aa r X i v : . [ m a t h . DG ] D ec K ¨AHLER MANIFOLDS WITH SOME KILLING TENSORS.
W lodzimierz Jelonek
Abstract.
The aim of this paper is to classify compact, simply connected K¨ahlermanifolds which admit J-invariant Killing tensor with two eigenvalues of multiplicity2 and n-2 and with constant eigenvalue corresponding to 2-dimensional eigendistri-bution.
0. Introduction.
The aim of the present paper is to classify compact K¨ahlermanifolds (
M, g, J ), dimM = 2 n >
2, admitting a J -invariant Killing tensor S withtwo eigenvalues λ, µ of multiplicity 2 and 2 n − λ corresponding to the 2-dimensional eigendistribution.We show that if µ is a nonconstant function then such manifold admit a complex,holomorphic, totally geodesic foliation F by curves on an open dense subset of M (corresponding to the eigendistribution D λ of S ) and is a CP -holomorphic bundle P ( L ⊕ O ) over a K¨ahler Hodge manifold ( N, g, J ) or is a projective space CP n . Inthe first case both eigenvalues of S are everywhere distinct and in the second casethey coincide in an exactly one point.If both eigenvalues of S are constant then the totally geodesic distribution cor-responding to D λ may not be holomorphic. If dimM = 4 then also in the case ofconstant eigenvalues F is holomorphic and in that case if µ is constant then thesimply connected covering space ˜ M of M is the product Σ × N and the lift of S to ˜ M has eigendistributions T Σ , T N . If S is a Killing tensor with eigenvalues λ, µ where λ is constant then S ′ = S − λI is a Killing tensor with eigenvalues 0 , µ ′ . Hence we canassume that λ = 0. If S is a J-invariant Killing tensor with eigenvalues 0 , µ of mul-tiplicity 2 , n − φ ( X, Y ) = S ( JX, Y ) − µω ( X, Y )is a Hamiltonian form if the form S ( JX, Y ) − µω ( X, Y ) is closed (see [A-C-G-1],p.407). This last condition is satisfied if dimM = 4. In fact in [A-C-G-2] it isproved that if dimM = 4 and S is a symmetric J invariant tensor with arbitraryeigenvalues λ, µ then φ ( X, Y ) = S ( JX, Y ) − ( µ + λ ) ω ( X, Y ) is a Hamiltonian formif and only if S is a Killing tensor. We shall prove in the present paper that if µ is non-constant eigenvalue of a Killing tensor with eigenvalues 0 , µ of multiplic-ity 2 , n − φ ( X, Y ) = S ( JX, Y ) − µω ( X, Y ) is aHamiltonian K¨ahler 2-form also if dim
M > µ if dimM = 4. However if dimM > µ is constant. Note that Hamiltonian forms areclassified. MS Classification: 53C55,53C25. Key words and phrases: K¨ahler manifold, holomorphic foli-ation, homothetic foliation, special K¨ahler-Ricci potential, special K¨ahler potentialTypeset by
AMS -TEX W LODZIMIERZ JELONEK
1. Killing tensors.
Let us recall that a (1,1) symmetric tensor S on a Rie-mannian manifold ( M, g ) is called a Killing tensor if g ( ∇ S ( X, Y ) , Z ) + g ( ∇ S ( Z, X ) , Y ) + g ( ∇ S ( Y, Z ) , X ) = 0for all X, Y, Z ∈ T M . Equivalently it means that g ( ∇ S ( X, X ) , X ) = 0 for all X ∈ T M . Define the integer-valued function E S ( x ) = ( the number of distincteigenvalues of S x ) and set M S = { x ∈ M : E S is constant in a neighborhood of x } .The set M S is open and dense in M and the eigenvalues λ i of S are distinct andsmooth in each component U of M S . Let us denote by D λ i the eigendistributionscorresponding to λ i . We have (see [J-3]) Proposition 1.1.
Let S be a Killing tensor on M and U be a component of M S and λ , λ , .., λ k ∈ C ∞ ( U ) be eigenfunctions of S . Then for all X ∈ D λ i wehave (1.1) ∇ S ( X, X ) = − ∇ λ i k X k and D λ i ⊂ ker dλ i . If i = j and X ∈ Γ( D λ i ) , Y ∈ Γ( D λ j ) then (1.2) g ( ∇ X X, Y ) = 12
Y λ i λ j − λ i k X k . If T ( X, Y ) = g ( SX, Y ) is a Killing tensor on (
M, g ) and c is a geodesic on M then the function φ ( t ) = T ( ˙ c ( t ) , ˙ c ( t )) is constant on the domain of c . In fact φ ′ ( t ) = ∇ ˙ c ( t ) T (( ˙ c ( t ) , ˙ c ( t )) = 0.A distribution D ⊂
T M we call umbilical if 2 ∇ X X |D ⊥ = g ( X, X ) ξ for every X ∈ Γ( D ) and certain ξ ∈ Γ( D ⊥ ). A distribution D is called totally geodesic if isumbilical with ξ = 0. Lemma 1.
Let D be a complex, umbilical distribution J D ⊂ D and dim D = 2 .Then ∇ X Y |D ⊥ = g ( X, Y ) ξ + ω ( X, Y ) Jξ for all X, Y ∈ Γ( D ) .Proof. It is clear that ( ∇ X Y + ∇ Y X ) |D ⊥ = g ( X, Y ) ξ . Since 2 ∇ JX JX |D ⊥ = g ( X, X ) ξ it follows that 2 ∇ JX X |D ⊥ = − g ( X, X ) Jξ . Hence[ X, JX ] |D ⊥ = g ( X, X ) Jξ and [ X, Y ] |D ⊥ = ω ( X, Y ) Jξ for all X, Y ∈ Γ( D ). ♦ Thus we obtain
Lemma 2.
Let D be a complex, totally geodesic distribution (i.e. ∇ X X |D ⊥ = 0 for all X ∈ Γ( D ) ) and dim D = 2 . Then D is a totally geodesic foliation.Proof. It follows from Lemma 1, in our case ξ = 0. ♦ Corollary.
Let S be a J -invariant Killing tensor S with two eigenvalues λ, µ of multiplicity 2 and n-2 and with a constant eigenvalue λ corresponding to the2 - dimensional eigendistribution D on a K¨ahler manifold ( M, g, J ). Then D isintegrable and totally geodesic complex distribution. ¨AHLER MANIFOLDS WITH SOME KILLING TENSORS. 3 Remark.
We can always assume that λ = 0 considering the Killing tensor S ′ ( X, Y ) = S ( X, Y ) − λg ( X, Y ).A hamiltonian 2-form is a J invariant 2-form on the K¨ahler manifold ( M, g, J )such that there exists a function σ on M such that(1.2) ∇ X φ = 12 ( dσ ∧ JX − d c σ ∧ X ) . It is easy to check that dσ = dtr ω φ and we shall assume that σ = tr ω φ . If φ is a hamiltonian form then the tensor S ( X, Y ) = φ ( X, JY ) − σg ( X, Y ) is aKilling tensor. In fact from (1.2) it easily follows that C X,Y,Z ∇ X φ ( Y, JZ ) = C X,Y,Z dσ ( X ) g ( Y, Z ) where C X,Y,Z means cyclic sum. Hence if φ is a Hamiltonianform such that the symmetric tensor φ ( X, JY ) has two eigenvalues λ, , n − µ = 0 constant then the corresponding Killingtensor S ( X, Y ) = φ ( X, JY ) − σg ( X, Y ) has eigenvalues 0 , − λ corresponding to thesame eigendistributions.
2. Complex homothetic foliations.
We start with (see [V], [J-1] ):
Definition.
A foliation F on a Riemannian manifold ( M, g ) is called conformalif L V g = θ ( V ) g holds on T F ⊥ where V ∈ Γ( T F ) and θ is a one form vanishing on T F ⊥ . A foliation F is called homothetic if it is conformal and dθ = 0. Definition.
A foliation F on a Riemannian manifold ( M, g ) is called holomor-phic if L V J ( T M ) ⊂ T F for every V ∈ Γ( T F ). Theorem 2.1
Let us assume that F is a conformal, homothetic , totally geodesicfoliation by curves. Then F is holomorphic in U = { x ∈ M : θ x = 0 } .Proof. We have 2 ∇ X X = − g ( X, X ) θ ♯ for X ∈ T F ⊥ . On the other handfor X, Y ∈ Γ( T F ⊥ ) θ ([ X, Y ]) = − dθ ( X, Y ) = 0 hence [
X, Y ] | T F = α ( X, Y ) Jθ ♯ for some two form α on T F ⊥ . Hence 2 ∇ X Y = − g ( X, Y ) θ ♯ + α ( X, Y ) Jθ ♯ and2 ∇ X JY = − g ( X, JY ) θ ♯ + α ( X, JY ) Jθ ♯ . On the other hand2 ∇ X JY = − g ( X, Y ) Jθ ♯ − α ( X, Y ) θ ♯ . Hence α ( X, Y ) = g ( X, JY ) = − ω ( X, Y ) and 2 ∇ X Y = − g ( X, Y ) θ ♯ − ω ( X, Y ) Jθ ♯ .Let us recall that F is holomorphic if(2.1) ∇ JX V − J ∇ X V ∈ Γ( D )for every X ∈ T M and V ∈ Γ( D ) where D = T F . If X ∈ Γ( D ) then (2.1) issatisfied since F is totally geodesic. Let X, Y ∈ Γ( D ⊥ ). We have to show that g ( ∇ JX V − J ∇ X V, Y ) = 0 or equivalently that g ( V, ∇ JX Y + ∇ X JY ) = 0. But wehave ∇ JX Y + ∇ X JY |D = − g ( X, JY ) θ − ω ( JX, Y ) Jθ − g ( X, JY ) θ − ω ( X, JY ) Jθ = 0which means that F is holomorphic. ♦ Remark.
Note that if dimM = 4 then F is also holomorphic when θ = 0.It follows from Lemma 2 that D ⊥ is then totally geodesic and integrable hence( ∇ JX Y + ∇ X JY ) |D = 0 for X, Y ∈ Γ( D ⊥ ). It follows that the following theoremholds W LODZIMIERZ JELONEK
Theorem 2.2
Let ( M, g, J ) be a K¨ahler manifold, dim M=4 and let F be acomplex, conformal, homothetic and totally geodesic foliation on M . Then F isholomorphic. Theorem 2.3
Let ( M, g, J ) be a compact K¨ahler manifold and let F be a com-plex, conformal, homothetic and totally geodesic foliation on M such that θ doesnot vanish identically on M . Then F is holomorphic.Proof. Let U = { x ∈ M : | θ x | 6 = 0 } . Then exactly as in [J-1], [J-2] we show that U is dense in M . Since F is holomorphic on an open and dense subset it followsthat it is holomorphic in the whole of M . ♦
3. Examples of Killing tensor fields on compact K¨ahler manifolds.
First we give a definition
Definition.
A nonconstant function τ ∈ C ∞ ( M ) , where ( M, g, J ) is a K¨ahlermanifold, is called a special K¨ahler potential if the field X = J ( ∇ τ ) is a Killingvector field and, at every point with dτ = 0 all nonzero tangent vectors orthogonalto the fields X, JX are eigenvectors of ∇ dτ . If dimM = 4 we shall additionallyassume that the field ∇ τ is an eigenfield of Ricci tensor Ric of ( M, g, J ) . K¨ahler compact manifolds which admit special K¨ahler potential are in fact clas-sified in [D-M-2] ( see [J-1]). These are CP holomorphic bundles P ( L ⊕ O ) overHodge K¨ahler manifolds, where L is a holomorphic line bundle over K¨ahler Hodgemanifold ( N, ω, J ) or CP n .Let τ be a special K¨ahler potential on a compact K¨ahler manifold ( M, g, J ).Then (see [D-M-1], Prop.11.5) τ has exactly two critical manifolds, which are the τ -preimages of its extremum values τ max , τ min and one of the following two casesholds:(a) both critical manifolds of τ are of complex codimension one;(b) one of the critical manifolds has complex codimension 1 and the other is asingle point.What is more the function Q = g ( ∇ τ, ∇ τ ) is a composite consisting of τ followedby a C ∞ function [ τ min , τ max ] ∋ τ → Q ∈ R which is positive on the open interval( τ min , τ max ) and vanishes at the endpoints while the values of dQdτ at the endpointsare mutually opposite and nonzero. We shall denote | dQdτ | = 2 a = 0. Let us denote V = span {∇ τ, J ∇ τ } on U = { x ∈ M : dτ ( x ) = 0 } and let F be a foliation on U given by the integrabledistribution V . By Θ we denote the eigenvalue of the Hessian H τ correspondingto the distribution H = V ⊥ . Then F is a totally geodesic, holomorphic complexconformal foliation. We have θ = 2 Θ Q dτ, ζ = 2 Θ Q ∇ τ , where θ ( X ) = g ( ζ, X ) and Q = g ( ∇ τ, ∇ τ ). There exists a constant c such that(3.1) Q Θ = 2( τ − c )or Θ = 0. We shall assume that Θ = 0 on U . Note also that(3.2) dQ = 2Λ dτ. ¨AHLER MANIFOLDS WITH SOME KILLING TENSORS. 5 Let us assume that Θ = 0 and define a tensor S on M by SX = 0 if X ∈ V and SX = ( τ − c ) X if X ∈ H . If M = P ( L ⊕ O ) then both distributions V , H extendsto smooth distributions over M and clearly S is a smooth tensor on M . In thatcase the foliation tangent to V is a conformal, homothetic foliations by curves and θ = d ln | τ − c | . If M = CP n at the point x where τ ( x ) = c the distributions H , V are not defined and we define S x = 0. If ( M, g, J ) with special Killing potential τ admits a critical submanifold consisting of one point x then τ ( x ) = c and g ( SJX, Y ) = ( τ − c ) ω − τ − cQ dτ ∧ d c τ which means that S is smooth also in a neighborhood of x since lim x → x τ − cQ = dQdτ = ± a is finite and τ − cQ is smooth in a certain neighborhood of x . Note thatat points where dτ ( x ) = 0 but τ ( x ) = c the distributions H , V are defined.We also define Killing tensors with two constant eigenvalues on the products M = Σ × N where Σ is a complex surface and N is a K¨ahler manifold with dimN = n − V = T Σ , H = T N and SX = 0 for X ∈ V and SX = µX for X ∈ H where µ ∈ R is any real number.We prove now that the killing tensors constructed above are related to Hamil-tonian forms namely the 2-form φ = ( τ − c ) dτ ∧ d c τQ is a Hamiltonian form (see also[A-C-G] p.368 ) which means that (note that σ = tr ω φ = τ − c )(3.4) ∇ X φ = 12 ( dτ ∧ JX − d c τ ∧ X ) . Note that φ is defined on U = { x ∈ M : dτ ( x ) = 0 } and has a smooth extension onthe whole of M . On the critical manifold N with dim C N = n − τ − c ) ω where ω is the volume form of the foliation F and F extendssmoothly to N . If the critical manifold is { x } then the function τ − cQ is smooth at x and hence φ extends smoothly to x . We have ∇ X φ = dτ ( X ) dτ ∧ d c τQ − dQ ( X ) Q ( τ − c ) dτ ∧ d c τ (3.5) + τ − cQ ∇ X dτ ∧ d c τ + τ − cQ dτ ∧ ∇ X d c τ. We consider two cases:(a)If X ∈ D ⊥ then (note that Qτ − c = 2Θ) ∇ X φ ( Y, Z ) = 12Θ (Θ g ( X, Y ) d c τ ( Z ) − Θ g ( Z, X ) d c τ ( Y ))+12Θ (Θ g ( JX, Z ) dτ ( Y ) − Θ g ( JX, Y ) dτ ( Z ))hence ∇ X φ ( Y, Z ) = ( dτ ∧ JX − d c τ ∧ X )( Y, Z ) . (b) If X ∈ D then we can consider two subcases:(1) X = J ∇ τ In this case one can easily check that ∇ X φ = 0 hence again (3.4)holds, W LODZIMIERZ JELONEK (2) X = ∇ τ Then ∇ X φ ( Y, Z ) = dτ ∧ d c τ ( Y, Z ) − dτ ∧ d c τ ( Y, Z )+ Λ2Θ ( dτ ( Y ) d c τ ( Z ) − dτ ( Z ) d c τ ( Y )) + Λ2Θ ( dτ ( Y ) d c τ ( Z ) − dτ ( Z ) d c τ ( Y ))= dτ ∧ d c τ ( Y, Z )and again (3.4) holds.
4. Classification of Killing tensors.
Let (
M, g, J ) be a compact K¨ahlermanifold of real dimension 2 n . Let S be a J-invariant Killing tensor on ( M, g, J )with two eigenvalues 0 , µ and let U = { x ∈ M : µ ( x ) = 0 } . We assume that in U dimkerS = 2 , dimker ( S − µI ) = 2 n −
2. Then D = kerS is a totally geodesiccomplex foliation defined in U which is holomorphic in the set V = { x ∈ U : ∇ µ ( x ) = 0 } in view of Th.2.1. In fact since in U we have (see (1.2)) ∇ X X | D µ = 0if X ∈ Γ( D ) and ∇ X X |D = − g ( X, X ) ∇ µ µ if X ∈ Γ( D µ ) it follows that in U D is totally geodesic, conformal and homothetic foliation with θ = d ln | µ | . Let K = { x ∈ M : µ ( x ) = 0 } .Just as in [J-1] we prove that in V = { x ∈ U : dµ ( x ) = 0 } there exist locallydefined holomorphic Killing vector field with special K¨ahler potential τ such that θ = d ln | τ − c | = d ln | µ | . We can assume that τ − c = µ . Hence we see that thefield ξ = J ( ∇ µ ) is a holomorphic Killing vector field defined in the whole of V . Proposition 4.1.
The set V is connected and the set M − V has an emptyinterior. Proof.
Note that the function µ is the special Killing potential in V . Note that Q Θ = 2 µ where Q = g ( ξ, ξ ) and Θ is an eigenvalue of H µ corresponding to thedistribution D ⊥ . The function Θ is bounded on M and | Θ | ≤ sup X ∈ SM | H µ ( X, X ) | where SM = { X ∈ T M : g ( X, X ) = 1 } is compact unit subbundle of T M . Next weshow that the interior of the set M − V is empty and the set M − V is connected.Let V be a connected component of V and let x ∈ V . Let x ∈ ( M − V ) ∪ ( V − V ).Let c be a geodesic joining the points c (0) = x and c ( k ) = x . Let J be the Jacobifield along c which in V coincides with ξ . We show that g ( J, ˙ c ) = 0. Note that c intersects the set K = { x ∈ M : µ ( x ) = 0 } or { x ∈ U : | θ | = 0 } . Let l ∈ domc be such a number that c ([0 , l )) ⊂ V and c ( l ) ∈ K . Since Q ( c ( t )) = µ Θ( c ( t ))and | Θ | is bounded on M it follows that lim t → l − Q ( c ( t )) = 0. On the other handif c ( l ) ∈ { x ∈ U : | θ | = 0 } and c ([0 , l )) ⊂ V then exactly as in [J-1], [J-2]] weprove that lim t → l − Q ( c ( t )) = 0. Since g ( J, ˙ c ) is constant along c it follows that g ( J, ˙ c ) = 0. Now let us assume that there exists an open neighborhood W of x such that W ⊂ ( M − V ) ∪ ( V − V ). Let us assume that x = exp x X forsome X ∈ T x M . Then g ( X , ξ ( x )) = 0. We find an open neighborhood W ′ of X such that exp x ( W ′ ) ⊂ W . We find X ∈ W ′ such that g ( ξ ( x ) , X ) = 0.Hence the geodesic d ( t ) = exp x tX intersects the boundary of V and g ( J, ˙ d ) = 0 acontradiction. It follows that int ( M − V ) ∪ ( V − V ) = ∅ which means that V = V and int ( M − V ) = ∅ . ♦ Proposition 4.2.
The field ξ = J ∇ µ is a holomorphic Killing vector field onthe whole of M with a special Killing potential µ . ¨AHLER MANIFOLDS WITH SOME KILLING TENSORS. 7 Proof.
Note that H µ ( JX, JY ) = H µ ( X, Y ) on the open and dense subset V of M . From the continuity of H µ it follows that H µ ( JX, JY ) = H µ ( X, Y ) in thewhole of M which proves that ξ = J ( ∇ µ ) is a holomorphic Killing vector field on M . Since on M − V the field ∇ µ vanishes and µ is a special Killing potential on V it follows that µ is a special Killing potential on M . ♦ Theorem 4.1.
Let assume that a Killing tensor S on a compact K¨ahler manifold ( M, g, J ) has two eigenvalues , µ of multiplicity , n − respectively, where n = dimM . Let us assume that µ is not constant. Then µ = 0 on the whole of M and M = P ( L ⊕ O ) where L is a holomorphic line bundle over Hodge K¨ahlermanifold ( N, h, J ) with Chern connection with curvature Ω = 2 aω h or there existsone point x ∈ M such that µ ( x ) = 0 and then M is biholomorphic to the projectivespace CP n . In both cases manifold M admits special K¨ahler potential µ and φ = S ( JX, Y ) − µω ( X, Y ) is a global Hamiltonian 2-form such that φ = µ dµ ∧ d c µQ on U = { x : dµ ( x ) = 0 } .Proof. In the papers [D-M-1], [D-M-2] in fact there is given a classification ofcompact K¨ahler manifolds admitting a special K¨ahler potential if dimM ≥ dimM = 4 function µ is a special K¨ahler- Ricci potential(see [D-M-2],[J-1]). Hence it follows that the Killing vector field J ∇ µ has exactlytwo critical submanifolds which are both of (complex) codimension 1 and then isbiholomorphic to the manifold P ( L ⊕ O ) where L is a holomorphic line bundle overHodge manifold ( N, ω N ) with Chern connection with curvature Ω = 2 aω h or oneof two critical manifold is a point { x } and then M = CP n . Note that the point x we have µ ( x ) = 0 and this is the only point with this property ( the specialK¨ahler potential τ = µ satisfies τ ( x ) = c = 0 at x ).If dim M = 4 and µ is constant then simply connected covering space ˜ M of M isa product CP × N since in that case the foliation F is holomorphic (see Th.2.2). ♦ Theorem 4.2.
Let assume that a Hamiltonian 2-form φ on a compact K¨ahlermanifold ( M, g, J ) has two eigenvalues µ, of multiplicity , n − respectively,where n = dimM . Let us assume that µ is not constant. Then µ = 0 on thewhole of M and M = P ( L ⊕ O ) where L is a holomorphic line bundle over HodgeK¨ahler manifold ( N, h, J ) with Chern connection with curvature Ω = 2 aω h or thereexists one point x ∈ M such that µ ( x ) = 0 and then M is biholomorphic to theprojective space CP n . In both cases manifold M admits the special K¨ahler potential τ such that a Hamiltonian 2-form φ = ( τ − c ) dτ ∧ d c τQ on U = { x : dτ ( x ) = 0 } .If µ is constant then the simply connected covering space ˜ M of M is the product ˜ M = Σ × N and the lift of φ to M has eigendistributions T Σ , T N .Remark. If a compact K¨ahler manifold (
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