A splitting result for real submanifolds of a Kahler manifold
aa r X i v : . [ m a t h . DG ] J a n A SPLITTING RESULT FOR REAL SUBMANIFOLDS OF A K ¨AHLERMANIFOLD
LEONARDO BILIOTTI
Abstract.
Let (
Z, ω ) be a connected K¨ahler manifold with an holomorphic action of the com-plex reductive Lie group U C , where U is a compact connected Lie group acting in a hamiltonianfashion. Let G be a closed compatible Lie group of U C and let M be a G -invariant connectedsubmanifold of Z . Let x ∈ M . If G is a real form of U C , we investigate conditions such that G · x compact implies U C · x is compact as well. The vice-versa is also investigated. We alsocharacterize G -invariant real submanifolds such that the norm square of the gradient map isconstant. As an application, we prove a splitting result for real connected submanifolds of ( Z, ω )generalizing a result proved in [7], see also [1, 3]. Introduction
Let (
Z, ω ) be a K¨ahler 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 withrespect to the given action of U on Z and the coadjoint action Ad ∗ of U on u ∗ and satisfyingthe following condition. Let ξ ∈ u . We denote by ξ Z the induced vector field on Z , i.e., ξ Z ( p ) = ddt | t =0 exp( tξ ) p . Let µ ξ be the function µ ξ ( z ) := µ ( z )( ξ ), i.e., the contraction of themoment map along ξ . Then d µ ξ = i ξ Z ω .Let G be a closed connected subgroup of U C compatible with respect to the Cartan decom-position of U C , i.e. G = K exp( p ), for K = U ∩ G and p = g ∩ i u [13, 15]. The inclusion i p ֒ → u induces by restriction a K -equivariant map µ i p : Z −→ ( i p ) ∗ [11, 12]).Let h· , ·i be a U -invariant scalar product on u . Let h· , ·i denote also the inner product on i u such that i be an isometry of u into i u . Hence we may identify u ∗ and u by means of h· , ·i and sowe view µ as a map µ : Z −→ u . Therefore, we may view µ i p as a map µ p : Z −→ p as follows: h µ p ( x ) , β i = −h µ ( x ) , iβ i . We call µ p the G -gradient map associated with µ . We also set µ β p := h µ p , β i . By definition,it follows that grad µ β p = β Z . If M is a G -stable locally closed real submanifold of Z , wemay consider µ p as a mapping µ p : M −→ p such that grad µ p = β M , where the gradient is Mathematics Subject Classification.
Key words and phrases. gradient map; real reductive Lie groups.The author was partially supported by Project PRIN 2017 “Real and Complex Manifolds: Topology, Geometryand holomorphic dynamics” and by GNSAGA INdAM.. 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 hasa closed orbit contained in U C · x [11]. On the other hand, if G · x is closed then it is not ingeneral 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 Lagrangiansubmanifold of U C · x . Finally, we study the case when Z is U C -semistable, M is G -semistableand is contained in the zero level set of the gradient map of K C . As an application we proof awell-known result of Birkes [2].A strategy for analyzing the G action on M is to view the function ν p : M −→ R , ν p ( x ) = k µ p ( x ) k 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 -stablesubmanifold of M that they are strata of a Morse type stratification of M [11, 14]. In Section3, we investigate under which condition ν p is constant. The following result has some interestitself. Proposition 1.
Let M be a G -stable connected submanifold of Z and let µ p : M −→ p be therestricted gradient map. Then the square of the gradient map ν p : M −→ R is constant if andonly if any G orbit is compact. By the stratification Theorem [11], it follows that M coincides with a maximal pre-stratumand µ p ( M ) = K · β . Moreover, M = K × K β µ − p ( β ), where K β = { k ∈ K : Ad( k )( β ) = β } . Let x ∈ µ − 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 semisimpleLie group and let ρ : U −→ SL( W ) be a complex representation. Let G be a noncompactconnected 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 themaximum of the norm squared of the G -gradient map restricted to O , then G · x is closed andit 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 µ p ( x ) .If M is a U -invariant compact connected complex submanifold of ( Z, ω ), then ν i u constant isequivalent to U is semisimple and M = U/U β × µ − ( β ). The above splitting is Riemannian [7](see also [1, 3] for the same result under the assumption that M is symplectic). In this paperwe prove this splitting result without any assumption on M . Theorem 2.
Let M be a U C -stable connected submanifold of Z and let µ : M −→ u be therestricted momentum map. Then the square of the momentum map k µ k is constant if andonly if U is semisimple and M is U -equivariantly isometric to the product of a flag manifoldand an embedded, closed submanifold which is acted on trivially by U . SPLITTING RESULT FOR REAL SUBMANIFOLDS OF A K ¨AHLER MANIFOLD 3
Assume that G is a real form of U . The momentum map of U on Z induces a gradient map µ i k of K C in Z . We say that M is G -semistable if M = { p ∈ M : G · p ∩ µ − p (0) = ∅} . Theorem 3.
Assume that Z is U C -semistable and M is a G -semistable real connected submani-fold of Z . Assume also M is contained in the zero fiber of µ i k . Then the square of the G -gradientmap k µ p k is constant if and only if G is semisimple and M is K -equivariantly isometric tothe 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 K¨ahler 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 compatiblesubgroup and let M be a G -invariant submanifold of ( Z, ω ) and let µ p : M −→ p be the associated G -gradient map. Lemma 4.
Let x ∈ M . Then: • if x realizes a local maximum of ν p , then G · x = K · x and so it is compact; • if G · x is compact, then G · x = K · x and x is a critical point of ν p .Proof. If x realizes a local maximum for ν p , then ν p : G · x −→ R has a local maximum at x . ByCorollary 6 . 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 . p.
21 in [11], we get G · x = K · x . We compute the differential of ν p at x . It iseasy to check d ν p ( v ) = 2 h (d µ p ) x ( v ) , µ p ( x ) i . Therefore, keeping in mind that Ker (d µ p ) x = ( p · x ) ⊥ , where p · x = { ξ Z ( x ) : ξ ∈ p } see [10], itfollows ( dν p ) x = 0 on ( p · x ) ⊥ . Since G · x = K · x , it follows p · x ⊂ k · x and so, keeping in mindthat ν p is K -invariant, (d ν p ) x = 0 on p · x as well, proving x is a critical point of ν p . (cid:3) Lemma 5.
Let x ∈ M be such that G · x is compact. Let β = µ p ( x ) . Then k · x = p · x ⊥ ⊕ k β · x. Therefore k · x = p · x if and only if dim K · x = dim K · β .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 thatKer (d µ p ) x = ( p · x ) ⊥ , we have ( p · x ) ⊥ ∩ k · x = k β · x. Since G · x = K · x , we get k · x = p · x ⊥ ⊕ (( p · x ) ⊥ ∩ k · x ) = p · x ⊥ ⊕ k β · x. This also implies k · x = p · x if and only if dim K · x = dim K · β , concluding the proof. (cid:3) LEONARDO BILIOTTI
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 irreduciblefaithful 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 maximalvector. On the other hand G could have more than one closed orbit in P ( V ) [9, Proposition4.28, p. 58]. The following result tells us that there exists a unique closed G -orbit contained inthe 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 existsexactly 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 ofWolf [19], it follows that G has a unique closed orbit in M . The G orbit is given by the orbitthroughout the maximum of the norm square of the gradient map [11]. (cid:3) The following result arises from Lemma 5.
Corollary 7.
Let x ∈ M be such that G · x is compact. If dim K · x = dim K · µ p ( x ) , then U C · x is closed.Proof. Since u = k ⊕ i p , it follows u · x = k · x + i p · x . By Lemma 5, k · x = p · x and so u C · x = u · x .This implies U · x is open and closed in U C · x . Therefore U C · x = U · x , concluding the proof. (cid:3) The following result gives a necessary and sufficient condition such that U C · x is closedwhenever G · x is. Proposition 8.
Let x ∈ M be such that G · x is compact. If G is a real form of U C , then U C · x is closed if and only if i k µ p ( x ) · x ⊆ u · x ∩ i ( p · x ) ⊥ . If M is Lagrangian, then U C · x is closed if andonly if µ p : K · x −→ K · µ p ( x ) is a covering map. Moreover, G · x is a Lagrangian submanifoldof U C · x .Proof. Set β = µ p ( x ). By Lemma 5, k · x = p · x ⊥ ⊕ k β · x . Therefore, keeping in mind u = k ⊕ i p ,we have u · x = p · x ⊥ ⊕ k β · x + i p · x. Since i k β · x is orthogonal to i p · x , it follows that u · x = u C · x, if and only if i k β · x ⊂ u · x ∩ i ( p · x ) ⊥ .If M is Lagrangian, then T x Z = T x M ⊥ ⊕ J ( T x M ). Therefore u · x = p · x ⊥ ⊕ k β · x ⊥ ⊕ i p · x. This implies u · x = u C · x if and only if i k β · x ⊆ i p · x . By the first part of the proof we get U C · x is compact if and only if k β · x = { } and so if and only if dim K · x = dim K · β . Inparticular p · x = k · x . This implies dim R G · x = dim C U C · x and so G · x is a compact Lagrangiansubmanifold of U C · x . (cid:3) SPLITTING RESULT FOR REAL SUBMANIFOLDS OF A K ¨AHLER MANIFOLD 5
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 Lagrangiansubmanifold of U C · x .Proof. Since M is Lagrangian, we have u · x = k · x ⊥ ⊕ i p · x. Therefore u · x = u C · x if and only if i k · x ⊆ i p · x and p · x ⊆ k · x hence if and only if k · x = p · x .This also implies G · x is compact, dim R G · x = dim C U C · x and so G · x is a compact Lagrangiansubmanifold of U C · x . (cid:3) Proposition 10.
Let x ∈ Z . Assume that both G · x and U C · x are compact. Then dim R U C · x ≤ G · x . If the equality holds then G · x is totally real.Proof. By Lemma 4 U C · x = U · x and G · x = K · x . Since u · x = k · x + i p · x and p · x ⊆ k · x ,it follows that dim R U C · x ≤ G · x. Note also that k C · x = u C · x . This implies K C · x is open in U C · x . This remark is not new, see[11, 12], and it arises from the Matsuki duality [18]. Finally, 2 dim G · x = dim R U C if and onlyif k · x = p · x and u · x = k · x ⊕ i p · x . In particular G · x is totally real in U C · x . (cid:3) The momentum map of U on Z induces a gradient map µ i k of K C in Z . Assume that M iscontained in the zero fiber of µ i k . Lemma 11.
Let x ∈ M . If U C · x is closed, then G · x is closed.Proof. Let y ∈ U C · x . Since µ = µ i k + µ p , it follows that k µ p ( y ) k ≤k µ ( y ) k = k µ ( x ) k = k µ p ( x ) k . Hence ν p : U C · x −→ R achieves its maximum in x . By Lemma 4, G · x is closed. (cid:3) We say that M is G -semistable if M = { p ∈ M : U C · p ∩ µ − p (0) } . In the papers [10, 11], theauthors proved if M is G -semistable then G · x is closed if and only if G · x ∩ µ − p (0) = ∅ . As anapplication we get the following result. Proposition 12.
Assume that ( Z, ω ) is U C -semistable and M is G -semistable and it is containedin the zero fiber of µ i k . Let x ∈ M . Then G · x is closed if and only if U C · x is closed.Proof. By the above result it is enough to prove if G · x is closed then U C · x is closed. If G · x is closed then G · x ∩ µ − p (0) = ∅ . Since µ − p (0) ∩ M = µ − (0) ∩ M , the result follows. (cid:3) 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 subspaceof V such that V = W C . Assume that G acts on W . Let w ∈ W . Then G · w is closed if andonly if U C · x is closed. LEONARDO BILIOTTI
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 resultfollows. (cid:3) norm square of the gradient map We investigate splitting results for G -invariant real submanifolds of ( Z, ω ). Proposition 14.
Let M be a G -stable connected submanifold of Z and let µ p : M −→ p be therestricted gradient map. Then the square of the gradient map ν p : M −→ R is constant if andonly if any G orbit is compact.Proof. Assume ν p is constant. Let x ∈ M . Then ν p : G · x −→ R is constant and so ν p has amaximum 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 (d ν p ) x = 0 for any x ∈ M . Since M is connected it follows ν p is constant. (cid:3) The following result is proved in [11]. For the sake of completeness we give a proof.
Proposition 15.
Let M be a G -stable connected submanifold of Z and let µ p : M −→ p be therestricted gradient map. If ν p is constant, then µ p ( M ) = K · β , µ − ( β ) is a submanifold and thefollowing splitting M = K × K β µ − p ( β ) , holds.Proof. Since ν p is constant, it follows that M = S β , where S β is the maximal strata, and µ p ( S β ) = µ p ( M ) = K · β [11, p. M = Kµ − p ( β ) and we may think µ p : M −→ K · β . Therefore β is a regular value and so µ − p ( β ) is a K β -invariant submanifold of M .Let x ∈ µ − p ( β ). By the K -equivariance of µ p , it is easy to check K · x ∩ µ − p ( β ) = K β · x .We claim that the same holds infinitesimally, i.e., T x µ − p ( β ) ∩ k · x = k β · x . Indeed, let v ∈ T x µ − p ( β ) ∩ k · x . Let ξ ∈ k such that v = ξ M ( x ). Since T x µ − p ( β ) = Ker (d µ p ) x , we get0 = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 µ p (exp( tξ ) x ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 Ad(exp( tξ )) β, and so v ∈ k β · x .We define the map Ψ : K × K β µ − p ( β ) −→ M [ k, x ] kx. It is easy to check that Ψ is K -equivariant and smooth. Since µ p ( M ) = K · β it follows M = K · µ − p ( β ) and so Ψ is surjective. It is also injective since kx = k ′ x if and only if k ′− k ∈ K β , proving it is bijective. Now, we proof that Ψ is a local diffeomorphism. Thisimplies that Ψ is a diffeomorphism concluding the proof. Note that it is enough to prove dΨ [ e,x ] is a diffeomorphism by the K -equivariance. Now, T x M = ( p · x ) ⊕ ( p · x ) ⊥ = ( p · x ) ⊕ T x µ − p ( β ) . SPLITTING RESULT FOR REAL SUBMANIFOLDS OF A K ¨AHLER MANIFOLD 7
By Proposition 14 any G orbit is a K orbit. This implies p · x ⊂ k · x . Since k β · x ⊂ ( p · x ) ⊥ , itfollows that the map p · x ֒ → k · x −→ k · x/ k β · x, is injective. Therefore dΨ [ e,x ] is surjective. Since Ψ is bijective it follows that dΨ [ e,x ] must bebijective. (cid:3) 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 iscompact 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. Bythe above proposition M = U/U β × µ − ( β ) and for very x ∈ µ − ( β ), U x = U β and so U x actstrivially on µ − ( β ). If x ∈ µ − ( β ), then T x M = ( i u · x ) ⊥ ⊕ T x µ − ( β ) = T x U · x ⊥ ⊕ T x µ − ( β ) . This implies that the U action on M is polar with section µ − ( β ) [6] and so µ − ( β ) is totallygeodesic. 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 − ( β ).Let x ∈ µ − ( p ) and v ∈ T x µ − ( 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 , . . . , ξ k ∈ u such that ( ξ ) M ( x ) , . . . ( ξ k ) M ( x ) is a basis of T x U · x . 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 ) M ( y ) , . . . , ( ξ k ) M ( y ) is a basis of T y U · y for any y ∈ W . Applying a Gram-Schmidt processwe get an orthonormal basis { Y , . . . , Y k } of T y U · y for any y ∈ W . Let ˜ X any local extensionof v . Then X = ˜ X − g ( Y , ˜ X ) Y − · · · − g ( Y k , ˜ X ) Y k , satisfies the above conditions. Moreover, for any z ∈ µ − p ( β ) ∩ W , the vector field X lies in T z µ − p ( β ) due to the orthogonal splitting T z M = T z U · z ⊥ ⊕ T z µ − p ( β ).Let ν M = − J ( ξ M ) Then J ( ν M ) = ξ M . Since M = U/U β × µ − ( p ), it follows [ X, ξ M ] =[ X, ν M ] = 0 along µ − p ( β ). By the closeness of ω , we haved ω ( v, ν M ( x ) , ξ M ( x )) = 0 . On the other hand, by the Cartan formula [16], we haved ω ( v, ν M ( x ) , ξ M ( x )) = Xω ( ν M , ξ M ) + ν M ω ( ξ M , X ) + ξ M ω ( X, ν M ) − ω ([ X, ν M ] , ξ M ) − ω ([ ν M , ξ N ] , X ) − ω ([ ξ M , X ] , Y ) . Now, ω ([ X, ν M ] , ξ M ) = ω ([ ξ M , X ] , Y ) = 0 due to the fact that [ X, ν M ]( x ) = [ ξ M , X ]( x ) = 0, Theterm ω ([ ν M , ξ N ] , X ) = 0, since ω ([ ν M , ξ N ] , X ) = g ( J ([ ν M , ξ M ] , X ) = 0 LEONARDO BILIOTTI due to the facts that the U orbit is complex and the splitting T x M = T x µ − ( β ) ⊥ ⊕ T x U · x holds.Finally, ν M ω ( ξ M , X ) = 0, respectively ξ M ω ( X, ν M ) = 0, due to the fact that ω ( ξ M , X ) = g ( J ξ M , X ) = 0 , respectively, ω ( X, ν M ) = g ( J X, ν M ) = − g ( X, J ν M ) = 0 , along U · x . Therefore0 = d ω ( v, ν M ( x ) , ξ M ( x )) = Xω ( ν M , ξ M ) = Xg ( J ( ν M ) , ξ M ) = Xg ( ξ M , ξ M ) , and so g ( ξ M , ξ M ) is constant along µ − p ( β ) and the result is proved. (cid:3) Proof of Theorem 3.
By Proposition 15 M = K × K β µ − p ( β ). By Proposition 14 it follows U C · x is compact for any x ∈ µ − p ( β ). Let x ∈ µ − p ( β ). By Proposition 12, U C · x is compact as welland µ p ( x ) = µ ( x ) = β . This implies K x = K ∩ U x = K ∩ U β = K β for any x ∈ µ − p ( β ) andso M = K/K β × µ − p ( β ). The Lie algebra of the center of G is contained in the Lie algebra ofthe center of U C . On the other hand, the Lie algebra of the center of U C is the complexificationof 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 − p ( β ), applyingthe same idea of the above proof we get the splitting M = K/K β × µ − p ( β ) is Riemannian. (cid:3) References [1]
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Bull. Amer. Math. Soc. (1969) 1121–1237. (Leonardo Biliotti) Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Universit`a diParma (Italy) Email address ::