A remark on the gradient map

Mathematische Annalen, 2011

We consider actions of reductive complex Lie groups G = K C on Kähler manifolds X such that the K -action is Hamiltonian and prove then that the closures of the G-orbits are complex-analytic in X . This is used to characterize reductive homogeneous Kähler manifolds in terms of their isotropy subgroups. Moreover we show that such manifolds admit K -moment maps if and only if their isotropy groups are algebraic.

2015

In this paper we provide a positive answer to a conjecture due to A. J. Di Scala, A. Loi, H. Hishi (see [3, Conjecture 1]) claiming that a simply-connected homogeneous Kähler manifold M endowed with an integral Kähler form μω, admits a holomorphic isometric immersion in the complex projective space, for a suitable μ>0. This result has two corollaries which extend to homogeneous Kähler manifolds the results obtained by the authors in [8] and in [12] for homogeneous bounded domains.

Annales Scientifiques de l’École Normale Supérieure, 2004

Annales de la faculté des sciences de Toulouse Mathématiques, 2008

We prove that every Kähler solvmanifold has a finite covering whose holomorphic reduction is a principal bundle. An example is given that illustrates the necessity, in general, of passing to a proper covering. We also answer a stronger version of a question posed by Akhiezer for homogeneous spaces of nonsolvable algebraic groups in the case where the isotropy has the property that its intersection with the radical is Zariski dense in the radical.

Inventiones mathematicae, 2009

Ann Sci Ecole Norm Super, 2004

2021

Let (Z, ω) be a Kähler manifold and let U be a compact connected Lie group with Lie algebra u acting on Z and preserving ω. We assume that the U -action extends holomorphically to an action of the complexified group U and the U -action on Z is Hamiltonian. Then there exists a U -equivariant momentum map μ : Z → u. If G ⊂ U is a closed subgroup such that the Cartan decomposition U = Uexp(iu) induces a Cartan decomposition G = Kexp(p), where K = U ∩ G, p = g ∩ iu and g = k ⊕ p is the Lie algebra of G, there is a corresponding gradient map μp : Z → p. If X is a G-invariant compact and connected real submanifold of Z, we may consider μp as a mapping μp : X → p. Given an Ad(K)-invariant scalar product on p, we obtain a Morse like function f = 1 2 ‖ μp ‖ 2 on X. We point out that, without the assumption that X is real analytic manifold, the Lojasiewicz gradient inequality holds for f . Therefore the limit of the negative gradient flow of f exists and it is unique. Moreover, we prove that ...

Journal of the Australian Mathematical Society, 2004

We study the harmonicity of maps to or from cosymplectic manifolds by relating them to maps to or from Kähler spaces.

Matematicki Vesnik

In this work we define generalized Kahlerian spaces and for them consider holomorphically projective mappings with an invariant complex structure. Also we consider eq-uitorsion holomorphically projective mappings and for them we find some invariant geometric objects.

International Mathematics Research Notices, 2015

In the case of a compact real analytic symplectic manifold (M, ω) we describe an approach to the complexification of Hamiltonian flows [Se, Do1, Th1] and corresponding geodesics on the space of Kähler metrics. In this approach, motivated by recent work on quantization, the complexified Hamiltonian flows act, through the Gröbner theory of Lie series, on the sheaf of complex valued real analytic functions, changing the sheaves of holomorphic functions. This defines an action on the space of (equivalent) complex structures on M and also a direct action on M. This description is related to the approach of [BLU] where one has an action on a complexification M C of M followed by projection to M. Our approach allows for the study of some Hamiltonian functions which are not real analytic. It also leads naturally to the consideration of continuous degenerations of diffeomorphisms and of Kähler structures of M. Hence, one can link continuously (geometric quantization) real, and more general non-Kähler, polarizations with Kähler polarizations. This corresponds to the extension of the geodesics to the boundary of the space of Kähler metrics. Three illustrative examples are considered. We find an explicit formula for the complex time evolution of the Kähler potential under the flow. For integral symplectic forms, this formula corresponds to the complexification of the prequantization of Hamiltonian symplectomorphisms. We verify that certain families of Kähler structures, which have been studied in geometric quantization, are geodesic families. Contents 1 Introduction 2 Complexification of analytic flows and action on complex structures 3 Complexification of Hamiltonian flows 4 Action on Kähler structures