loop deformed c-map space. As an application, we show that this construction increases the cohomogeneity of the automorphism groups by at most one. In particular, if the initial manifold is homogeneous then the one-loop deformed metric is of cohomogeneity at most one. As an example, we consider the one-loop deformation of the symmetric quaternionic Kähler metric on SU (n, 2)/S(U (n) × U (2)), which we prove is of cohomogeneity exactly one. This family generalizes the so-called universal hypermultiplet (n = 1), for which we determine the full isometry group.

Russian Academy of Sciences Sbornik Mathematics, 2003

Let G/K be an irreducible Hermitian symmetric space of compact type with the standard homogeneous complex structure. Then the real symplectic manifold (T * (G/K), Ω) has the natural complex structure J −. We construct all Ginvariant Kähler structures (J, Ω) on homogeneous domains in T * (G/K) anticommuting with J −. Each such a hypercomplex structure, together with a suitable metric, defines a hyper-Kählerian structure. As an application, we obtain a new proof of the Harish-Chandra and Moore theorem for Hermitian symmetric spaces. Bibliography: 13 titles.

Annales Polonici Mathematici, 2012

For each non-compact quaternion-Kähler symmetric space of dimension 8, all of its descriptions as a homogeneous Riemannian space (obtained through the Witte's refined Langlands decomposition) and the associated homogeneous quaternionic Kähler structures are studied. * Partially supported by Dgicyt, Spain, under Grant mtm-.  ams Subject Classification: Primary c; Secondary c, c.

Proceedings of the Edinburgh Mathematical Society, 2010

We study homogeneous Kähler structures on a non-compact Hermitian symmetric space and their lifts to homogeneous Sasakian structures on the total space of a principal line bundle over it, and we analyse the case of the complex hyperbolic space.

Handbook of Pseudo-Riemannian Geometry and Supersymmetry

Under the action of the c-map, special Kähler manifolds are mapped into a class of quaternion-Kähler spaces. We explicitly construct the corresponding Swann bundle or hyperkähler cone, and determine the hyperkähler potential in terms of the prepotential of the special Kähler geometry. Contents 1 Introduction 1 2 The c-map 2 3 Hyperkähler cones and the Legendre transform 3 4 Hyperkähler cones from the c-map 5 4.

Journal of Geometric Analysis, 1999

A general method for calculation of the full isometry group of a Riemannian solvmanifold is presented. Using it we determine the full isometry group of the non-symmetric quaternionic K ahler solvmanifolds M : T-, Wand V-spaces. As an application we prove that the isometry group acts transitively on the twistor space and on the S O(3)-principal (\3-Sasakian") bundle of M and that the manifold M does not admit quotients of nite volume. As other application, we give a simple description of the quaternionic K ahler solvmanifolds in terms of a certain spinorial module S of the group S pin(3; 3 + k). The Lie bracket is de ned by means of the unique embedding of the vector module V = R 3;3+k into^2S. We describe also the group of isometries which preserves the principal K ahler submanifold U M .

Mathematical Proceedings of the Cambridge Philosophical Society, 2000

The Kähler case of Riemannian homogeneous structures [3, 15, 18] has been studied in [1, 2, 6, 7, 13, 16], among other papers. Abbena and Garbiero [1] gave a classification of Kähler homogeneous structures, which has four primitive classes [Kscr ]1, …, [Kscr ]4 (see [6, theorem 5·1] for another proof and Section 2 below for the result). The purpose of the present paper is to prove the following result:THEOREM 1·1. A simply connected irreducible homogeneous Kähler manifold admits a nonvanishing Kähler homogeneous structure in Abbena–Garbiero's class [Kscr ]2 [oplus ] [Kscr ]4if and only if it is the complex hyperbolic space equipped with the Bergman metric of negative constant holomorphic sectional curvature.