Ali Ben

Abdelghani Zeghib
email: abdelghani.zeghib@ens-lyon.fr

D Épartement

ON HOMOGENEOUS HERMITE-LORENTZ SPACES

We define naturally Hermite-Lorentz metrics on almost-complex manifolds as special case of pseudo-Riemannian metrics compatible with the almost complex structure. We study their isometry groups.

CONTENTS

1. Introduction 1 2. Results 4 3. Some preparatory facts 8 4. Subgroups of U(1, n) 11 5. Proof of Theorem 2.1 15 6. Proof of Theorem 2.2: Preliminaries 19 7. Proof of Theorem 2.2: non-amenable isotropy case 21 8. Proof of Theorem 2.2 in the amenable case 23 References 25

INTRODUCTION

Simplest pseudo-Hermitian structures. Let us call a quadratic form q on a complex space of dimension n + 1 of Hermite-Lorentz type if it is C-equivalent to the standard form q 0 = -|z 0 | 2 + |z 1 | 2 + . . . + |z n | 2 on C n+1 . In other words, q is Hermitian, and as a real form, it has a signature --+ . . . +. Here, Lorentz refers to the occurrence of exactly one negative sign (in the complex presentation). Classically, this one negative sign distinguishes, roughly, between time and space components. (A "complex-Lorentz" form could perhaps be an equally informative terminology?)

One can then define Hermite-Lorentz metrics on almost complex manifolds. If (M, J) is an almost complex manifold, then g is a Hermite-Lorentz metric if g is a tensor such that for any x ∈ M , (T x M, J x , g x ) is a Hermite-Lorentz linear space.

Hermite-Lorentz metrics generalize (definite) Hermitian metrics, and they are the nearest from them, among general pseudo-Hermitian structures, in the sense that they have the minimal (non-trivial) signature. Our point of view here is to compare Hermite-Lorentz metrics, on one hand with (definite) Hermitian metrics in complex geometry, and with Lorentz metrics in (real) differential geometry.

Date: January 2, 2017. 1 H-structure. Let U(1, n) ⊂ GL n+1 (C) be the unitary group of q 0 . Then, a Hermite-Lorentz structure on a manifold M of real dimension 2n + 2 is a reduction of the structural group of T M to U(1, n). They are different from "complex Riemannian" metrics which are reduction to O(n + 1, C).

Kähler-Lorentz spaces. As in the positive definite case, the Kähler form ω is defined by ω(u, v) = g(u, Jv). It is a J-invariant 2-differential form. A Kähler-Lorentz metric corresponds to the case where J is integrable and ω is closed. A Kähler-Lorentz manifold is in particular symplectic.

Conversely, from the symplectic point of view, a symplectic manifold (M, ω) is Kähler-Lorentz if ω can be calibrated with a special complex structure J. Let us generalize the notion of calibration by letting it to mean that J satisfies that g(u, v) = ω(u, Jv) is non-degenerate, i.e. g is a pseudo-Hermitian metric. Now, the classical Kähler case means that g is Hermitian and in addition J is integrable. So Kähler-Lorentz means that g is "post-Hermitian" in the sense that it has a Hermite-Lorentz signature.

Holomorphic sectional curvature. Differential geometry can be developed for general pseudo-Hermitian metrics exactly as in the usual Hermitian as well as the usual pseudo-Hermitian cases. In particular, there is a Levi-Civita connection and a Riemann curvature tensor R. For a tangent vector u, the holomorphic sectional curvature K(u) is the sectional curvature of the real 2-plane Cu; K(u) = g(R(u,Ju)Ju,u) g(u,u) 2 (this requires u to be non isotropic g(u, u) = 0, in order to divide by the volume u ∧ Ju). So, K is a real function on an open set of the projectivization bundle of T M (which fibers over M with fiber type P n (C)).

In fact, K determines the full Riemann tensor in the pseudo-Kähler case [START_REF] Kobayashi | Foundations of differential geometry[END_REF][START_REF] Barros | Indefinite Kähler manifolds[END_REF] (but not in the general pseudo-Hermitian case). In particular, the case K constant in the definite Kähler case corresponds to the most central homogeneous spaces: C n , P n (C) and H n (C) (the complex hyperbolic space). Kähler-Lorentz spaces of constant curvature are introduced below.

Examples.

We are going to give examples of homogeneous spaces M = G/H, where the natural (generally unique) G-invariant geometric structure is a Kähler-Lorentz metric.

1.1.1. Universal Kähler-Lorentz spaces of constant holomorphic curvature. If a Kähler-Lorentz metric has constant holomorphic sectional curvature, then it is locally isometric to one of the following spaces:

(1) The universal (flat Hermite-Lorentz) complex Minkowski space Mink n (C) (or C 1,n-1 ), that is C n endowed with

q 0 = -|z 1 | 2 + |z 2 | 2 + . . . + |z n | 2 .
(2) The complex de Sitter space dS n (C) = SU(1, n)/U(1, n -1) 1 It has a positive constant holomorphic sectional curvature.

(3) The complex anti de Sitter space AdS n (C) = SU(2, n -1)/U(1, n -1). It has negative curvature.

• As said above, Hermite-Lorentz metrics are generalizations of both Hermitian metrics (from the definite to the indefinite) and Lorentz metrics (from the real to the complex). Let us draw up in the following table the analogous of our previous spaces in both Hermitian and Lorentzian settings.

Kähler-Lorentz spaces

Hermitian (positive definite) (Real) Lorentz of constant curvature counterpart counterpart Mink n (C)

C n Mink n (R) dS n (C) = P n (C) = dS n (R) = SU(1, n)/U(1, n-1) SU(1 + n)/U(n) SO 0 (1, n)/SO 0 (1, n -1) AdS n (C) = H n (C) = AdS n (R) = SU(2, n -1)/U(1, n -1) SU(1, n)/U(n) SO(2, n -1)/SO 0 (1, n -1)
(Of course, we also have as Riemannian counterparts of constant sectional curvature, respectively, the Euclidean, spherical and hyperbolic spaces, R n , S n and H n ).

• The Kähler-Lorentz spaces of constant holomorphic sectional curvature are pseudo-Riemannian symmetric spaces (see below for further discussion). They are also holomorphic symmetric domains in C n . Indeed, dS n (C) is the exterior of a ball in the projective space P n (C). It is strictly pseudo-concave. The ball of P n (C) is identified with the hyperbolic space H n (C), and then dS n (C) is the space of geodesic complex hypersurfaces of H n (C).

As for AdS n (C), it can be represented as the open set q < 0 of P n (C), where

q = -|z 0 | 2 -|z 1 | 2 + |z 2 | 2 + . . . + |z n | 2 .
1.1.2. Irreducible Kähler-Lorentz symmetric spaces. Let M = G/H be a homogeneous space. Call p the base point 1.H. The isotropy representation at p is identified with the adjoint representation ρ : H → GL(g/h), where g and h are the respective Lie algebras of G and H. The homogeneous space is of Hermite-Lorentz type if ρ is conjugate to a representation in U(1, n) (where the real dimension of

G/H is 2n + 2).
The space G/H is symmetric if -Id TpM belongs to the image of ρ. This applies in particular to the two following spaces:

CdS n = SO 0 (1, n + 1)/SO 0 (1, n -1) × SO(2)
1. Here U(1, n -1) as a subgroup of SU(1, n) stands for matrices of the form

λA 0 0 λ -1 , |λ| = 1, A ∈ SU(1, n -1).
In general U(1, n -1) designs a group isomorphic to a product U(1) × SU(1, n -1), where the embedding U (1) depends on the context.

CAdS n = SO(3, n -1)/SO(2) × SO 0 (1, n -1)
Complexification. The isotropy representation of these two spaces is the complexification of the SO 0 (1, n -1) standard representation in R n , i.e. its diagonal action on C n = R n +iR n ; augmented with the complex multiplication by U(1) ∼ = SO(2).

If one agrees that a complexification of a homogeneous space X is a homogeneous space CX whose isotropy is the complexification of that of X, then CdS n and CAdS n appear naturally as complexification of dS n (R) and AdS n (R) respectively. In contrast, dS n (C) and AdS n (C) are the set of complex points of the same algebraic object as dS n (R) and AdS n (R). As another example, the complexification of S n is not P n (C) but rather the Kähler Grassmanian space SO(n + 2)/SO(n) × SO(2)?! 1.1.3. List. There are lists of pseudo-Riemannian irreducible symmetric spaces, see for instance [START_REF] Besse | Einstein manifolds[END_REF][START_REF] Berger | Les espaces symétriques non compacts[END_REF][START_REF] Kath | On the structure of pseudo-Riemannian symmetric spaces[END_REF]. (Here irreducibility concerns isotropy, but for symmetric spaces, besides the flat case, the holonomy and isotropy groups coincide. In particular, holonomy irreducible symmetric spaces are isotropy irreducible) . It turns out that the five previous spaces are all the Kähler-Lorentz (or equivalently Hermite-Lorentz) ones. Our theorem 2.1 below will give in particular a non list-checking proof of this classification.

RESULTS

2.0.1. Convention. It is sometimes a nuisance and with no real interest to deal with "finite objects". We will say, by the occasion, that some fact is true up to finite index, if it is not necessarily satisfied by the given group itself, say H, but for another one H ′ commensurable to it, that is H ∩ H ′ has finite index in both. We also use "up to finite cover" for a similar meaning. 2.0.2. Objective. Our aim here is the study of isometry groups Iso(M, J, g) of Hermite-Lorentz manifolds. They are Lie groups acting holomorphically on M . If g were (positive definite) Hermitian, then Iso(M, J, g) acts properly on M , and is in particular compact if M is compact. This is no longer true for g indefinite.

In the real case, that is without the almost complex structure, there have been many works tending to understand how and why the isometry group of a Lorentz manifold can act non-properly (see for instance [START_REF] Arouche | On Lorentz dynamics: From group actions to warped products via homogeneous spaces[END_REF][START_REF] Deffaf | Zeghib Actions of noncompact semisimple groups on Lorentz manifolds[END_REF]). The Lorentz case is the simplest among all the pseudo-Riemannian cases, since, with its one negative sign, it lies as the nearest to the Riemannian case. For instance, the situation of signature --+ . . . + presents more formidable difficulties. With this respect, the Hermite-Lorentz case seems as an intermediate situation, which besides mixes in a beautiful way pseudo-Riemannian and complex geometries.

2.1. Homogeneous vs Symmetric. We are going to prove facts characterizing these Kähler-Lorentz symmetric spaces by means of a homogeneity hypothesis (as stated in Theorems 2.1 and 2.2 below).

In pseudo-Riemannian geometry, it is admitted that, among homogeneous spaces, the most beautiful are those of constant sectional curvature, and then the symmetric ones, and so on... This also applies to pseudo-Kähler spaces, where the sectional curvature is replaced by the holomorphic sectional curvature.

In general, being (just) homogeneous is so weaker than being symmetric which in turn is weaker than having constant (sectional or holomorphic sectional) curvature.

For instance, Berger spheres are homogeneous Riemannian metrics on the 3sphere that have non constant sectional curvature and are not symmetric. On the other hand different Grassmann spaces are irreducible symmetric Riemannian (or Hermitian) spaces but do not have constant sectional (or holomorphic) curvature.

Our first theorem says that in the framework of Hermite-Lorentz spaces, being homogeneous implies essentially symmetric! Theorem 2.1. Let (M, J, g) be a Hermite-Lorentz almost complex space, homogeneous under the action of a Lie group G. Suppose that the isotropy group G p of some point p acts C-irreducibly on T p M , and dim C M > 3. Then M is a global Kähler-Lorentz symmetric space, and it is isometric, up to a cyclic cover, to

Mink n (C), dS n (C), AdS n (C), CdS n or CAdS n .
• The content of the theorem is: 1. Irreducible isotropy =⇒ symmetric, 2. The list of Hermite-Lorentz symmetric spaces with irreducible isotropy are the five mentioned ones. This fact may be extracted from Berger's classification of pseudo-Riemannian irreducible symmetric spaces. Here, we provide a direct proof.

• In the (real) Lorentz case, there is a stronger version, which states that an isotropy irreducible homogeneous space has constant sectional curvature [START_REF] Boubel | Isometric actions of Lie subgroups of the Moebius group[END_REF] (the fact that irreducible and symmetric implies constancy of the curvature was firstly observed in [START_REF] Cahen | Lorentz manifolds modeled on a Lorentz symmetric space[END_REF] by consulting Berger's list).

• The theorem is not true in the Riemannian case. As an example of a compact irreducible isotropy non-symmetric space, we have M = G/K, where G = SO( n(n-1)

2

), and K is the image of the representation of SO(n) in the space of trace free symmetric 2-tensors on R n (see [START_REF] Besse | Einstein manifolds[END_REF] Chap 7).

2.2.

Actions of semi-simple Lie groups. Let now (M, J, g) be an almost Hermite-Lorentz manifold and G a Lie group acting (not necessarily transitively) on M by preserving its structure. We can not naturally make a hypothesis on the isotropy in this case, since it can be trivial (at least for generic points). It is however more natural to require dynamical properties on the action. As discussed in many places (see for instance [START_REF] Arouche | On Lorentz dynamics: From group actions to warped products via homogeneous spaces[END_REF][START_REF] Deffaf | Zeghib Actions of noncompact semisimple groups on Lorentz manifolds[END_REF]), non-properness of the G-action is a reasonable condition allowing interplay between dynamics and the geometry of the action. For instance, without it everything is possible; a Lie group G acting by left translation on itself can be equipped by any type of tensors by prescribing it on the Lie algebra.

The literature contains many investigations on non-proper actions preserving Lorentz metrics [START_REF] Adams | Orbit nonproper actions on Lorentz manifolds[END_REF][START_REF] Adams | Dynamics of semisimple Lie groups on Lorentz manifolds[END_REF][START_REF] Arouche | On Lorentz dynamics: From group actions to warped products via homogeneous spaces[END_REF][START_REF] Deffaf | Zeghib Actions of noncompact semisimple groups on Lorentz manifolds[END_REF] and specially [START_REF] Kowalsky | Noncompact simple automorphism groups of Lorentz manifolds[END_REF]. We are going here to ask similar questions on the Hermite-Lorentz case. We restrict ourselves here to transitive actions, since the general idea, within this geometric framework, is that a G-nonproper action must have non-proper G-orbits, i.e. orbits with non-precompact stabilizer. The natural starting point is thus the study of non-proper transitive actions.

Theorem 2.2. Let G be a non-compact simple (real) Lie group of finite center not locally isomorphic to SL 2 (R), SL 2 (C) or SL 3 (R). Let G act non-properly transitively holomorphically and isometrically on an almost complex Hermite-Lorentz space (M, J, g), with dim M > 3. Then, M is a global Kähler-Lorentz irreducible symmetric space, and is isometric, up to a cyclic cover, to dS n (C), AdS n (C), CdS n or CAdS n .

2.3. Some Comments.

2.3.1.

Integrabilities. Observe that in both theorems, we do not assume a priori neither that J is integrable, nor g is Kähler.

2.3.2. The exceptional cases. The hypotheses dim M > 3 and G different form SL 2 (R), SL 2 (C) and SL 3 (R) are due on the one hand to "algebraic" technical difficulties in proofs and on the other hand to that statements become complicated in this cases.

As an example, SL 2 (C) with its complex structure admits a left invariant Hermite-Lorentz metric g which is moreover invariant by the right action of SL 2 (R). So, its isometry group is G = SL 2 (C) × SL 2 (R) and its isotropy is SL 2 (R) acting by conjugacy. On the Lie algebra g is defined as: a, b = tr(a b), where a, b ∈ sl 2 (C) ⊂ Mat 2 (C). This metric is not Kähler, neither symmetric, although the isotropy is C-irreducible, and so Theorem 2.1 does not apply in this case.

In the case of SL 3 (R) one can construct an example of a left invariant Hermite-Lorentz structure (J, g), with J non-integrable, invariant under the action by conjugacy of a one parameter group, and therefore Theorem 2.2 does not apply to the SL 3 (R)-case. Notice on the other hand that, although SL 3 (R) is a not a complex Lie group, it admits left invariant complex structures. This can be seen for instance by observing that its natural action on P 2 (C) × P 2 (C) has an open orbit on which it acts freely. We hope to come back to this discussion elsewhere.

2.4. About the proof. The tangent space at a base point of M is identified to C n+1 , and the isotropy H to a subgroup of U(1, n).

(Henceforth, in all the article, the complex dimension of the manifold M will be n + 1).

Subgroups of U(1, n).

We state in Proposition 4.1 a classification of such subgroups (when connected and non pre-compact ) into amenable and (essentially) simple ones.

In fact there have been many works on the (recent) literature about these groups, generally related to the study of the holonomy of pseudo-Riemannian and pseudo-Kähler spaces. Indeed, A. Di Scala and T. Leistner classified connected irreducible Lie subgroups of SO(2, m) [START_REF] Di Scala | Connected subgroups of SO(2, n) acting irreducibly on R 2,n[END_REF] (which contains our case U(1, n), for m = 2n). On the other hand, the case of non necessarily irreducible connected subgroups of U(1, n) was considered by A. Galaev in [START_REF] Galaev | Classification of connected holonomy groups of pseudo-Kählerian manifolds of index 2[END_REF] and Galaev-Leistner in [START_REF] Galaev | Recent developments in pseudo-Riemannian holonomy theory[END_REF]. There, the authors used the term "pseudo-Kähler of index 2" for our "Kähler-Lorentz" here.

A proof of Proposition 4.1 could be extracted from these references, but for reader easiness we give here our independent (and we think more geometric) proof! More important, in our proofs (and hypotheses) of Theorems 2.1 and 2.2, we deal with non-necessarily connected groups (that is the isotropy H is not assumed to be connected)! The analysis of connectedness occupies in fact a large part of §4.

2.4.2. Regarding Theorem 2.1, since it acts irreducibly, the possibilities given for H (more precisely its Zariski closure) are U(1, n), SU(1, n), U(1) × SO 0 (1, n), and SO 0 (1, n) (the last acts C-irreducibly but not R-irreducibly). Geometric and algebraic manipulations yield the theorem, that is the explicit possible G and H, §5. At one step of the proof, we show that M is a symmetric space, but we do not refer to the Berger's classification to get its form.

Here also, an alternative, but highly more algebraic approach, would be to use elements of the theory of reductive homogeneous spaces to show that M is symmetric, and in a next step to consult Berge's list, by showing which members of it are Kähler-Lorentz spaces! Here again, the most difficulty comes from the a priori non-connectedness of H.

2.4.3.

As for Theorem 2.2, the idea is to apply Theorem 2.1 by showing that H is irreducible (assuming it non-precompact and G simple).

-One starts proving that H is big enough, §6.

-If H is simple, irreducibility consists in excluding the intermediate cases

SO 0 (1, k) ⊂ H ⊂ U(1, k), for k < n, §7.
Here again, theory of reductive homogeneous spaces could apply, but not to the general non-amenable (non connected) case.

• However, the most delicate situation to exclude is the amenable one, §8. Observe in fact that, in general, homogeneous pseudo-Riemannian manifolds with a semi-simple Lie group may have, for instance, an abelian isotropy. Take for example dim H = 1, that is H a one parameter group, and assume its Lie subalgebra h is non-degenerate (i.e. non-isotropic) in g, the Lie algebra of G endowed with its Killing form. The G-action on the quotient space G/H will then preserve a pseudo-Riemannian metric given by the restriction of the Killing form on h ⊥ . (More complicated constriction are surely possible!) This part of proof, i.e that H can not be amenable (in our Hermite-Lorentz case) might be the essential mathematical (i.e. from the point of view of proof) contribution of the present article. Very briefly, amenability allows one to associate to any point of M a lightlike complex hypersurface, fixed by its isotropy group. Now, the point is to prove that this determines a foliation, that is two such hypersurfaces are disjoint or equal. The contradiction will come from that the quotient space of such a foliation is a complex surface on which the group G acts non-trivially, which was excluded by the hypothesis that G is different from SL 2 (R), SL 2 (C) or SL 3 (R).

SOME PREPARATORY FACTS

C n+1 is endowed with the standard Hermite-Lorentz form

q 0 = -|z 0 | 2 +|z 1 | 2 + . . . + |z n | 2 . The Hermitian product is denoted , . Recall that u is lightlike (or isotropic) if q(u) = 0. A C-hyperplane is lightlike if it equals the orthogonal Cu ⊥ of a lightlike vector u.
It is also sometimes useful to consider the equivalent form

q 1 = z 0 zn + z0 z n + |z 1 | 2 + . . . + |z n-1 | 2 .
As usually, U(q 0 ) is denoted U(1, n), and

SU(1, n) its special subgroup. The Lorentz group SO 0 (1, n) is a subgroup of SU(1, n); it acts diagonally on C n+1 = R n+1 + iR n+1 , by A(x + iy) = A(x) + iA(y). 3.1. Some SO 0 (1, n)-invariant theory. Levi form. FACT 3.1. 1. For n > 1, there is no non-vanishing SO 0 (1, n)-invariant anti- symmetric form R n+1 × R n+1 → R. 2. Let n > 1 and b : (R n+1 + iR n+1 ) × (R n+1 + iR n+1 ) → R be a SO 0 (1, n)- invariant anti-symmetric bilinear form. Then, up to a constant, b(u+iv, u ′ +iv ′ ) = u, v ′ -v, u ′ . (That is, up to a constant, b coincides with the Kähler form i(-dz 0 ∧ d z 0 + dz 1 ∧ d z 1 + . . . + dz n ∧ d z n )).
Proof.

1. Let b : R n+1 × R n+1 → R be such a form, and u a timelike vector, that is u, u < 0. Thus, the metric on Ru ⊥ is positive and the action of the stabilizer (in

SO 0 (1, n)) of u on it is equivalent to the usual action of SO(n) on R n . The linear form v ∈ Ru ⊥ → b(u, v) ∈ R is SO(n)-invariant,
and hence vanishes (since its kernel is invariant, but the SO(n)-action is irreducible). Thus u belongs to the kernel of b, and so is any timelike vector, and therefore b = 0

2. Let now b : (R n+1 +iR n+1 )×(R n+1 +iR n+1 ) → R. From the previous point b(u+i0, v ++i0) = b(0+iu, 0+iv) = 0. It remains to consider b(u, iv). It can be written b(u, iv) = u, Av , for some A ∈ End(R n ), commuting with SO 0 (1, n).
By the (absolute) irreducibility of SO 0 (1, n), A is scalar. (Indeed, by irreducibility, A has exactly one eigenvalue λ with eigenspace the whole R n+1 . If λ is pure imaginary, then SO 0 (1, n) preserves a complex structure, but this is impossible (for instance hyperbolic elements of SO 0 (1, n) have simple real eigenvalues). Thus A is a real scalar).

The rest of the proof follows.

Kähler form.

FACT 3.2. For n > 2, there is no non-vanishing (real) exterior 3-form α on

C n+1 (= R n+1 + iR n+1 ) invariant under the SO 0 (1, n)-action.
Proof. Let α be such a form. Let e ∈ R n+1 be spacelike: e, e > 0, and consider α e = i e α. First, α(e, ie, z) is a linear form on Ce ⊥ invariant under a group conjugate to SO 0 (1, n -1), and hence vanishes. On Ce ⊥ , α e is a 2form as in the fact above. It then follows that for any u ∈ Ce, and v, w ∈ Cu ⊥ , α(u, v, w) = φ(u)ω(v, w), where ω is the Kähler form on Ce ⊥ , φ : Ce → R is a function, necessarily linear. There is u ∈ Ce such that φ(u) = 0, and hence u ∈ ker α. This kernel is a SO 0 (1, n)-invariant space. If it is not trivial, then it has the form {au + biu, u ∈ R n+1 }, where a and b are constant. But α induces a form on the quotient C n+1 / ker α which vanishes for same reasons. Hence α = 0.

Nijenhuis tensor.

FACT 3.3. For n > 2, there is no non-trivial anti-symmetric bilinear form C 1+n × C 1+n → C 1+n , equivariant under SO 0 (1, n). Proof. Let b : C n+1 × C n+1 → C n+1 be a SO 0 (1, n)-invariant anti-symmetric form.
Let us first consider the restriction of b to R n+1 . Let u, v ∈ R n+1 two linearly independent lightlike vectors and w in the orthogonal space

Span C (u, v) ⊥ ∩ R n+1 . Consider H the subgroup of A ∈ SO 0 (1, n) such that there exists λ ∈ R, A(u) = λu, A(v) = λ -1 v, and A(w) = w. By equivariance b(u, v) is fixed by H. But H is too big; its fixed point set is Cw. Indeed its action on Span C (u, v, w) ⊥ is equivalent to the action of SO(n -2) on R n-2 . Therefore, b(u, v) ∈ Cw. But, since n ≥ 3, we have freedom to choose w in Span C (u, v) ⊥ ∩ R n+1 . Hence, b(u, v) = 0.
Last, observe that R n+1 is generated by lightlike vectors and hence b = 0 on R n+1 .

One can prove in the same meaner that b(u, iv) = 0, for u, v ∈ R n+1 . It then follows that b = 0.

Remark 3.4. [Dimension 3] For

n = 2, the vector product R 2+1 × R 2+1 → R 2+1 is anti-symmetric and SO 0 (1, 2)-equivariant. One can equally define a vector product on C 2+1 equivariant under SU(1, 2). For given u, v, u ∧ v is such that det(w, u, v) = w, u ∧ v (
here , is the Hermitian product on C 2+1 ). Observe nevertheless that this vector product is not equivariant under U(1, 2).

Parabolic subgroups. By definition, a maximal parabolic subgroup of SU(1, n)

is the stabilizer of a lightlike direction. It is convenient here to consider the form

q 1 = z 0 zn + z0 z n + |z 1 | 2 + . . . + |z n-1 | 2 .
Thus, e 0 is lightlike and the stabilizer P ′ of Ce 0 in U(1, n) consists of elements of the form:

  a 0 0 0 A 0 0 0 ā-1   where, a ∈ C * , A ∈ U(n -1), aā -1 detA = 1. Thus P is a semi-direct product S(C * × U(n -1)) ⋉ Heis. The stabilizer P of Ce 0 in SU(1, n) is P ′ ∩SU(1, n) = S((C * ×U(n-1))⋉Heis)
The Heisenberg group is the unipotent radical of P and consists of:

  1 t -t 2 2 + is 0 1 - t 0 0 1   where t ∈ C n-1 and s ∈ R.
We see in particular that P is amenable. Recall here that a topological group is amenable if any continuous action of it on a compact metric space, preserves some probability measure. For instance, as it will seen in §4, our group P will act on the boundary at infinity of the complex hyperbolic space and preserves there a Dirac measure. (In general, a Lie group is amenable iff its semi-simple part is compact).

3.3. Lightlike geodesic hypersurfaces. We will meet (especially in §8.1) special complex hypersurfaces L ⊂ M . We say that L is lightlike if for any y ∈ L, T y L is a lightlike complex hyperplane of (T y M, g y ). The kernel of (T y L, g y ) defines a complex line sub-bundle N of T L (not necessarily holomorphic). The metric on T L/N is positive.

We say that L is (totally) geodesic if for any u ∈ T L, the geodesic γ u tangent to u, is locally contained in L (there exists ǫ, such that γ u (]ǫ, +ǫ[) ⊂ L). This is equivalent to invariance of T L by the Levi-Civita connection; if X and Y are vector fields defined in a neighbourhood of L, and are tangent to L (i.e. X(y), Y (y) ∈ T L, for y ∈ L), then ∇ X Y (y) ∈ T L, for y ∈ L.

Let us prove in this case that N is parallel along L and thus it is in particular integrable. For this, consider three vector fields X, Y and Z tangent to L, with X tangent to N . We have X, Z = 0, and thus 0 = Y X, Z = ∇ Y X, Z + X, ∇ Y Z = ∇ Y X, Z (since X is tangent to N ). This is true for any Z, and therefore ∇ Y X is tangent to N , which means that N is a parallel 2-plane field.

Denote by N the so defined foliation of L. The leaves are complex curves. Transversally, N is a Riemannian foliation, that is, there is a well defined projected Riemannian metric on the leaf (local) quotient space Q = L/N . Equivalently, the Lie derivative L X h = 0, where h is the metric restricted to L and X is tangent to N . This is turn is equivalent to that, for any Y invariant under the

X-flow, i.e. [X, Y ] = 0, the product Y, Y is X-invariant. To check this, observe that X Y, Y = ∇ X Y, Y = ∇ Y X, Y = 0, since as we have just proved, N is parallel (that is ∇ Y X is tangent to N ). Corollary 3.5. Let f be an isometry of M preserving L and fixing a point x ∈ L. Assume D x f ∈ GL(T x M ) is unipotent (i.e. D x f -Id is nilpotent). Then f preserves (individually) each leaf of N .
Proof. f acts as an isometry f of the (local) quotient space L/N endowed with its projected Riemannian metric. The derivative Dx f at the projection of x is unipotent. But the orthogonal group O(n) contains no non-trivial unipotent elements. Therefore, D x f = Id T xQ , and hence as a Riemannian isometry, f = Id Q (of course, we are tacitly assuming everything connected).

SUBGROUPS OF U(1, n)

The following proposition says roughly that, up to compact objects, a subgroup of SU(1, n) is either contained in a parabolic group, or conjugate to one of the standard subgroups SO 0 (1, k) or SU(1, k). Proposition 4.1. Let H be a non-precompact connected Lie subgroup of SU(1, n) (i.e. its closure in not compact). Then:

(1) H is amenable iff it preserves a lightlike hyperplane (that is, by definition, H is contained in a maximal parabolic subgroup).

(2) In opposite, if H acts C-irreducibly on C n+1 (there is no non-trivial complex invariant subspace), then H equals SO 0 (1, n) or SU(1, n) (See also [START_REF] Di Scala | Connected subgroups of SO(2, n) acting irreducibly on R 2,n[END_REF]).

(3) In the general (intermediate) case, when H is not amenable, it acts Rirreducibly on some non-trivial subspace E, such that:

(a) Either E is totally real, and up to a conjugacy in [START_REF] Karpelevic | Surfaces of transitivity of a semisimple subgroup of the group of motions of a symmetric space[END_REF] and Mostow [START_REF] Mostow | Some new decomposition theorems for semi-simple groups[END_REF]. It states that a given Cartan decomposition of a Lie subgroup extends to a Cartan decomposition of the ambient simple Lie group. An essentially geometric (algebraic free) approach is also available in the case of SO 0 (1, n), see [START_REF] Boubel | Isometric actions of Lie subgroups of the Moebius group[END_REF][START_REF] Di Scala | The geometry of homogeneous submanifolds of hyperbolic space[END_REF].

SU (1, n), E = R k+1 ⊂ R n+1 ⊂ R n+1 + iR n+1 = C n+1 ,
-Observe finally that we do not assume H to be closed.

Proof. Let H ⊂ SU(1, n) be as in the proposition.

4.0.1. Hyperbolicity. Let G = SU(1, n), K = U(1, n -1)
and consider X = G/K = H n (C) the associated Riemannian symmetric space. We let SU(1, n) act on the (visual Hadamard) boundary ∂ ∞ X, which is identified to the space of complex lightlike directions of C 1+n . (See [START_REF] Goldman | Complex hyperbolic geometry[END_REF] to learn about the geometry of

H n (C)).
By definition, a maximal parabolic subgroup P is the stabilizer of a lightlike direction, or equivalently a point of ∂ ∞ X. From §3.2, P is amenable (the fact that maximal parabolic groups are amenable characterizes rank 1 groups). Therefore, any group fixing a point at ∂ ∞ X is amenable.

We have to prove conversely that a non-precompact connected amenable group fixes some point at ∂ ∞ X.

Recall that elements of SU(1, n) are classified into elliptic, parabolic or hyperbolic.

An isometry is elliptic if it fixes some point in X, and thus lies in its stabilizer which is a compact subgroup of SU(1, n). Conversely, any compact subgroup of SU(1, n) fixes some point of X. Thus an element is elliptic iff it generates a precompact subgroup.

A parabolic element has exactly one fixed point in ∂ ∞ X, and a hyperbolic one has two fixed points. Furthermore, in both cases, by iteration, all points of ∂ ∞ X tend to these fixed points. In particular, in both cases, any invariant measure has support a set F with cardinality #(F ) ≤ 2.

Now, H is amenable and hence leaves invariant a probability measure ν on ∂ ∞ X.

If H contains a parabolic or a hyperbolic element, then the support of ν consists of a set F of one or two points, and hence, H preserves such F . If F has cardinality 2, since H is assumed to be connected, it fixes each of the points of F . Therefore, in all cases, H has a fixed point in ∂ ∞ X.

In order to prove that indeed H contains a parabolic or hyperbolic element, one uses the classical fact that a non-compact connected Lie group contains some nonprecompact one parameter group (see for instance [START_REF] Dambra | Isometry groups of Lorentz manifolds[END_REF] for a proof of this). No element of such a one parameter group can be elliptic.

This completes the proof of (1) in the proposition. (Actually, a more selfcontained proof, say without using this fact on non-compact Lie groups, is available, but needs more details!). 4.0.2. Non-amenable case. H is a semi-direct product (S × C) ⋉ R (up to finite index) where S is semi-simple with no compact factor, C is compact semi-simple, and R is the (solvable) radical. Observe that R must be precompact. Indeed, if not, from the above proof, the fixed point set F of R in ∂ ∞ X has cardinality 1 or 2. Since R is a normal subgroup of H, F is preserved by H. By connectedness, H fixes each of the points of F , hence H is contained in a parabolic group, and is thus amenable.

This implies that the semi-direct product is in fact direct, up to finite index (H acts by conjugacy on the the compact torus R. But the identity component of the automorphism group of a torus is trivial). Let us say that H is a product S × C ′ where C ′ is precompact. We now investigate S and come back later on to S × C ′ . Observe first that S is simple. Indeed, if S = S 1 ×S 2 , then any non-elliptic f ∈ S 2 , will centralize S 1 , which implies S 1 has a fixed point at ∂ ∞ X and hence amenable. 4.0.3. Simple Lie subgroups. In order to understand the geometry of S, we investigate the symmetric space X (rather than its boundary as in the previous case). Let p be a base point, say that fixed by the maximal compact K. We have a Cartan decomposition of the Lie algebra of G: g = p ⊕ k, where k is the Lie subalgebra of K, and p is (the unique) K-invariant supplementary space of k in g. Geometrically, p is identified with T p M , and if u ∈ p, the orbit exp(tu).p is the geodesic of X determined by u.

More other properties are: K acts irreducibly on p, and [p, p] = k.

If S ⊂ G is a simple Lie subgroup, then Karpelevich-Mostow's theorem [START_REF] Karpelevic | Surfaces of transitivity of a semisimple subgroup of the group of motions of a symmetric space[END_REF][START_REF] Mostow | Some new decomposition theorems for semi-simple groups[END_REF] states, up to a conjugacy in G (or equivalently a modification of the base point), we get a Cartan decomposition by taking intersection:

s = s ∩ p ⊕ s ∩ k. Observe that s ∩ p determines s, since s ∩ k = [s ∩ p, s ∩ p].
In our case, p = T p X is identified to C n . The subspace E = s ∩ p is either complex or totally real, since E ∩ iE is S ∩ K-invariant, and this last group acts irreducibly on E. Now, K = U(n) acts transitively on the set of totally real (resp. complex) planes of a given dimension k. Thus, up to conjugacy, s ∩ p is the canonical R k+1 or C k+1 in C n . Candidate for s in these cases are the Lie algebras of the standard subgroups SO 0 (1, k) or SU(1, k), respectively. But since s ∩ p determines completely s, there are the unique possibilities.

End. We have thus proved (2) and (3) of the proposition at the group level: H is conjugate in SU(1, n) to S × C, with S = SO 0 (1, k) or SU(1, k). Since the precompact factor C commutes with the non-compact S, it is contained in SO(nk) or SU(nk), respectively. In particular H preserves R k+1 or C k+1 . This completes the proof of the proposition.

Corollary 4.3. Let L be a subgroup of SU(1, n) (not necessarily connected or closed) acting C-irreducibly on C n+1 . Then, the identity component of its Zariski closure equals SU(1, n) or SO 0 (1, n). If the identity component L 0 is not trivial, L itself equals SU(1, n) or SO 0 (1, n).
Proof. Let L Zar be the Zariski closure of L. It is a closed subgroup of SU(1, n) with finitely many connected components. It is non-compact, since otherwise L will be precompact and can not act irreducibly.

The identity component H of L Zar is non-precompact too. If H is amenable, then its fixed point set F in ∂ ∞ X is preserved by L, since L normalizes H. If F consists of one point, then L fixes it, seen as a lightlike direction in C 1+n contradicting the fact that it acts irreducibly.

If F consists of two lightlike directions, then L preserves the (timelike) 2-plane that they generate in C 1+n , again contradicting its irreducibility.

We infer from this that H is not amenable. Apply Proposition 4.1 to get that H is essentially SO 0 (1, k) or SU(1, k). Our group L itself is then contained in the normalizer of one of these groups. On easily sees that such a normalizer can not act irreducibly unless k = n (for instance the normalizer of SO 0 (1, k) or SU(1, k) preserves the space of their fixed vectors which is non-trivial for k < n). . We have more:

Corollary 4.4. Let L be a subgroup of SU(1, n). If L is non-amenable, then its Zariski closure contains a copy of SO 0 (1, k) or SU(1, k) for some k > 0. If furthermore L 0 is non-pre-compact, then L itself contains SO 0 (1, k) or SU(1, k). (Of course SO 0 (1, k) ⊂ SU(1, k),

but we prefer our formulation here for a later use ).

Proof. The first part is obvious.

For the second one, it suffices to show that L 0 is non-amenable. But this L 0 is normalized by L Zar (L itself normalizes L 0 and by algebraicity, L Zar too preserves it). But SO 0 (1, k) or SU(1, k) can normalize no amenable non-compact connected subgroup of SU(1, n).

4.1. Subgroups of U(1, n). We will now deal with subgroups L of of U(1, n) = U(1) × SU(1, n). The following lemma will help to understand them and maybe has its own interest:

Lemma 1. Let L be a subgroup of U(1, n).
1) If L is non-precompact, then: (i) either L preserves a unique lightlike direction (ii) or L preserves a unique timelike (i.e. on which the restriction of q 0 on it is of Hermite-Lorentz type) 2-plane, and also each of the two lightlike directions inside it.

(iii) or L preserves a unique timelike subspace on which it acts C-irreducibly. In all cases, this lightlike direction, or timelike subspace are invariant under the normalizer of L in U(1, n).

2) L acts irreducibly iff L ∩ SU(1, n) acts irreducibly.

Proof. 1) If L acts C-irreducibly, then we are done. So, assume it preserves some proper subspace E, and thus also E ⊥ . If E is degenerate, then E ∩ E ⊥ is an invariant lightlike direction. If E is spacelike then E ⊥ is timelike, and vice versa.

Assume, we are not in case (i), so either there is no invariant lightlike direction at all, or there are many. In all cases, we can find an invariant timelike subspace (by taking sums if there are many lightlike directions).

Let E be an invariant timelike subsapce of minimal dimension. Let us prove that either E is irreducible, or we are in case (ii).

Assume there exists E ′ a proper invariant subspace of E. By definition, neither E ⊥ ∩ E nor E ′ ⊥ ∩ E are timelike, and thus E ′ is degenerate, and hence D = E ′ ∩ E ′ ⊥ is an invariant lightlike direction. It is not unique, because we are not in case (i). So, there is another similar one D ′ . It must be contained in E since otherwise its projection on E would be an invariant timelike direction. Let P = D ⊕ D ′ . By minimality, E = P . To show that we are in case (ii), let us prove uniqueness of E. If D ′′ is another invariant direction not in P , then its projection on P will give a timelike invariant direction. This implies that the action on P is equicontinuous, but since P ⊥ is spacelike, the group L will be precompact in this case.

It remains to consider the case where E is irreducible, and show it is unique. Assume by contradiction that E ′ is analogous to E. Let R = E ⊥ ⊕ E ′ ⊥ . The L-action on R is equicontinuous (since both E ⊥ and E ′ ⊥ are spacelike). Let P = R ∩ E. Then P = 0, unless E = E ′ . Furthermore P = E since otherwise E = C 1+n and L will be precompact. This contradicts the irreducibility of E.

2) Assume L irreducible. Consider the projections π 1 and π 2 of U(1, n) onto U(1) and SU(1, n) respectively.

If L ∩ SU(1, n) = {1}, then π 1 sends injectively L in U(1) and hence L is abelian, and can not act irreducibly.

Observe that π 2 (L) acts C-irreducibly. Indeed, L is contained in U(1) × π 2 (L) and U(1) preserves any C-subspace.

Observe also that π 2 (L) and L∩SU(1, n) normalizes each one the other, and that the commutator group

[π 2 (L), π 2 (L)] is contained in L ∩ SU(1, n). If L ∩ SU(1, n)
is not precompact, then the previous step implies it is irreducible. Finally, from Corollary 4.4, one infers that the commutator group of an irreducible subgroup of SU(1, n) is not precompact, and therefore L ∩ SU(1, n) acts irreducibly. If furthermore L 0 = 1, and its Zariski closure does not contain SU(1, n), then, either, L 0 equals SO 0 (1, n), or L 0 ⊃ U(1).

Proof. The first part is obvious from the discussion above, let us prove the second one. In this case L is a subgroup of U(1) × SO 0 (1, n).

-

If L 0 ∩ SO 0 (1, n) = 1, then its equals SO 0 (1, n) by irreducibility of L ∩ SO 0 (1, n), in particular L ⊃ SO 0 (1, n). Furthermore, if a product ab, a ∈ U(1), b ∈ SO 0 (1, n) belongs to L, then b ∈ L, that is π(L) = L ∩ U(1)
. This last group is either U(1) (in which case L = U(1) × SO 0 (1, n)), or totally discontinuous, in which case L 0 = SO 0 (1, n).

-Assume now that L 0 ∩ SO 0 (1, n) = 1. Let l t = a t b t be a one parameter group in L 0 , and c ∈ L ∩ SO 0 (1, n). The commutator [c, a t b t ] equals [c, b t ]. This is a one parameter group in L ∩ SO 0 (1, n), and hence must be trivial. But since we can choose c in a Zariski dense set in SO 0 (1, n), the one parameter group b t must be trivial. This means that l t ∈ U(1), and hence L 0 ⊃ U(1).

PROOF OF THEOREM 2.1

Let (M, J, g) be an almost complex Hermite-Lorentz space on which a group G acts transitively with C-irreducible isotropy.

Let p be a base point of M , and call H its isotropy group in G. The tangent space T p M is identified to C 1+n and H to a subgroup of U(1, n). By hypothesis H acts C-irreducibly on C 1+n .

The first part of Theorem 2.1, that is J is integrable and g is Kähler will be proved quickly. Indeed, by Corollary 4.5 the Zariski closure of

H (in U(1, n)) contains SO 0 (1, n).
Kähler Character. Let ω be the Kähler form of g. Its differential at p, α = dω p is an H-invariant 3-form on C n+1 . By Corollary 4.5, α is SO 0 (1, n)-invariant. By Fact 3.2, α = 0, that is, M is Kähler.

Integrability of the complex structure. The (Nijenhuis, obstruction to) integrability tensor at p is an R-antisymmetric bilinear vectorial form C 1+n × C 1+n → C 1+n . The same argument, using Fact 3.3 yields its vanishing, that is J is integrable.

Remark 5.1. Observe that we need dim M > 3 in order to apply Facts 3.2 and 3.3.

5.0.1.

Classification. The rest of this section is devoted to the identification of M as one of theses spaces: Mink n+1 (C), dS n+1 (C), AdS n+1 (C), CdS n+1 or CAdS n+1 (up to a central cyclic cover in some cases).

The identity component H

0 . Let us prove that H 0 = 1. If not G is a cov- ering of M , in particular T p M ∼ = C n+1
is identified to the Lie algebra g, and H acts by conjugacy. The bracket is an R-bilinear form like the integrability tensor, and hence vanishes, that is g is abelian. This contradicts the fact that H acts nontrivially by conjugacy. Therefore H 0 is non-trivial. Applying subsection 4.1, we get three possibilities:

1. The Zariski closure of [START_REF] Kobayashi | Foundations of differential geometry[END_REF][START_REF] Barros | Indefinite Kähler manifolds[END_REF]. We will see below ( §5.4) that M is (globally) isometric to Mink n+1 (C), dS n+1 (C) or to a cover of AdS n+1 (C).

H contains SU(1, n) 2. H 0 = SO 0 (1, n) 3. H 0 contains U(1).

Case 2:

H 0 = SO 0 (1, n).
The final goal here is to show that M is Mink n+1 (C).

First we replace M = G/H by G/H 0 which enjoys all the properties of the initial M . In other words, we can assume

H = H 0 = SO 0 (1, n).
Invariant distributions. SO 0 (1, n) acts C-irreducibly but not R-irreducibly. We set a G-invariant distribution S on M as follows. Define S to be equal to R n+1 at p. For x = gp, define S x = D p g(S p ). This does not depend on the choice of g since S p is H-invariant.

The orthogonal distribution S ⊥ is in fact determined similarly by means of the H-invariant space iR n+1 .

Integrability of distributions.

The obstruction to the integrability of S is encoded in the anti-symmetric Levi form II : S × S → S ⊥ , where II(X, Y ) equals the projection on S ⊥ of [X, Y ], for X and Y sections of S. At p, we get an anti-symmetric bilinear form R n+1 × R n+1 → R n+1 , equivariant under the SO 0 (1, n)-action. By Fact 3.3, this must vanish and hence S and analogously S ⊥ are integrable.

We denote by S and S ⊥ the so defined foliations.

Observe that since G preserves these foliations, then each leaf of them is homogeneous. If F is such a leaf, x, y ∈ F , and g ∈ G is such that y = gx, then g sends the distribution at x to that at y, and hence, gF = F .

Leaves of S or S ⊥ are (real) homogeneous Lorentz manifolds with (maximal) isotropy SO 0 (1, n). They are easy to handle du to the following fact, the proof of which is standard: FACT 5.2. Let F = A/B be a homogeneous Lorentz manifold of dimension n + 1 such that the action of B on the quotient a/b of Lie algebras is equivalent to the standard action of SO 0 (1, n) on R n+1 . Then F has constant sectional curvature. If F is flat, then F = Mink n+1 and A = SO 0 (1, n) ⋉ R n+1 . If F has positive curvature then it equals dS n+1 and A = SO 0 (1, n + 1). Finally, in the negative curvature case, F is a cover of AdS n+1 , and A covers SO(2, n).

Let A be the stabilizer of S p . If leaves of S are not flat, then A is SO 0 (1, n + 1) in case of positive curvature, and A = SO(2, n) in the negative curvature case. Consider the (local) quotient space Q = M/S, it has dimension n + 1. The group A acts by fixing F , seen as a point of Q. But SO 0 (1, n + 1) and SO(2, n)) have no linear representation of dimension n + 1. Therefore, A acts trivially on the tangent space T F Q. But this tangent space is identified to S ⊥ p . There, SO 0 (1, k), as an isotropy subgroup, acts non-trivially. This contradiction implies that the leaves of S and analogously S ⊥ are flat. Now, we need to study further the geometry of our foliations. We claim that their leaves are in fact totally geodesic. Indeed, there is a symmetric Levi form measuring the obstruction of geodesibility. More exactly, it is given by II * (X, Y ) = the orthogonal projection of the covariant derivative ∇ X Y . From 3.3, since equivariant symmetric bilinear forms do not exist, the foliations S and S ⊥ are geodesic. It is classical that the existence of a couple of orthogonal geodesic foliations implies a metric splitting of the space, see for example [START_REF] Kobayashi | Foundations of differential geometry[END_REF] about the proof of the de Rham decomposition Theorem (one starts observing that . That is, at least locally, M is isometric to the product S p × S ⊥ p . In particular, M is a flat Hermite-Lorentz manifold, that is M is locally isometric to Mink 1+n (C).

One can moreover prove that G is a semi-direct product SO 0 (1, n) ⋉ R n+1 and M = Mink n+1 (C) (see §5.4 below for details in a similar situation).

Case 3:

U(1) ⊂ H ⊂ U(1) × SO 0 (1, n).
The goal here is to prove that M is flat or isomorphic to one of the two spaces

CdS n = SO 0 (1, n + 1)/SO 0 (1, n - 1) × SO(2) or CAdS n = SO(3, n -1)/SO(2) × SO 0 (1, n -1).
The crucial observation is that M is a (pseudo-Riemannian) symmetric space, that is there exists f ∈ G, such that D p f = -Id TpM . Indeed, -Id ∈ U(1).

There is a de Rham decomposition of M into a product of a flat factor and irreducible symmetric spaces. In our case, there exists a subgroup of the isotropy that acts irreducibly. It follows that M is either flat, or irreducible. There is nothing to prove in the first case, we will therefore assume that M is irreducible. We can also assume that G is the full isometry group of M (the hypotheses in Theorem 2.1 on the G-action are also valid for the full isometry group). It is known that isotropy groups of symmetric spaces have finitely many connected components. Thus, up to a finite cover (say assuming it connected), H must be U(1) × SO 0 (1, n).

Consider a Cartan decomposition g = h + p, where p is identified with C n+1 . Consider the bracket [, ] :

p × p → h = so(1, n) + u(1).
Its second component is a SO 0 (1, n)-invariant anti-symmetric scalar bilinear form α :

C n+1 × C n+1 → u(1) = R. By Fact 3.1, α vanishes on R n+1 , that is if X, Y ∈ R n+1 , then [X, Y ] ∈ so(1, n). On the other hand, SO(1, n) preserves R n+1 , and hence if T ∈ so(1, n) and Z ∈ R n+1 , then [T, Z] ∈ R n+1 . Summarizing, if X, Y, Z ∈ R n+1 , then [[X, Y ], Z] ∈ R n+1 .
It is known that this implies that R n+1 determines a totally geodesic submanifold, say F . It has dimension n + 1 and isotropy SO 0 (1, n). From Fact 5.2, F is a Lorentz space of constant curvature. It can not be flat since in that case, the bracket [, ] vanishes on R n+1 , but this implies it vanishes on the whole of C n+1 . So M is the de Sitter or the anti de Sitter space.

The two cases are treated identically, let us assume F = dS n+1 . Its isometry group SO 0 (1, n + 1) is thus contained in G.

The goal now is to show that G = SO 0 (1, n + 2). For this, we consider the homogeneous space N = G/SO 0 (1, n + 1). Since we know the dimensions of G/U(1) × SO 0 (1, n) and SO 0 (1, n + 1)/SO 0 (1, n), we can compute that of G/SO 0 (1, n + 1), and find it equals n + 2.

Thus SO 0 (1, n + 1) has an isotropy representation ρ in the (n + 2)-dimensional space E, the tangent space at the base point of N . In a direct way, we prove that this is the standard representation of SO 0 (1, n+1) in R n+2 . For this, we essentially use that ρ restricted to SO 0 (1, n) is already known.

From Fact 5.2, G is SO 0 (1, n + 2) or SO(2, n + 1). Again, in a standard way, we exclude the case G = SO(2, n + 1) (just because it does not contain the isotropy U(1) × SO 0 (1, n)). We have thus proved that M = SO 0 (1, n + 2)/SO 0 (1, n) × SO(2). 5.4. Global symmetry. It was proved along the investigation of cases ( 2) and (3) that M is (globally) symmetric (the global isometry with Mink n+1 (C) in case (2) can be handled following the same next argument). It remains to consider the first case, that is when the Zariski closure of H contains SU(1, n).

Exactly as previously, by Corollary 4.5, we have SU(1, n) ⊂ H, or U(1) ⊂ H. The last case is globally symmetric, let us focus on the first one, SU(1, n) ⊂ H.

M is locally isometric to a universal space X of constant holomorphic sectional curvature. We let the universal cover G act on X. Since the isotropy SU(1, n) of M has codimension 1 in the isotropy U(1, n) of X, G has codimension 1 in Iso(X). However, if X is not flat, Iso(X) is a simple Lie group with no codimension 1 subgroup, since it is not locally isomorphic to SL 2 (R) since dim X ≥ 3 (the unique simple Lie group having a codimension 1 subgroup is SL 2 (R)). Therefore, dim G = dim(Iso(X)), and in particular the isotropy of M is U(1, n), in particular M is (globally) symmetric.

Let us now consider the case of X = Mink n+1 (C). Thus Iso(X) = U(1, n) ⋉ C n+1 . Since G acts (locally) transitively, it must contain some translation, that This finishes the proof of Theorem 2.1. 2

is A = G ∩ C n+1 = 1. The subgroup A is normal in G,

PROOF OF THEOREM 2.2: PRELIMINARIES

Let M be a Hermite-Lorentz space homogeneous under the holomorphic isometric action of a semi-simple Lie group G of finite center.

For x in M , we denote by G x its stabilizer in G, g the Lie subalgebra of G, and g x the Lie subalgebra of G x . The goal in this section is to show that g x is big; it contains nilpotent elements.

6.1. Stable subalgebras, actions on surfaces.

Notations.

An element X in the Lie algebra g is R-split (or hyperbolic) if ad X is diagonalizable with real eigenvalues. Thus g = Σ α g α , where α runs over the set of eigenvalues of ad X . Let

W s X = Σ α(X)<0 g α , W u X = Σ α(X)>0 g α and W s0 X = Σ α(X)≤0
g α be respectively, the stable, unstable and weakly-stable sub-algebras of X. We have in particular g = W s0

X ⊕ W u

X

The stable and unstable subalgebras are nilpotent in the sense that, for Y ∈ W s X (or W u X ), ad Y is a nilpotent element of Mat(g), equivalently, exp ad Y is a unipotent element of GL(g) (this follows from relations [g α , g β ] ⊂ g α+β ). It then follows that if h is an ad Y -invariant subspace, then exp ad Y determines a unipotent element of GL(g/h).

It is known that W s X and W u X are isomorphic; an adapted Cartan involution sends one onto the other. In particular the codimension of W s0 X in G equals the dimension of W s X . Assuming (to simplify) that G is simply connected, it acts on G/L, where L is the Lie subgroup determined by W s0 X . Then, dim(G/L) = dim W s X ; summarizing: FACT 6.1. If for some X, dim W s X = 2, then G acts on a surface, that is there exists a G homogeneous space of (real) dimension 2.

Semi-simple Lie groups satisfying the fact can be understood: FACT 6.2. A semi-simple Lie group G acting (faithfully) on a surface is locally

isomorphic to SL 2 (R), SL 2 (R) × SL 2 (R), SL 2 (C) or SL 3 (

R). (it is well known that acting on dimension 1 implies being locally isomorphic to SL 2 (R)).

Proof. This can be derived from the classification theory of simple Lie groups. One starts observing that the problem can be complexified, that is complexified groups act on complex surfaces; algebraically, they possess codimension 2 complex subalgebras in their complexified algebras. Let the isotropy group have a Levi decomposition S ⋉ R. Since S has a faithful 2-dimensional representation, it is locally isomorphic to SL 2 (C). If S ′ is the affine subgroup of SL 2 (C), then S ′ ⋉ R is solvable and has codimension 3 in G. Therefore, a Borel group of G has codimension ≤ 3. This implies that the cardinality of the set of positive roots is ≤ 3 (for any associated root system). With this restriction, one observes that the (complex) rank is ≤ 2, and consult a list of root systems to get our mentioned groups.

Example 6.3. These actions on surfaces are in fact classified (up to covers). We have the projective action of SL 2 (R) (resp. SL 3 (R)) on the real projective space P 1 (R) (resp. P 2 (R)). There is also the action of SL 2 (C) on the Riemann sphere P 1 (C), and the product action of SL 2 (R) 2 on P 1 (R) 2 . Finally, the hyperbolic, de Sitter and (the punctured) affine planes are obtained as quotients of SL 2 (R) by suitable one parameter groups. It is finally possible, in some cases, to take covers or quotients by discrete (cyclic) groups of the previous examples.

6.2. Non-precompactness. FACT 6.4. Let M = G/H be a homogeneous space where G is semi-simple of finite center and acts non-properly (and faithfully) on M . Then H seen as the isotropy group of a base point, say p, in not precompact in GL(T p M ).

Proof. By contradiction, if H is precompact then it preserves a Euclidean scalar product on T p M , and hence G preserves a Riemannian metric on G/H (of course H is closed in G since it equals the isotropy of p). Let us show that H is compact. Indeed, let L be the isometry group of the Riemannian homogeneous space X = G/H, and K its isotropy group in L, which is compact since the homogeneous space is of Riemannian type. Now, H = K ∩ G. It is known that a semi-simple Lie group of finite center is closed in any Lie group where it lies. Therefore, H is a closed subgroup of K, and hence compact.

Dynamics vs Isotropy.

For V be a subspace (in general a subalgebra) of g, its evaluation at x is the tangent subspace V (x) = {v(x) ∈ T x M, v ∈ V } (here v is seen as a vector field on M ). FACT 6.5. (Kowalsky [START_REF] Kowalsky | Noncompact simple automorphism groups of Lorentz manifolds[END_REF]) There exists X ∈ g (depending on x), an R-split element, such that W s X (x) is isotropic. In the sprit of Kowalsky's proof, we have the following precise statement.

Y, Z] = X) satisfies that (RX ⊕ W s X )(x) is isotropic.
Proof. Let L be the subgroup of G determined by {X, Y, Z}. It is isomorphic up to a finite cover to SL 2 (R). A Cartan KAK decomposition yields exp(tY ) = L t exp(s(t)X)R t , where L t and R t belong to the compact SO(2).

Write X t = Ad(R -1 t )(X) (for t fixed), it generates the one parameter group s → exp sX t = (R t ) -1 exp sXR t . Thus, exp(tY ) = D t exp s(t)X t , where

D t = L t R t ∈ SO(2).
Let u α t and v β t be the eigenvectors for ad Xt (acting on g) associated to two roots α and β. Since exp tY preserves , , we have

u α t , v β t = exp tY u α t , exp tY v β t = e s(t)(α+β) D t u α t , D t v β t
• The point now is that X t converges to X, when t → ∞. This follows from a direct computation of the KAK decomposition in SL 2 (R). It follows for the eigenvectors of ad X that u α , v β is dominated by a function of the form e s(α+β) . Thus u α , u β = 0, whence α < 0, and β ≤ 0. In particular W s X (x) is isotropic and orthogonal to X(x) (since ad X X = 0).

• It remains to verify that X(x) is isotropic. For this, consider M ′ the SL 2 (R)orbit (of the base point of M ). If the isotropy group of the SL 2 (R)-action on M ′ is exactly generated by exp tY , then M ′ is the affine punctured plane R 2 -{0}. The unique SL 2 (R)-invariant (degenerate) metric is 0 or a multiple of dθ 2 in polar coordinates (θ, r), and therefore X(x) is isotropic since it coincides with ∂ ∂r .

In the case where the isotropy group is bigger, the metric on M ′ must vanish (see for instance §2 in [START_REF] Bekkara | Actions of semisimple Lie groups preserving a degenerate Riemannian metric[END_REF] for details). FACT 6.7. g x contains a nilpotent element unless for any X as in Fact 6.5, dim W s X ≤ 2. In particular, if G does not act (locally) on surfaces, then g x ∩ W s X = 0, and g x contains nilpotent elements.

Proof. Consider the evaluation map V ∈ W s X → V (x) ∈ T x M . Its image is isotropic, and thus has at most dimension 2 (since the metric is Hermite-Lorentz). Its kernel I x = g x ∩ W s X consists of nilpotent elements and satisfies dim(I x ) ≥ dim(W s X ) -2, which is positive if for some X, dim W s X > 2, in particular if G does not act (locally) on surfaces by Fact 6.1.

PROOF OF THEOREM 2.2: NON-AMENABLE ISOTROPY CASE

Let (M, J, g) and G be as in Theorem 2.2, that is G is simple, not locally isomorphic to SL 2 (R), SL 2 (C) or SL 3 (R), and acts non-properly by preserving the almost complex and Hermite-Lorentz structures on M . Theorem 2.2 states that M is exactly as in Theorem 2.1, that is M is a global symmetric Kähler-Lorentz space. It is thus natural to prove Theorem 2.2 by showing that its hypotheses imply those of Theorem 2.1, i.e. if the acting group is simple, and the action in non-proper, then the isotropy is irreducible.

As previously, this isotropy H is a subgroup of U(1, n). Let us assume by contradiction that H does not act irreducibly on C n+1 . By Fact 6.7, the identity component H 0 is non-precompact, which allows us using Proposition 4.1.

The goal of the present section is to get a contradiction assuming H is nonirreducible and non-amenable. The amenable case will be treated in the next section.

Remark 7.1. Actually, in the present non-amenable case, all what we use from the preliminaries of §6, is that H 0 is non pre-compact (Fact 6.7). For instance if we start assuming H connected, then the proof will be independent of these preliminaries and follows from Theorem 2.1 by the short proof below.

By Proposition 4.1, up to conjugacy, H preserves C k+1 , for some 1 < k < n, and its non-compact semi-simple part is SO 0 (1, k) or SU(1, k). Let us assume here that it is SO 0 (1, k), since the situation with SU(1, k) is even more rigid! Integrability of distributions. As in §5.2 during the proof of Theorem 2.1, we define a G-invariant distribution S on M , by declaring S p = C k+1 .

We first show that the distribution S ⊥ is integrable. The obstruction to its integrability is encoded in the anti-symmetric Levi form II : S ⊥ × S ⊥ → S, where II(X, Y ) equals the projection on S of [X, Y ], for X and Y sections of S ⊥ .

At p, we get a skew-symmetric form II : C n-k × C n-k → C k+1 , equivariant under the actions of SO 0 (1, k) on C n-k and C k+1 respectively. Observe however that SO 0 (1, k) acts trivially on C n-k . Therefore, the image of II in C k+1 consists of fixed points, which is impossible since SO 0 (1, k) has no such points (in C k+1 ).

• We denote by S ⊥ the so defined foliation. Before going further, let us notice that SO 0 (1, k) acts trivially on the leaf S ⊥ p . Indeed, it preserves the induced (positive definite) Hermitian metric on S ⊥ p . But, the derivative action of SO 0 (1, k) on T p S ⊥ p = S ⊥ p is trivial, and hence SO 0 (1, k) acts trivially on S ⊥ p . • Let us now study S itself from the point of view of integrability. We consider a similar Levi form. This time, we get an equivariant form C k+1 × C k+1 → C n-k . Since SU(1, k) acts trivially on C n-k , this form is SO 0 (1, k)-invariant. However, up to a constant, the Kähler form ω is the unique scalar SO 0 (1, k)-invariant form (Fact 3.1). It follows that there exists v ∈ C n-k , such that II = ωv. This determines a vector field V on M such that V (p) = v, and a distribution S ′ = S ⊕ RV .

Of course, it may happen that V = 0, in which case S is integrable. We claim that S ′ is integrable. Indeed, by construction, the bracket [X, Y ] of two sections of S belongs to S ′ . It remains to consider a bracket of the form [V, X]. As previously, consideration of an associated Levi form leads us to the following linear algebraic fact: an SO 0 (1, k)-invariant bilinear form C k+1 ×R → R×C n-k-1 must vanish. Its proof is straightforward.

Contradiction. Now, we have two foliations S ⊥ and S ′ . The group G acts by preserving each of them. It also acts on Q, the (local) quotient space of S ′ , i.e. the space of its leaves. However, SO 0 (1, k) acts trivially on Q. Indeed, as we have seen, SO 0 (1, k) acts trivially on S ⊥ p , and this is a kind of cross section of the quotient space Q; say, S ⊥ p meets an open set of leaves of S. On this open set, SO 0 (1, k) acts trivially. By analyticity, SO 0 (1, k) acts trivially on Q.

Thus the G-action on Q has a non-trivial connected Kernel, and is therefore trivial since G is a simple Lie group. This means Q is reduced to one point, that is k = n, which contradicts our hypothesis that H is not irreducible.

PROOF OF THEOREM 2.2 IN THE AMENABLE CASE

We continue the proof of Theorem 2.2 started in the previous section, with here the hypothesis (by contradiction) that the isotropy H is amenable. The idea of the proof is as follows. To any x we associate, in a G-equivariant meaner, F x , the asymptotic leaf of the isotropy group G x at x. It is a (complex) codimension 1 lightlike geodesic hypersurface in M . This is got by widely general considerations (see for instance [START_REF] Dambra | Lectures on transformation groups: geometry and dynamics[END_REF][START_REF] Zeghib | Remarks on Lorentz symmetric spaces[END_REF][START_REF] Zeghib | Isometry groups and geodesic foliations of Lorentz manifolds, Part I: Foundations of Lorentz dynamics[END_REF]). Next, the point is to check that x → F x is a foliation: F x ∩F y = ∅ =⇒ F x = F y . Its (local) quotient space would be a (real) surface with a G-action, which is impossible by hypotheses of Theorem 2.2; leading to that H can not be amenable. 8.0.1. Notation and Dimension. For x in M , we denote by G x its stabilizer in G, g x its Lie sub-algebra, and I x = g x ∩ W s X , where X is a fixed R-split element as in Fact 6.6 (associated to x). FACT 8.1. 1) X(x) = 0.

2) dim(I x ) ≥ 2.

Proof. 1) By contradiction, if X ∈ g x , then one first proves directly (an easy case of Fact 6.5) that also the unstable W u X is isotropic at x and for the same reasons W u X ∩ g x = 0, say Z ∈ W u X ∩ g x , and also Y ∈ W s X ∩ g x . Thus {X, Y, Z} ⊂ g x . Now, the isotropy G x embeds in the unitary group U(T x M, , x ) identified with U(1, n). The element exp tX is an R-split one parameter group that acts on g x with both contracted and an expanded eigenvectors. But, by the amenability hypothesis on G x , it is contained in a maximal parabolic subgroup P of U(1, n). However, P (see §3.2) has no such elements.

2) Since I x = 0, we apply Fact 6.6 to modify X if necessary and get that (RX ⊕ W s X )(x) is isotropic. Now the kernel I ′ x of the evaluation RX ⊕ W s X → T x M has dimension at least (1 + dim W s X ) -2 ≥ 4 -2 = 2 (since we assumed that G can not act on surfaces, and hence dim W s X ≥ 3).

It remains to check that I ′

x is contained in W s X to conclude that I ′ x = I x , and obtain the desired estimation. For this assume by contradiction that

X ′ = X + u ∈ I ′ x , with u ∈ W s X . It is known that any such X ′ is conjugate to X in RX ⊕ W s X
(this is the Lie algebra of a semi-direct product of R by R k , with R acting on R k by contraction. For k = 1, we get the affine group of R). Therefore, we are led to the situation X(x) = 0 (for some other x), which we have just excluded.

8.1. Asymptotic leaf. (see [START_REF] Dambra | Lectures on transformation groups: geometry and dynamics[END_REF][START_REF] Zeghib | Remarks on Lorentz symmetric spaces[END_REF][START_REF] Zeghib | Isometry groups and geodesic foliations of Lorentz manifolds, Part I: Foundations of Lorentz dynamics[END_REF] for a similar situation).

Endow M × M with the metric (+g) ⊕ (-g). Let f : M → M be a diffeomorphism and Graph(f ) = {(x, f (x)), x ∈ M }. By definition, f is isometric iff Graph(f ) is isotropic for g ⊕ (-g). Furthermore, in this case, Graph(f ) is a (totally) geodesic submanifold.

Let f n be a diverging sequence in G x , i.e. no sub-sequence of it converges in G x . Consider the sequence of graphs Graph(f n ). In order to avoid global complications, let us localize things by taking E n the connected component of (x, x) in a (small) convex neighbourhood (O × O) ∩ Graph(f n ), where O is a convex neighbourhood of x, that is, two points of it can be joined within it by a unique geodesic.

Let

V n = Graph(D x f n ) ⊂ T x M × T x M .
Then, E n is the image by the exponential map exp (x,x) of an open neighbourhood of 0 in V n .

If V n converge to V in the Grassmanian space of planes of T x M × T x M , then E n converge in a natural way to a geodesic submanifold E in M × M . Let V 1 be the projection on the first factor T x M . It is no longer a graph, since otherwise it would correspond to the graph of an element of G x which is a limit of a subsequence of (f n ) (in fact the map f ∈ G x → Graph(D x f ) is a homeomorphism onto its image in the Grassmann space).

Since a sequence of isometries converge iff the sequence of inverse isometries converge, V intersects both T x M × 0 and 0 × T x M non-trivially. Since V n is a complex (resp. isotropic) subspace, also is V ∩ (T x M × 0). Hence, because the metric on M is Hermite-Lorentz, V ∩ (T x M × 0) is a complex line. Furthermore, since V is isotropic, the projection V 1 is a lightlike complex hyperplane, with orthogonal direction V ∩ (T x M × 0).

Define similarly E 1 to be the projection of E on M . It equals the image by exp x of an open subset of V 1 . It is a lightlike geodesic complex hypersurface (see §3.3).

Finally, without assuming that V n converge, we consider all the limits obtained by means of sub-sequences of (f n ). Any so obtained space V 1 (resp. E 1 ) is called asymptotic space (resp. leaf) of (f n ) at x. (Observe that different limits V may have a same projection V 1 ). FACT 8.2. Let H be a non-precompact amenable subgroup of U(1, n). There is exactly one or two degenerate complex hyperplanes which are asymptotic spaces for any sequence of H, and are furthermore invariant under H. Similarly, there exist one or two asymptotic leaves for G x (assuming it amenable and non-compact).

Proof. H is contained in a maximal parabolic group P . By definition P is the stabilizer of a lightlike direction u ∈ C 1+n . One then observes that (Cu) ⊥ is a common asymptotic space for all diverging sequences in P , and hence for H.

Assume now that H preserves two other different degenerate complex hyperplanes (Cv) ⊥ and (Cw) ⊥ . Let us prove that H is pre-compact in this case. Indeed H preserves the complex 3-space W = Span C (u, v, w) and 3-directions inside it, and also W ⊥ . Since W ⊥ is spacelike (the metric on it is positive), it suffices to consider the case W ⊥ = 0. So the statement reduces to the compactness of the subgroup of U(1, 2) preserving 3 different C-lines. This is a classical fact related to the definition of the cross ratio. At our fixed x, we have one or two asymptotic leaves. By homogeneity, we have the same property, one or two asymptotic leaves, for any y ∈ M . If there are two, we arrange to choose an asymptotic leaf denoted F y , in order to insure continuity (at least) in a neighbourhood of x. Proof. Let (f n ) is a diverging sequence in G y ∩ G z . The fixed point set of each f n is a geodesic submanifold containing y and z. The intersection of all of them when n varies is a geodesic submanifold S containing y and z, fixed by all the f n . The graph of f n above S is the diagonal of S × S. Hence, S is contained in the projection of any limit of Graph(f n ). Therefore, any asymptotic leaf at y is also asymptotic at z.

Geometry.

FACT 8.4. (Remember the notation I x = g x ∩ W s X ). If X can be chosen such that dim I x ≥ 3, then for any y ∈ F x , g y ∩ I x = 0. In particular F y = F x . Proof. By Corollary 3.5, the group generated by I x preserves each leaf of the characteristic foliation N of F x . Equivalently, the evaluation I x (y) is contained in the tangent space of the characteristic leaf N y , for any y ∈ F x . Since dim I x ≥ 3, and dim N y = 2, we conclude that I x ∩ g y = 0, and hence F x = F y by Fact 8.3. 8.2.1. Same conclusion in the other case. Assume now that for any choose of X, dim I x = 2, say I x = Span{a, b}, and let A = exp a and B = exp b. The set of fixed points of a nilpotent element, say a (or equivalently a unipotent element A) is a geodesic submanifold F ix(a) of complex codimension 2 (one cheeks this for elements of U(1, n)).

Choose y ∈ F ix(a), then by Fact 8.3, F y = F x . Since a ∈ g y , we can apply Facts 6.6 and 6.7 and get for the same X that I y = g y ∩ W s X has dimension ≥ 2. Let I be the subalgebra of W s X generated by I x and I y . It is nilpotent since contained in W s X . Assume furthermore that y ∈ F ix(a) -F ix(b), then I x = I y , and hence dim I ≥ 3.

Since it is generated by I x and I y , I preserves individually the leaves of the characteristic foliation N on F x . As above, by the inequality on dimensions, any z ∈ F x is fixed by a non-trivial element of I. Therefore, by Fact 8.3, F z = F x . 8.3. End, Contradiction. The previous conclusion means that two asymptotic leaves are disjoint or equal, that is they define a foliation of M , of (real) codimension 2. This foliation is G invariant. Therefore, G acts on the (local) quotient space of the foliation. This contradicts our hypothesis that G does not act (locally) on surfaces.

Corollary 4 . 5 .

 45 Let L be a subgroup of U(1, n) acting irreducibly on C n+1 . Then, its Zariski closure contains SO 0 (1, n) or SU(1, n).

5. 1 .

 1 Case 1: the Zariski closure of H contains SU(1, n). The holomorphic sectional curvature at p is an H-invariant function on the open subset in P n (C) of nonlightlike C-lines of C n+1 . But SU(1, n) acts transitively on this set. It follows that this holomorphic sectional curvature is constant. Therefore M is a Kähler-Lorentz manifold of constant holomorphic sectional curvature, and thus M is locally isometric to one the universal spaces Mink n+1 (C), dS n+1 (C) or AdS n+1 (C)

  and is in particular SU(1, n)-invariant. By irreducibility, A = C n+1 , and thus G = SU(1, n) ⋉ C n+1 . The group G is a quotient of G by a discrete central subgroup. But G has no such a subgroup. It then follows that M = SU(1, n) ⋉ C n+1 /SU(1, n), and hence M = Mink n+1 (C).

FACT 6 . 6 .

 66 If the stabilizer algebra g x contains a nilpotent Y , then any R-split element X of an sl 2 -triplet {X, Y, Z} (i.e. [X, Y ] = -Y, [X, Z] = +Z, and [

8. 1 . 1 .

 11 Varying x.

FACT 8 . 3 .

 83 Assume y and z near x.If G y ∩ G z is non-compact, then F y = F z .

Acknowledgments. We would like to thank Benedict Meinke for his careful reading and valuable remarks on the article.


Franoise André

Lilia Hinde

Jérémy Bouziane

Jean-Louis Buisson

Christian Pazat

Pérez

Towards dynamic adaptability support for the master-worker paradigm in component based applications

Keywords: Software components, Grid, Master-worker, Dynamic evolution, Adaptability framework

When executing scientific applications, resources that may be used can vary from multi-core processors to grids. Therefore, abstracting the programming model enables portability on various resource infrastructures. Furthermore, software component technology appears to be a very promising approach to deal with the growing complexity of scientific applications. Hence, we proposed a model to improve the support of master-worker paradigm in component models. Capitalizing on our experience of adaptability frameworks, we propose to enhance our model so that master-worker applications can adapt at runtime to varying conditions. This report studies how to introduce transparently adaptability in our model for master-worker applications, what impact it has on the model, and what requirements it expects from the adaptability framework.

Vers un support d'adaptation dynamique pour le paradigme matre-travailleur dans les applications base de composants

Résumé :

Les plate-formes d'exécution d'applications scientifiques peuvent varier de processeurs multi-core aux grilles de calcul en passant par des grappes. Pour assurer la portabilité de ces applications sur ces différentes plate-formes, il est nécessaire d'avoir des modèles de programmation capables de fournir un niveau d'abstraction suffisamment élevé par rapport aux ressources utilisées. Pour cela, la programmation à base de composants apparaît très prometteuse. De plus, elle permet de simplifier la programmation des applications scientifiques, dont le code ne cesse d'accroître en complexité. Dans des travaux précédents, on a proposé d'améliorer le support du paradigme maître-travailleur dans les applications à base de composants logiciels.En nous basant sur notre expérience en adaptabilité, notre objectif est d'étudier l'adaptation d'une application maître-travailleur aux variations dynamiques des conditions d'exécution. Ce rapport étudie l'intégration d'un canevas d'adaptation de manière transparente, son impacte sur le modèle maître-travailleur proposé ainsi que les services attendus d'un canevas d'adaptabilité.

Mots-clés : Composants logiciels, Grille, Maître-travailleur, évolution dynamique, canevas d'adaptation

Introduction

While computing Grids are becoming more and more common, the question of their programmability is raising attention. The underlying motivation not only stems from the high complexity of grids that shall be hidden to programmers but it also comes from the increasing complexity of applications. In order to take advantage of the huge possibilities of grids, more complex applications like code coupling applications are getting popular.

Software component technology appears very promising to handle the complexity of both grids and applications. Code reuse enables to build complex applications based on validated building blocks while component composition provides a mechanism to support complex relationships independently of the architecture of the execution platform.

An example of such a relationship is the master-worker paradigm. While it is an algorithmic concept, its implementation varies quite a lot depending on the execution platform. Hence, we defined a high level master-worker relationship between components [START_REF] Bouziane | Modeling and executing master-worker applications in component models[END_REF][START_REF] Antoniu | Combining data sharing with the master-worker paradigm in the common component architecture[END_REF]. While it provides a model close to the abstract concept to the programmers, it can be configured by the execution environment to fit to the actual resources. However, this previous work did not consider dynamic adaptation. For example, the number of workers may change depending on the number of incoming requests or the number of available machines. The goal of this report is to study how to introduce adaptability support in a master-worker paradigm and to evaluate the impact on adaptation frameworks.

The remainder of this report is organized as follow. Section 2 briefly presents the model to handle master-worker relationship between components as well as an analysis of the various levels of adaptability. Section 3 introduces a framework of adaptability while Section 4 discusses different strategies to introduce adaptability within the master-worker relationship. An example is presented in Section 5. Section 6 concludes the report and presents some future works.

A high-level master-worker composition model

We proposed in [START_REF] Bouziane | Modeling and executing master-worker applications in component models[END_REF] to increase the abstraction level of component models with respect to the master-worker (M-W) paradigm. Our motivation is twofold. First, we aim to relieve programmers of dealing with resource dependencies, such as the number of workers to instantiate or request transport concerns. Second, we target to reuse existing master-worker environments, like DIET [START_REF] Caron | A Scalable Approach to Network Enabled Servers[END_REF], as they implement advanced request transport and scheduling algorithms.

Overview

We present in a Fractal formalism our generic model, which is already projected to specific component models Fractal [START_REF] Bouziane | Modeling and executing master-worker applications in component models[END_REF], Ccm and Cca [START_REF] Antoniu | Combining data sharing with the master-worker paradigm in the common component architecture[END_REF].

The model is based on the concept of collection, which is defined as a set of exposed ports, bound to some internal component type ports. A collection behaves like a component: it can be connected to other components. However, such a composition is done by an abstract architecture description, which represents the user's view of the application. Ideally at deployment time, a collection is turned into a composite by defining an initial number of internal component instances and by selecting a request transport pattern. A pattern represents a requests transport algorithm that may be used between master and worker components. It is a composite whose implementation should be done by some experts and can or can not be based on software components, such as DIET [START_REF] Caron | A Scalable Approach to Network Enabled Servers[END_REF]. Request transport patterns are defined independently of a collection. Fig. 1 presents an overview of the concepts of the proposed model.

RT n 0333

Need for dynamic behavior

The proposed model dealt with building a static master-worker application because the transition from the abstract collection to a concrete composite fixes the number of workers as well as a pattern at deployment time. However, such choices have to be dynamic to take into account modifications of the application behavior and/or of the resources. The application behavior encompasses collection level behaviors like the frequency and the kind of incoming requests, the number of requests waiting for a worker, or the number of connected masters. It also comprises application's level behavior when there are several collections within an application. Resource behaviors are made of standard considerations like availability, end of a resource reservation, etc.

For a collection, there are three elements that may be dynamically modified: 1) the number of workers, 2) the used pattern and 3) the tuning of the pattern.

For example, let consider an increase of the number of waiting requests. If the pattern is not the bottleneck, the solution is to add more workers if there are available resources. However, if the pattern is the bottleneck, either the pattern may by optimized or it has to be changed by a more scalable one. In order to help decision making, validity constraints may be attached to a pattern. For instance, a round-robin pattern can be adequate for one or a few connected masters, for equivalent request load and for homogeneous processors. If at least one of these conditions are not met, another pattern should be considered, like for example a load-balacing pattern or DIET.

A collection can also be modified to optimize resources usage. For instance, if there is a lot of workers compared to the number of requests, it can be suitable to remove some workers to release resources. Last, more complex situations occurs when an application contains several instances of the master-worker paradigm. In such a case, re-structuring a collection should be coordinated with other collections.

A framework for adaptability

In a previous work [START_REF] Aldinucci | An abstract schema modelling adaptivity management[END_REF][START_REF] Buisson | Afpac: Enforcing consistency during the adaptation of a parallel component[END_REF], we have studied how to make applications suit varying conditions, relying on the notion of adaptability. This work led us to develop a generic component framework for adaptability, Dynaco. Benefiting from a joint work with the university of Pisa [START_REF] Aldinucci | An abstract schema modelling adaptivity management[END_REF], this framework splits adaptability into four sub-functionalities: 1) the framework has to be able to observe characteristics of the environment in order to trigger adaptability; 2) when a change is detected, the framework has to decide an adaptation strategy according to observed measures; 3) once a strategy has been decided, the framework has to plan actions to implement it; at last, 4), planned actions have to be executed synchronously with the execution of applicative code. In Fig. 2, each sub-functionality is captured by a Fractal component.

Rather than reimplementing the components of the framework specifically to each application, developers are encouraged to focus on application-specific issues, thanks to the reuse of existing generic engines. For instance, we have experimented 3 generic engines for the RT n 0333 decide component: 1) a Java virtual machine, such that the decision procedure is implemented with a general-purpose language, allows easy implementations of intuitive decision procedures; 2) the Jess [START_REF]the rule engine for the java platform[END_REF] expert system, such that the decision procedure is expressed with a domain-specific rule language (i.e. as a collection of ordered rules that looks like the following statement: "decide a given strategy when an associated condition becomes true"), allows efficient implementations of complex rule-based decision procedures; and 3) a genetic algorithm, such that the decision procedure is expressed as a function to optimize (e.g. the performance model of the application), allows to implement straightforwardly decision procedures when the application's behavior is well formalized, possibly with a higher runtime cost. As described, each engine proposes a different trade-off.

The same applies to the plan and execute components. For the former, for instance, a pattern matching based mapping from strategies to predefined plans suits well simple cases; while more sophisticated formalisms such as Strips [START_REF] Nilsson | STRIPS: a new approach to the application of theorem proving to problem solving[END_REF] (developers only declare the collection of possible actions as pre-and post-conditions) may be relevant when developers cannot predefine plans by hand. Similarly, synchronizing adaptation actions with the applicative code depends mostly on the applicative programming model: we have proposed an algorithm (Afpac [START_REF] Buisson | Afpac: Enforcing consistency during the adaptation of a parallel component[END_REF]) for any SPMD application. Other algorithms could be used such as Assist [START_REF] Aldinucci | Dynamic reconfiguration of grid-aware applications in assist[END_REF] when using its parmod skeleton programming model.

The observe component does not adhere to the same design: monitor s are facilities provided by the environment itself that are wrapped into adapter components, which gather, aggregate and preprocess raw measures and events to their expected formats. The whole observe component is almost independent of the application and does not need any particular specialization.

As shown in Fig. 3, application-specific code is captured in policy, guide and a collection of actions, which respectively specialize the decide, plan and execute components. Using generic engines in that way is what makes Dynaco highly generic and open, while it encourages effective code reuse.

Design choices for adaptability in the M-W paradigm

This section studies how to make use of an adaptability framework such as Dynaco in the master-worker paradigm. It analyzes two major design choices we have identified: the choice of the adaptability strategy and of the architecture. The discussion is done with respect to three criteria: modularity, accuracy, and scalability. Modularity measures the possibility to compound strategies such as at the collection level and at the pattern level. Accuracy stands for the kinds of allowed adaptations while scalability refers to the number of components in the collection.

Strategy level

The first choice concerns the way to logically design the adaptation strategy, which can be monolithic, independent or coordinated.

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Considering a single monolithic strategy, the global strategy should handle any possible situation and adaptation for the whole collection. Especially, it should consistently handle the adaptation at the level of the collection, the pattern, and the pattern implementation. For instance, observing that the request queue lengthens, instantiating new workers may increase the heterogeneity of processors, such that the pattern should be replaced by a more suited one (e.g. switching from round-robin to DIET). A monolithic strategy is able to handle those two adaptations at once. Assuming now that the bottleneck is the pattern, which may not be able to perform better, not even with a different implementation nor with additional resources. Being aware of all of the implemented patterns, a monolithic strategy has sufficient knowledge to detect such a situation and prevent useless workers. Therefore, high accuracy is provided. However, the major drawback is the poor modularity. Indeed, the tight entanglement between adaptations makes it particularly difficult to add incrementally the support for new patterns, as well as to maintain the strategy, as any local modification may have an impact on the whole strategy. Worse, in the case of a multicollection application, adaptations for all of the collections have to be handled by a single strategy at the level of the whole application.

Rather than designing the strategy as a whole, it may be better to decompose it such that the specification of the strategy for each adaptation is close to what is adapted. Basically, in order to allow good modularization, 3 sub strategies would be designed: the first one, attached to the collection composite, adapts the number of workers; the second one, attached to the pattern, selects a convenient pattern; and the last one, attached to the pattern implementation, optimizes the pattern. Two alternatives can be derived from this compound strategy. Each sub strategy may be independent or otherwise it may be coordinated. In the former case, independence means that no explicit interaction occurs from one sub strategy to the others. The latter case allows explicit interactions between sub strategies such that they can coordinate the adaptations of the elements of the collection. Any technique can be used to implement the coordination, such as triggering adaptations from other adaptations (propagating adaptations) or running a negotiation protocol (agreeing on adaptations).

Focusing on the independent approach, let us consider first the above example of adding worker instances that increase heterogeneity, which may result from different processors or from different implementations. Independence implies that the pattern switches its implementation on its own when it observes that heterogeneity increases, once the collection (independently) has instantiated new workers. Thus, despite their independence, the sub strategies achieve together the same adaptations as the single monolithic strategy. However, that way of observing effects of adaptations is not always enough to implement accurate adaptations. Consider that the queue lengthens. An accurate strategy does not instantiate new workers if the pattern would not be able to dispatch requests at a sufficient pace; and it does not optimize the pattern if there can't be enough workers to handle requests. However, independence of the sub strategies prevents the collection from knowing whether the pattern would be able or not to dispatch requests to additional workers; and it prevents the pattern from knowing whether the collection would be able to instantiate new workers. In such a situation, this strategy would desperately preserve the status quo, even if the collection would be able to perform better; while lowering accuracy may lead to instantiate useless workers or to over-optimize the pattern. Last, the coordinated strategy promises to bring the advantages of the two other strategies without their drawbacks (Fig. 4). It preserves compound strategies for modularity and scalability while letting a global vision to be built for accuracy. However, several adaptation modules have to be interconnected.

Achitecture level

The second design choice concerns the architecture of the adaptability. Two alternatives are identified: centralized and distributed.

A centralized architecture locates the whole adaptability management into a single location. With respect to the model presented in Section 2, it has to be into the membrane of the collection. Bindings are, nevertheless, present to enable it to control the whole collection. The centralized approach is compatible with all adaptation strategies described in Section 4.1. It also simplifies the implementation of the coordinated strategy as the communication between the different strategies may be embedded into the same adaptation framework. However, it raises an issue for the compound strategies with respect to the composition of components: the adaptation part of sub components needs to be injected into the adaptation part of the collection. Hence, the connection operator of the component model turns out to be more complex. As far as we know, there is no standard component models that permits it.

With a distributed architecture, the adaptability management is spread over the whole collection, and in particular in the membranes of the collection, of the pattern and of the pattern implementation components. The distributed architecture is not straightfowardly compatible with the monolithic strategy. However, it perfectly fits with the counpound strategies provided that the communications of the coordinated strategy are quite simply done through some ports. Considering the advantages of the coordinated strategy, we conclude that this strategy with a distributed architecture appears to be the best choice to deal with dynamic change in a collection.

Positionning

Only a couple of adaptability frameworks address the problem of coordinating and distributing adaptations. Dynaco is neutral as it does not prevent policies to coordinate on their own, but it does not provide any specific support.

Among the other frameworks, Aceel [START_REF] Chefrour | Plate-forme de composants logiciels pour la coordination des adaptations multiples en environnement dynamique[END_REF] and Plasma [START_REF] Layaida | Designing self-adaptive multimedia applications through hierarchical reconfiguration[END_REF], are fairly close to the constructive approach defended here, i.e. building global adaptations as the collaboration of individual local adaptations. With our previous framework Aceel, each component contacts other components before adapting, in order to ensure consistent and synchronized adaptation of the whole assembly. With Plasma, components impose their adaptations to the other ones through a simpler propagation mechanism. The contract-driven approach of Assist [START_REF] Aldinucci | Autonomic qos in assist grid-aware components[END_REF] is different: considering a hierarchical component model, composite components divide their contract in order to assign recursively subcontracts to their sub components. Coordination of adaptation is enforced by the submission of contracts that are consistent with another. However, this approach requires composite components to have precise understanding of the composition of their immediate subcomponents, in order to devise subcontracts.

Those frameworks are however tied to the programming models for which they have been specifically designed, often restricted to a fixed collection of predefined adaptations; while focusing only on adaptability, Dynaco integrates gracefully to any programming model. Consequently, Dynaco is a better start point.

Impact on the master-worker model

Section 4.2 concludes with the choice of a coordinated strategy with a distributed architecture. Let us consider now the impact of such choice on the model presented in Section 2. Achieving the objective of transparent dynamic management, there is no need to modify the model at the user view level. The collection instantiation process seems to be more appropriate to introduce an adaptability framework. A collection implementation, in particular the collection and pattern membranes, are determined at this stage. Adding adaptability RT n 0333 framework as controllers in appropriate membranes appears to be straightforward. However, the diversity of resource infrastructures and resource management systems lead to various adaptability policies. For instance, a policy can be more constrained by resource availability when resource sharing is privileged, otherwise it can be more constrained by application requirements. As a consequence, similarly as for patterns, the framework has to do a selection from a set of adaptability implementations. Fortunately, the specificity of Dynaco to be component-based allows the use of different implementations. The master-worker model extended with adaptability support is presented in Fig. 5.

Example of master-worker application adaptation

Based on the preceding analysis, this section presents a concrete example of design for the adaptability of a master-worker application. As outlined in Section 2.2, the adaptation aims at preventing the request queue from unacceptably growing, while making the queue contain enough requests to feed continuously the workers. In order to enforce that objective, we propose the following intuitive compound strategy, using the coordinated approach:

• at the level of the collection: if the request queue lengthens beyond a threshold, if the pattern is able to increase its dispatch rate accordingly and if there are available resources, then instantiate new workers; if the request queue shortens under a threshold, then terminate some workers.

• at the level of the pattern: if the number of masters or the variability of request durations increases above a threshold, or if the heterogeneity of workers increases beyond a threshold, then switch to the DIET pattern; otherwise, under a threshold switch to roundrobin.

In this strategy, coordination occurs before the collection instantiates new workers. It actually asks the pattern whether it would be able to dispatch requests at a sufficient rate, for instance involving a contract renegotiation protocol. The length of the request queue cannot always be observed directly; lengthenings and shortenings can nevertheless be deduced from the comparison between arrival and service rates. Other observations are almost obvious. The examples shows that the coordinated and distributed design suits well and that the necessary monitoring does not breach encapsulation.

Conclusion

The report analyses how to design dynamic adaptability support for component-based master-worker applications. Among the discussed possibilities, coordinating several distributed adaptations appears to be the best-suited solution with regard to modularity, scalability and accuracy. In addition, integrating adaptability at the level of the master-worker abstraction achieves the goal of hiding the management of execution resources from the developers' sight.

Among adaptability frameworks, none fully meets the requirements of our proposal. Its genericity and openness make Dynaco be the best start point. Based on the experience INRIA we gained in our previous work on Aceel [START_REF] Chefrour | Plate-forme de composants logiciels pour la coordination des adaptations multiples en environnement dynamique[END_REF][START_REF] Chefrour | Développement d'applications en environnements mobiles à l'aide du modèle de composant adaptatif ACEEL[END_REF], we plan to extend Dynaco with specific support for the coordination of distributed adaptations, so that it meets the requirements. We will also evaluate the proposed model on synthetic master-worker benchmarks as well as the possibilities to write generic adaptation policies at the collection and application levels.

Figure 1 :

 1 Figure 1: Overview of the master-worker model from the user and framework points of view.

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Figure 2 :

 2 Figure 2: Overview of the Dynaco architecture as an assembly of Fractal components.

Figure 3 :

 3 Figure 3: Detailed architecture of the Dynaco framework: entities and dependencies belonging to each Fractal component.

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Figure 5 :

 5 Figure 5: Introducing dynamic management in the master-worker model.