A splitting result for real submanifolds of a Kahler manifold

Let $(Z,\omega)$ be a connected Kahler manifold with an holomorphic action of the complex reductive Lie group $U^{\mathbb C}$, where $U$ is a compact connected Lie group acting in a hamiltonian fashion. Let $G$ be a closed compatible Lie group of $U^{\mathbb C}$ and let $M$ be a $G$-invariant connected submanifold of $Z$. Let $x\in M$. If $G$ is a real form of $U^{\mathbb C}$, we investigate conditions such that $G\cdot x$ compact implies $U^{\mathbb C} \cdot x$ is compact as well. The vice-versa is also investigated. We also characterize $G$-invariant real submanifolds such that the norm square of the gradient map is constant. As an application, we prove a splitting result for real connected submanifolds of $(Z,\omega)$ generalizing a result proved in \cite{pg}, see also \cite{bg,bs}.


Introduction
Let (Z, ω) be a Kähler manifold. Assume that U C acts holomorphically on Z, that U preserves ω and that there is a momentum map for the U action on Z. This means there is a map µ : Z −→ u * , where u is the Lie algebra of U and u * is its dual, which is U equivariant with respect to the given action of U on Z and the coadjoint action Ad * of U on u * and satisfying the following condition. Let ξ ∈ u. We denote by ξ Z the induced vector field on Z, i.e., ξ Z (p) = d dt | t=0 exp(tξ)p. Let µ ξ be the function µ ξ (z) := µ(z)(ξ), i.e., the contraction of the moment map along ξ. Then dµ ξ = i ξ Z ω. Let G be a closed connected subgroup of U C compatible with respect to the Cartan decomposition of U C , i.e. G = K exp(p), for K = U ∩ G and p = g ∩ iu [13,15]. The inclusion ip ֒→ u induces by restriction a K-equivariant map µ ip : Z −→ (ip) * [11,12]).
Let ·, · be a U -invariant scalar product on u. Let ·, · denote also the inner product on iu such that i be an isometry of u into iu. Hence we may identify u * and u by means of ·, · and so we view µ as a map µ : Z −→ u. Therefore, we may view µ ip as a map µ p : Z −→ p as follows: We call µ p the G-gradient map associated with µ. We also set µ β p := µ p , β . By definition, it follows that gradµ β p = β Z . If M is a G-stable locally closed real submanifold of Z, we may consider µ p as a mapping µ p : M −→ p such that gradµ p = β M , where the gradient is computed with respect to the induced Riemannian metric on M . Since M is G-stable it follows β Z (p) = β M (p) for any p ∈ M .
Assume that G is a real form of U C . If U C · x is compact, then it is well-known that G has a closed orbit contained in U C · x [11]. On the other hand, if G · x is closed then it is not in general true that U C · x is closed as well [9]. In Section 2, we investigate conditions such that G · x compact implies U C · x is compact. If G · x is compact then we give a necessary condition to U C · x be compact. If M is Lagrangian, then U C · x being compact implies G · x is a Lagrangian submanifold of U C · x. Finally, we study the case when Z is U C -semistable, M is G-semistable and is contained in the zero level set of the gradient map of K C . As an application we proof a well-known result of Birkes [2].
A strategy for analyzing the G action on M is to view the function ν p : M −→ R, as a Morse like function. The function ν p is called the norm square of the gradient map. If M is compact or µ p is proper, then associated to the critical points of ν p we have G-stable submanifold of M that they are strata of a Morse type stratification of M [11,14]. In Section 3, we investigate under which condition ν p is constant. The following result has some interest itself. By the stratification Theorem [11], it follows that M coincides with a maximal pre-stratum and µ p (M ) = K · β. Moreover, M = K × K β µ −1 p (β), where K β = {k ∈ K : Ad(k)(β) = β}. Let x ∈ µ −1 p (β). By the K-equivariance of µ p , it follows that the stabilizer K x ⊆ K β . Although G · x is closed, it is not true in general K x = K β . Indeed, let U be a connected, compact semisimple Lie group and let ρ : U −→ SL(W ) be a complex representation. Let G be a noncompact connected semisimple real form of U C . It is well known that U C has a closed orbit in P(W ), which is a complex U -orbit [8]. Let O denote a closed orbit of U C . If x ∈ O realizes the maximum of the norm squared of the G-gradient map restricted to O, then G · x is closed and it is a K orbit [11]. Now, K x = K ∩ U µ(x) and U µ(x) = U x since U · x is complex [8]. However, µ(x) / ∈ p and so K x does not coincide in general with K µp(x) . If M is a U -invariant compact connected complex submanifold of (Z, ω), then ν iu constant is equivalent to U is semisimple and M = U/U β × µ −1 (β). The above splitting is Riemannian [7] (see also [1,3] for the same result under the assumption that M is symplectic). In this paper we prove this splitting result without any assumption on M . Assume that G is a real form of U . The momentum map of U on Z induces a gradient map Theorem 3. Assume that Z is U C -semistable and M is a G-semistable real connected submanifold of Z. Assume also M is contained in the zero fiber of µ ik . Then the square of the G-gradient map µ p 2 is constant if and only if G is semisimple and M is K-equivariantly isometric to the product of a real flag and an embedded closed submanifold which is acted on trivially by K.

Closed orbits and gradient map
Let (Z, ω) be a Kähler manifold. Assume that U C acts holomorphically on Z, that U preserves ω and that there is a momentum map for the U action on Z. Let G ⊂ U C be a closed compatible subgroup and let M be a G-invariant submanifold of (Z, ω) and let µ p : M −→ p be the associated G-gradient map. • if x realizes a local maximum of ν p , then G · x = K · x and so it is compact; Proof. If x realizes a local maximum for ν p , then ν p : G · x −→ R has a local maximum at x. By Corollary 6.12, p.21 in [11], it follows G · x = K · x.
Assume G · x is compact. Then ν p : G · x −→ R has a local maximum. Applying, again, Corollary 6.12 p.21 in [11], we get G · x = K · x. We compute the differential of ν p at x. It is easy to check Proof. Since G · x is compact, by the above Lemma G · x = K · x. By the K-equivariance of µ p , it follows that µ p : K · x −→ K · β is a smooth fibration. Therefore, keeping in mind that This also implies k · x = p · x if and only if dim K · x = dim K · β, concluding the proof.
Assume that G is a real form of U C . If G · x is closed then it is not in general true that U C · x is closed. Indeed, let V be a complex vector space and let τ : G −→ PGL(V ) be an irreducible faithful projective representation. Since the center of G acts trivially, we may assume that G is semisimple. The representation τ extends to an irreducible projective representation of U C . It is well-known that U C has a unique closed orbit [8]. It is the orbit throughout a maximal vector. On the other hand G could have more than one closed orbit in P(V ) [9, Proposition 4.28, p. 58]. The following result tells us that there exists a unique closed G-orbit contained in the unique closed orbit of U C . Proposition 6. Let M = U C · x be a compact orbit. If G is a real form of U , then there exists exactly one closed G-orbit in M .
Proof. U C · x = U · x and it is a flag manifold [11,8]. Applying a beautiful old Theorem of Wolf [19], it follows that G has a unique closed orbit in M . The G orbit is given by the orbit throughout the maximum of the norm square of the gradient map [11].
The following result arises from Lemma 5.
This implies U ·x is open and closed in U C ·x. Therefore U C ·x = U ·x, concluding the proof.
The following result gives a necessary and sufficient condition such that U C · x is closed whenever G · x is.
Proof. Set β = µ p (x). By Lemma 5, k · x = p · x ⊥ ⊕ k β · x. Therefore, keeping in mind u = k ⊕ ip, we have Since ik β ·x is orthogonal to ip·x, it follows that u·x = u C ·x, if and only if ik β ·x ⊂ u·x∩i(p·x) ⊥ .
This implies u · x = u C · x if and only if ik β · x ⊆ ip · x. By the first part of the proof we get U C · x is compact if and only if k β · x = {0} and so if and only if dim K · x = dim K · β. In particular p·x = k·x. This implies dim R G·x = dim C U C ·x and so G·x is a compact Lagrangian submanifold of U C · x. Proposition 9. Let M be a G-invariant Lagrangian submanifold of (Z, ω). Let x ∈ M . Then U C · x is closed if and only if k · x = p · x. In particular G · x is closed and it is a Lagrangian submanifold of U C · x.
Proof. Since M is Lagrangian, we have This also implies G · x is compact, dim R G · x = dim C U C · x and so G · x is a compact Lagrangian submanifold of U C · x.
Proposition 10. Let x ∈ Z. Assume that both G·x and U C ·x are compact. Then dim R U C ·x ≤ 2 dim G · x. If the equality holds then G · x is totally real.
This remark is not new, see [11,12], and it arises from the Matsuki duality [18].
The momentum map of U on Z induces a gradient map µ ik of K C in Z. Assume that M is contained in the zero fiber of µ ik .
Proof. Let y ∈ U C · x. Since µ = µ ik + µ p , it follows that Hence ν p : U C · x −→ R achieves its maximum in x. By Lemma 4, G · x is closed.
In the papers [10,11], the authors proved if M is G-semistable then G · x is closed if and only if G · x ∩ µ −1 p (0) = ∅. As an application we get the following result.
Proposition 12. Assume that (Z, ω) is U C -semistable and M is G-semistable and it is contained in the zero fiber of µ ik . Let x ∈ M . Then G · x is closed if and only if U C · x is closed.
A corollary we prove a well-known result of Birkes [2], see also [5] Corollary 13. Let G be a real form of U . Let V be complex vector space and W be real subspace of V such that V = W C . Assume that G acts on W . Let w ∈ W . Then G · w is closed if and only if U C · x is closed.
Proof. It is well-known that V , respectively W , is U C -semistable, respectively G-semistable [17], see also [4]. Since W is a Lagrangian subspace of V , applying the above Proposition the result follows.

norm square of the gradient map
We investigate splitting results for G-invariant real submanifolds of (Z, ω). Proof. Assume ν p is constant. Let x ∈ M . Then ν p : G · x −→ R is constant and so ν p has a maximum on x. By Lemma 4 G · x = K · x and so it is compact. Vice-versa, assume that any G orbit is compact. By Lemma 4 (dν p ) x = 0 for any x ∈ M . Since M is connected it follows ν p is constant.
The following result is proved in [11]. For the sake of completeness we give a proof.
By Proposition 14 any G orbit is a K orbit. This implies p · x ⊂ k · x. Since k β · x ⊂ (p · x) ⊥ , it follows that the map is injective. Therefore dΨ [e,x] is surjective. Since Ψ is bijective it follows that dΨ [e,x] must be bijective.
We are ready to prove the splitting results.
Proof of Theorem 2. Since ν is constant, applying Lemma 14 it follows that any U C orbit is compact and it is a complex U orbit. Then for any x ∈ M , we have U x = U µ(x) [8]. Since U µ(x) is a centralizer of a torus, then the center of U does not act on M and so U is semisimple. By the above proposition M = U/U β × µ −1 (β) and for very x ∈ µ −1 (β), U x = U β and so U x acts trivially on µ −1 (β). If x ∈ µ −1 (β), then This implies that the U action on M is polar with section µ −1 (β) [6] and so µ −1 (β) is totally geodesic. We claim that the above splitting is Riemannian. Let ξ ∈ u and let ξ M the induced vector field. It is enough to prove that the function g(ξ M , ξ M ) is constant when restricted to µ −1 (β). Let x ∈ µ −1 (p) and v ∈ T x µ −1 (p). We may extend v to a vector field on a neighborhood of p, that we denote by X, such that g(X, ξ M ) = 0 for any z ∈ W and for any ξ ∈ u. Indeed, let ξ 1 , . . . , ξ k ∈ u such that ( Since the U action on M has only one type of orbit, it follows that there exists a neighborhood W of x such that (ξ 1 ) M (y), . . . , (ξ k ) M (y) is a basis of T y U · y for any y ∈ W . Applying a Gram-Schmidt process we get an orthonormal basis {Y 1 , . . . , Y k } of T y U · y for any y ∈ W . LetX any local extension of v. Then satisfies the above conditions. Moreover, for any z ∈ µ −1 p (β) ∩ W , the vector field X lies in T z µ −1 p (β) due to the orthogonal splitting . By the closeness of ω, we have On the other hand, by the Cartan formula [16], we have Proof of Theorem 3. By Proposition 15 M = K × K β µ −1 p (β). By Proposition 14 it follows U C · x is compact for any x ∈ µ −1 p (β). Let x ∈ µ −1 p (β). By Proposition 12, U C · x is compact as well and µ p (x) = µ(x) = β. This implies K x = K ∩ U x = K ∩ U β = K β for any x ∈ µ −1 p (β) and so M = K/K β × µ −1 p (β). The Lie algebra of the center of G is contained in the Lie algebra of the center of U C . On the other hand, the Lie algebra of the center of U C is the complexification of the Lie algebra of the center of U which acts trivially on M . This implies G is semisimple. Finally, keeping in mind that ω is closed and U C · x is compact for any x ∈ µ −1 p (β), applying the same idea of the above proof we get the splitting M = K/K β × µ −1 p (β) is Riemannian.