Symplectic maps of complex domains into complex space forms
aa r X i v : . [ m a t h . S G ] M a r Symplectic maps of complex domainsinto complex space forms Andrea LoiDipartimento di Matematica e Informatica – Universit`a di Cagliari – Italye-mail address: [email protected] ZuddasDipartimento di Matematica e Informatica – Universit`a di Cagliari – Italye-mail address: [email protected]
Abstract
Let M ⊂ C n be a complex domain of C n endowed with a rotationinvariant K¨ahler form ω Φ = i ∂ ¯ ∂ Φ. In this paper we describe sufficientconditions on the K¨ahler potential Φ for (
M, ω Φ ) to admit a symplecticembedding (explicitely described in terms of Φ) into a complex spaceform of the same dimension of M . In particular we also provide con-ditions on Φ for ( M, ω Φ ) to admit global symplectic coordinates. Asan application of our results we prove that each of the Ricci flat (butnot flat) K¨ahler forms on C constructed by LeBrun in [15] admitsexplicitely computable global symplectic coordinates. Keywords : K¨ahler metrics; diastasis function; complex space form;symplectic coordinates; Darboux theorem.
Subj.Class : 53C55, 58C25, 53D05, 58F06.
Let (
M, ω ) and ( S, Ω) be two symplectic manifolds of dimension 2 n and 2 N , n ≤ N , respectively. Then, one has the following natural and fundamentalquestion. Question 1.
Under which conditions there exists a symplectic embedding
Ψ : (
M, ω ) → ( S, Ω) , namely a smooth embedding Ψ : M → S satisfying Ψ ∗ (Ω) = ω ? During the preparation of this article the authors were supported by the M.I.U.R.Project “Geometric Properties of Real and Complex Manifolds”. N -dimensional complexspace form S , namely ( S, Ω) is either the complex Euclidean space ( C N , ω ),the complex hyperbolic space ( C H N , ω hyp ) or the complex projective space( C P N , ω F S ) (see below for the definition of the symplectic (K¨ahler) forms ω , ω hyp and ω F S ). Indeed these theorems are consequences of Gromov’sh-principle [12] (see also Chapter 12 in [9] for a beautiful description ofGromov’s work ).
Theorem A (Gromov [12], see also [10])
Let ( M, ω ) be a contractiblesymplectic manifold. Then there exist a non-negative integer N and a sym-plectic embedding Ψ : (
M, ω ) → ( C N , ω ) , where ω = P Nj =1 dx j ∧ dy j de-notes the standard symplectic form on C N = R N . This was further generalized by Popov as follows.
Theorem B (Popov [20])
Let ( M, ω ) be a symplectic manifold. Assume ω is exact, namely ω = dα , for a -form α . Then there exist a non-negativeinteger N and a symplectic embedding Ψ : (
M, ω ) → ( C N , ω ) . Observe that the complex hyperbolic space ( C H N , ω hyp ), namely theunit ball C H N = { z = ( z , . . . , z N ) ∈ C N | P Nj =1 | z j | < } in C N endowedwith the hyperbolic form ω hyp = − i ∂ ¯ ∂ log(1 − P Nj =1 | z j | ) is globally sym-plectomorphic to ( C N , ω ) (see (20) in Lemma 2.2 below) hence Theorem B immediately implies Theorem C
Let ( M, ω ) be a symplectic manifold. Assume ω is exact.Then there exist a non-negative integer N and a symplectic embedding Ψ :(
M, ω ) → ( C H N , ω hyp ) . The following theorem, further generalized by Popov [20] to the non-compact case, deals with the complex projective C P N , equipped with theFubini–Study form ω F S . Recall that if Z , . . . , Z N denote the homogeneouscoordinates on C P N , then, in the affine chart Z = 0 endowed with coordi-nates z j = Z j Z , j = 1 , . . . , N , the Fubini-Study form reads as ω F S = i ∂ ¯ ∂ log(1 + N X j =1 | z j | ) . heorem D (Gromov [10], see also Tischler [22] ) Let ( M, ω ) be acompact symplectic manifold such that ω is integral. Then there exist a non-negative integer N and a symplectic embedding Ψ : (
M, ω ) → ( C P N , ω F S ) . At this point a natural problem is that to find the smallest dimensionof the complex space form where a given symplectic manifold (
M, ω ) can besymplectically embedded. In particular one can study the case of equidimen-sional symplectic maps, as expressed by the following interesting question.
Question 2.
Given a n -dimensional symplectic manifold ( M, ω ) underwhich conditions there exists a symplectic embedding Ψ of ( M, ω ) into ( C n , ω ) or ( C P n , ω F S ) ? Notice that locally there are not obstructions to the existence of such Ψ.Indeed, by a well-known theorem of Darboux for every point p ∈ M thereexist a neighbourhood U of p and an embedding Ψ : U → R n = C n suchthat Ψ ∗ ( ω ) = ω . In order to get a local embedding into ( C P n , ω F S ) wecan assume (by shrinking U if necesssary) that Ψ( U ) ⊂ C H n . Therefore f ◦ Ψ : U → ( C n , ω F S ) ⊂ ( C P n , ω F S ), with f given by Lemma 2.2 below,is the desired embedding satisfying ( f ◦ Ψ) ∗ ( ω F S ) = Ψ ∗ ( ω ) = ω . Observealso that Darboux’s theorem is a special case of the following Theorem E (Gromov [13]) A n -dimensional symplectic manifold ( M, ω ) admits a symplectic immersion into ( C n , ω ) if and only if the following threeconditions are satisfied: a) M is open, b) the form ω is exact, c) the tangentbundle ( T M, ω ) is a trivial Sp (2 n ) -bundle. (Observe that a), b), c) aresatisfied if M is contractible). It is worth pointing out that the previous theorem is not of any help in or-der to attack Question 2 due to the existence of exotic symplectic structureson R n (cfr. [11]). (We refer the reader to [1] for an explicit construction ofa 4-dimensional symplectic manifold diffeomorphic to R which cannot besymplectically embedded in ( R , ω )).In the case when our symplectic manifold ( M, ω ) is a K¨ahler manifold,with associated K¨ahler metric g , one can try to impose Riemannian or holo-morphic conditions to answer the previous question. From the Riemannianpoint of view the only complete and known result (to the authors’ knowl-edge) is the following global version of Darboux’s theorem. Theorem F (McDuff [19])
Let ( M, g ) be a simply-connected and complete n -dimensional K¨ahler manifold of non-positive sectional curvature. Then here exists a diffeomorphism Ψ : M → R n such that Ψ ∗ ( ω ) = ω . (See also [4], [5], [6] and [8] for further properties of McDuff’s symplecto-morphism).The aim of this paper is to give an answer to Question 2 in terms ofthe K¨ahler potential of the K¨ahler metric of complex domains (open andconnected) M ⊂ C n equipped with a K¨ahler form ω which admits a rotationinvariant K¨ahler potential. More precisely, throughout this paper we assumethat there exists a K¨ahler potential for ω , namely a smooth function Φ : M → R such that ω = i ∂ ¯ ∂ Φ, depending only on | z | , . . . , | z n | , where z , . . . , z n are the standard complex coordinates on C n . Therefore, thereexists a smooth function ˜Φ : ˜ M → R , defined on the open subset ˜ M ⊂ R n given by˜ M = { x = ( x , . . . , x n ) ∈ R n | x j = | z j | , z = ( z , . . . z n ) ∈ M } (1)such that Φ( z , . . . , z n ) = ˜Φ( x , . . . , x n ) , x j = | z j | , j = 1 , . . . , n. We set ω := ω Φ and call ω Φ a rotation invariant symplectic (K¨ahler) formwith associated function ˜Φ. It is worth pointing out that many interestingexamples of K¨ahler forms on complex domains are rotation invariant (evenradial, namely depending only on r = | z | + · · · + | z n | ), since they oftenarise from solutions of ordinary differential equations on the variable r (seeExample 3.3 below and also [3] in the case of extremal metrics).Our first result is Theorem 1.1 below where we describe explicit condi-tions in terms of the potential Φ for the existence of an explicit symplecticembedding of a rotation invariant domain ( M, ω Φ ) into a given complexspace form ( S, ω Ξ ) of the same dimension. In particular we find conditionson Φ for the existence of global symplectic coordinates of ( M, ω Φ ). Theorem 1.1
Let M ⊆ C n be a complex domain such that condition M ∩ { z j = 0 } 6 = ∅ , j = 1 , . . . , n (2) is satisfied and let ω Φ = i ∂ ¯ ∂ Φ be a rotation invariant K¨ahler form on M with associated function ˜Φ : ˜ M → R . Then Obviously (2) is satisfied if 0 ∈ M , but there are other interesting cases, see Examples3.2 and 3.3 below, where this condition is fulfilled. i) there exists a uniquely determined special symplectic immersion Ψ : ( M, ω Φ ) → ( C n , ω ) (resp. Ψ hyp : ( M, ω Φ ) → ( C H n , ω hyp ) ) if and only if, ∂ ˜Φ ∂x k ≥ , k = 1 , . . . , n. (3) (ii) there exists a uniquely determined special symplectic immersion Ψ F S : (
M, ω Φ ) → ( C n , ω F S ) , if and only if ∂ ˜Φ ∂x k ≥ , k = 1 , . . . , n a nd n X j =1 ∂ ˜Φ ∂x j x j < , (4) where we are looking at C n i ֒ → C P n as the affine chart Z = 0 in C P n endowed with the restriction of the Fubini–Study form ω F S .Moreover, assume that ∈ M . If (3) (resp.(4)) is satisfied then Ψ (resp. Ψ F S ) is a global symplectomorphism (and hence i ◦ Ψ F S : M → C P n is asymplectic embedding) if and only if ∂ ˜Φ ∂x k > , k = 1 , . . . , n (5) and lim x → ∂M n X j =1 ∂ ˜Φ ∂x j x j = + ∞ ( resp. lim x → ∂M n X j =1 ∂ ˜Φ ∂x j x j = 1) . (6) See (7) in the next section for the definition of special map between complex domains. For a rotation invariant continuous map F : M → R we writelim x → ∂M ˜ F ( x ) = l ∈ R ∪ {∞} , x = ( x , . . . , x n ) , if, for k x k → + ∞ or z → z ∈ ∂M , we have k ˜ F ( x ) k → l , where ∂M denotes the boundaryof M ⊂ C n and ˜ F : ˜ M → R , ˜ M given by (1), is the continuous map such that F ( z , . . . , z n ) = ˜ F ( x , . . . , x n ) , x j = | z j | . , Ψ hyp and Ψ F S can be described explicitely(see (23), (24) and (25) below). This is a rare phenomenon. In fact the proofsof Theorems A, B, C, D and E above are existential and the explicit formof the symplectic embedding or symplectomorphism into a given complexspace form is, in general, very hard to find.Theorem 1.1 is an extension and a generalization of the results obtainedby the first author and Fabrizio Cuccu in [7] for complete Reinhardt domainsin C . Actually, all the results obtained there become a straightforwardcorollary of our Theorem 1.1 (see Example 3.1 in Section 3).Our second result is Theorem 1.2 below where we describe geometricconditions on Φ, related to Calabi’s work on K¨ahler immersions, which im-plies the existence of a special symplectic immersion of ( M, ω Φ ) in ( R n , ω ), n = dim C M (and in particular the existence of global symplectic coordi-nates of ( M, ω Φ )). Theorem 1.2
Let M ⊆ C n be a complex domain such that ∈ M endowedwith a rotation invariant K¨ahler form ω Φ . Assume that there exists a K¨ahler(i.e. a holomorphic and isometric) immersion of ( M, g Φ ) into some finite orinfinite dimensional complex space form, where g Φ is the metric associatedto ω Φ . Then, (5) is satisfied and hence there exists a special symplecticimmersion Ψ of ( M, ω Φ ) into ( C n , ω ) , which is a global symplectomorphismif and only if lim x → ∂M P nj =1 ∂ ˜Φ ∂x j x j = + ∞ . If P nj =1 ∂ ˜Φ ∂x j x j < then thereexists a symplectic immersion Ψ F S of ( M, ω Φ ) into ( C P n , ω F S ) which is anembedding if and only if lim x → ∂M P nj =1 ∂ ˜Φ ∂x j x j = 1 . The paper is organized as follows. In the next section we prove Theorem1.1 and Theorem 1.2. The later will follow by an application of Calabi’sresults, which will be briefly recalled in that section. Finally, in Section 3we apply Theorem 1.1 to some important cases. In particular we recoverthe results proved in [7] and we prove that each of the Ricci flat (but notflat) K¨ahler forms on C constructed by LeBrun in [15] admits explicitelycomputable global symplectic coordinates. Observe that this last resultcannot be obtained by Theorem F above (see Remark 3.6 below).6 Proof of the main results
The following general lemma, used in the proof of our main results Theorem1.1 and Theorem 1.2, describes the structure of a special symplectic immer-sion between two complex domains M ⊂ C n and S ⊂ C n endowed withrotation invariant K¨ahler forms ω Φ and ω Ξ respectively. In all the paper weconsider smooth maps from M into S of the formΨ : M → S, z (Ψ ( z ) = ˜ ψ ( x ) z , . . . , Ψ n ( z ) = ˜ ψ n ( x ) z n ) , (7) z = ( z , ..., z n ), x = ( x , . . . , x n ) , x j = | z j | for some real functions ˜ ψ j : ˜ M → R , j = 1 , . . . , n , where ˜ M ⊂ R n is given by (1). A smooth map like (7) willbe called a special map. Lemma 2.1
Let M ⊆ C n and S ⊆ C n be complex domains as above. Aspecial map Ψ : M → S, z (Ψ ( z ) , . . . , Ψ n ( z )) , is symplectic, namely Ψ ∗ ( ω Ξ ) = ω Φ , if and only if there exist constants c k ∈ R such that thefollowing equalities hold on ˜ M : ˜ ψ k ∂ ˜Ξ ∂x k (Ψ) = ∂ ˜Φ ∂x k + c k x k , k = 1 , . . . , n, (8) where ˜Φ (resp. ˜Ξ ) is the function associated to ω Φ (resp. ω Ξ ), and ∂ ˜Ξ ∂x k (Ψ) = ∂ ˜Ξ ∂x k ( ˜ ψ x , . . . , ˜ ψ n x n ) , k = 1 , . . . , n. Proof:
From ω Ξ = i n X i,j =1 ∂ ˜Ξ ∂x i ∂x j ¯ z j z i + ∂ ˜Ξ ∂x i δ ij ! x = | z | ,...,x n = | z n | dz j ∧ d ¯ z i one getsΨ ∗ ( ω Ξ ) = i n X i,j =1 ∂ ˜Ξ ∂x i ∂x j (Ψ)Ψ i ¯Ψ j + ∂ ˜Ξ ∂x j (Ψ) δ ij ! x = | z | ,...,x n = | z n | d Ψ j ∧ d ¯Ψ i , where ∂ ˜Ξ ∂x i ∂x j (Ψ) = ∂ ˜Ξ ∂x i ∂x j ( ˜ ψ x , . . . , ˜ ψ n x n ) .
7f one denotes byΨ ∗ ( ω Ξ ) = Ψ ∗ ( ω Ξ ) (2 , + Ψ ∗ ( ω Ξ ) (1 , + Ψ ∗ ( ω Ξ ) (0 , the decomposition of Ψ ∗ ( ω Ξ ) into addenda of type (2 , , (1 ,
1) and (0 ,
2) onehas:Ψ ∗ ( ω Ξ ) (2 , = i n X i,j,k,l =1 ∂ ˜Ξ ∂x i ∂x j (Ψ)Ψ i ¯Ψ j + ∂ ˜Ξ ∂x j (Ψ) δ ij ! ∂ Ψ j ∂z k ∂ ¯Ψ i ∂z l dz k ∧ dz l (9)Ψ ∗ ( ω Ξ ) (1 , = i n X i,j,k,l =1 ∂ ˜Ξ ∂x i ∂x j (Ψ)Ψ i ¯Ψ j + ∂ ˜Ξ ∂x j (Ψ) δ ij ! (cid:18) ∂ Ψ j ∂z k ∂ ¯Ψ i ∂ ¯ z l − ∂ Ψ j ∂ ¯ z l ∂ ¯Ψ i ∂z k (cid:19) dz k ∧ d ¯ z l (10)Ψ ∗ ( ω Ξ ) (0 , = i n X i,j,k,l =1 ∂ ˜Ξ ∂x i ∂x j (Ψ)Ψ i ¯Ψ j + ∂ ˜Ξ ∂x j (Ψ) δ ij ! ∂ Ψ j ∂ ¯ z k ∂ ¯Ψ i ∂ ¯ z l d ¯ z k ∧ d ¯ z l . (11)(Here and below, with a slight abuse of notation, we are omitting the factthat all the previous expressions have to be evaluated at x = | z | , . . . , x n = | z n | . ) Since Ψ j ( z ) = ˜ ψ j ( | z | , ..., | z n | ) z j , one has: ∂ Ψ i ∂z k = ∂ ˜ ψ i ∂x k z i ¯ z k + ˜ ψ i δ ik , ∂ Ψ i ∂ ¯ z k = ∂ ˜ ψ i ∂x k z k z i (12)and ∂ ¯Ψ i ∂ ¯ z k = ∂ ˜ ψ i ∂x k z k ¯ z i + ˜ ψ i δ ik , ∂ ¯Ψ i ∂z k = ∂ ˜ ψ i ∂x k ¯ z k ¯ z i , (13)By inserting (12) and (13) into (9) and (10) after a long, but straightforwardcomputation, one obtains:Ψ ∗ ( ω Ξ ) (2 , = i n X k,l =1 A kl z k ¯ z l dz k ∧ dz l (14)and 8 ∗ ( ω Ξ ) (1 , = i n X k,l =1 " ( A kl + A lk ∂ ˜Ξ ∂x k ∂x l (Ψ) ˜ ψ k ˜ ψ l )¯ z k z l + ∂ ˜Ξ ∂x k (Ψ) δ kl ˜ ψ k dz k ∧ d ¯ z l , (15)where A kl = ∂ ˜Ξ ∂x k (Ψ) ∂ ˜ ψ k ∂x l + ˜ ψ k n X j =1 ∂ ˜Ξ ∂x j ∂x k (Ψ) ∂ ˜ ψ j ∂x l | z j | . (16)Now, we assume thatΨ ∗ ( ω Ξ ) = ω Φ = i n X k,l =1 ∂ ˜Φ ∂x k ∂x l ¯ z k z l + ∂ ˜Φ ∂x l δ lk ! x = | z | ,...,x n = | z n | dz k ∧ d ¯ z l . Then the terms Ψ ∗ ( ω Ξ ) (2 , and Ψ ∗ ( ω Ξ ) (0 , are equal to zero. This isequivalent to the fact that (16) is symmetric in k, l .Hence, by setting Γ l = ˜ ψ l ∂ ˜Ξ ∂x l (Ψ) , l = 1 , . . . , n equation (15) becomesΨ ∗ ( ω Ξ ) (1 , = i n X k,l =1 " ( A kl + ∂ ˜Ξ ∂x k ∂x l (Ψ) ˜ ψ k ˜ ψ l )¯ z k z l + ∂ ˜Ξ ∂x k (Ψ) δ kl ˜ ψ k dz k ∧ d ¯ z l == i n X k,l =1 (cid:18) ∂ Γ l ∂x k ¯ z k z l + Γ k δ kl (cid:19) dz k ∧ d ¯ z l . (17)So, Ψ ∗ ( ω Ξ ) = ω Φ implies i n X k,l =1 (cid:18) ∂ Γ l ∂x k ¯ z k z l + Γ k δ lk (cid:19) dz k ∧ d ¯ z l = i n X k,l =1 ∂ ˜Φ ∂x k ∂x l ¯ z k z l + ∂ ˜Φ ∂x l δ kl ! dz k ∧ d ¯ z l . In this equality, we distinguish the cases l = k and l = k and get respec-tively ∂ Γ l ∂x k = ∂ ˜Φ ∂x k ∂x l ( k = l )and 9 Γ k ∂x k x k + Γ k = ∂ ˜Φ ∂x k x k + ∂ ˜Φ ∂x k . By defining A k = Γ k − ∂ ˜Φ ∂x k , these equations become respectively ∂A k ∂x l = 0 ( l = k )and ∂A k ∂x k x k = − A k . The first equation implies that A k does not depend on x l and so by thesecond one we have A k = Γ k − ∂ ˜Φ ∂x k = c k x k , (18)for some constant c k ∈ R , i.e.Γ k = ˜ ψ k ∂ ˜Ξ ∂x k (Ψ) = ∂ ˜Φ ∂x k + c k x k , k = 1 , . . . , n, namely (8).In order to prove the converse of Lemma 2.1, notice that by differentiat-ing (8) with respect to l one gets: ∂ ˜Φ ∂x k ∂x l − c k x k δ kl = A kl + ∂ ˜Ξ ∂x k ∂x l ˜ ψ k ˜ ψ l with A kl given by (16). By ∂ ˜Φ ∂x k ∂x l = ∂ ˜Φ ∂x l ∂x k and ∂ ˜Ξ ∂x k ∂x l ˜ ψ k ˜ ψ l = ∂ ˜Ξ ∂x l ∂x k ˜ ψ l ˜ ψ k one gets A kl = A lk . Then, by (14), the addenda of type (2,0) (and (0,2)) inΨ ∗ ( ω Ξ ) vanish. Moreover, by (16) and (17), it follows that Ψ ∗ ( ω Ξ ) = ω Φ . ✷ In the proof of Theorem 1.1 we also need the following lemma whoseproof follows by Lemma 2.1, or by a direct computation.
Lemma 2.2
The map f : C H n → C n given by ( z , . . . , z n ) z p − P ni =1 | z i | , . . . , z n p − P ni =1 | z i | ! (19)10 s a special global diffeomorphism satisfying f ∗ ( ω ) = ω hyp (20) and f ∗ ( ω F S ) = ω , (21) where, in the second equation, we are looking at C n i ֒ → C P n as the affinechart Z = 0 in C P n endowed with the restriction of the Fubini–Study form ω F S and where ω denotes the restriction of the flat form of C n to C H n ⊂ C n . We are now in the position to prove our first result.
Proof of Theorem 1.1
First of all observe that under assumption (2), the c k ’s appearing in the statement of Lemma 2.1 are forced to be zero. So, theexistence of a special symplectic immersion Ψ : M → S is equivalent to˜ ψ k ∂ ˜Ξ ∂x k (Ψ) = ∂ ˜Φ ∂x k , k = 1 , . . . , n. (22)If we further assume ( S = C n , ω Ξ = ω ), namely ˜Ξ = P nj =1 x j , then condi-tion (8) reduces to ˜ ψ k = ∂ ˜Φ ∂x k , k = 1 , . . . , n, and hence (3) follows by Lemma 2.1. Further Ψ is given by:Ψ ( z ) = s ∂ ˜Φ ∂x z , · · · , s ∂ ˜Φ ∂x n z n x i = | z i | (23)In order to prove (i) when ( S = C H n , ω Ξ = ω hyp ) observe that since thecomposition of two special maps is a special map it follows by (20) that theexistence of a special symplectic map Ψ : ( M, ω Φ ) → ( C H n , ω hyp ) gives riseto a special symplectic map f ◦ Ψ : (
M, ω Φ ) → ( C n , ω ). The later is uniquelydetermined by the previous case, i.e. Ψ = f ◦ Ψ. So Ψ hyp := Ψ = f − ◦ Ψ and since the inverse of f is given by f − : C n → C H n , z z p P ni =1 | z i | , . . . , z n p P ni =1 | z i | ! , hyp ( z ) = vuut ∂ ˜Φ ∂x P nk =1 ∂ ˜Φ ∂x k x k z , . . . , vuut ∂ ˜Φ ∂x n P nk =1 ∂ ˜Φ ∂x k x k z n x i = | z i | . (24)In order to prove (ii), notice that by (21) a special symplectic map Ψ :( M, ω Φ ) → ( C n , ω F S ) is uniquely determined by the special symplectic mapΨ = f − ◦ Ψ : (
M, ω Φ ) → ( C H n , ω ) ⊂ ( C n , ω ) and therefore (4) is astraightforward consequence of the previous case (i). Furthermore Ψ F S isgiven byΨ
F S ( z ) = vuut ∂ ˜Φ ∂x − P nk =1 ∂ ˜Φ ∂x k x k z , . . . , vuut ∂ ˜Φ ∂x n − P nk =1 ∂ ˜Φ ∂x k x k z n x i = | z i | (25)Finally, notice that conditions (5) and (6) for the special map (23) (resp.(25)) are equivalent to Ψ − ( { } ) = { } (resp. Ψ − F S ( { } ) = { } ) and to theproperness of Ψ (resp. Ψ F S ). Hence, the fact that under these conditionsΨ (resp. Ψ F S ) is a global diffeomorphism follows by standard topologicalarguments. ✷ Remark 2.3
Observe that, by Theorem 1.1, if (
M, ω Φ ) admits a specialsymplectic immersion into ( C n , ω F S ), then it admits a special symplecticimmersion in ( C n , ω ) (or ( C H n , ω hyp )). The converse is false even if onerestricts to an arbitrary small open set U ⊆ M endowed with the restrictionof ω Φ (see Remark 3.4 below).In order to prove our second result (Theorem 1.2) we briefly recall Cal-abi’s work on K¨ahler immersions and his fundamental Theorem 2.9. Werefer the reader to [2] for details and further results (see also [17] and [18]). Calabi’s work
In his seminal paper Calabi [2] gave a complete answerto the problem of the existence and uniqueness of K¨ahler immersions of aK¨ahler manifold (
M, g ) into a finite or infinite dimensional complex spaceform. Calabi’s first observation was that if such K¨ahler immersion existsthen the metric g is forced to be real analytic being the pull-back via aholomorphic map of the real analytic metric of a complex space form. Thenin a neighborhood of every point p ∈ M , one can introduce a very specialK¨ahler potential D gp for the metric g , which Calabi christened diastasis .12he construction goes as follows. Take a a real-analytic K¨ahler potential Φaround the point p (it exists since g is real analytic). By duplicating thevariables z and ¯ z Φ can be complex analytically continued to a function ˆΦdefined in a neighborhood U of the diagonal containing ( p, ¯ p ) ∈ M × ¯ M (here ¯ M denotes the manifold conjugated to M ). The diastasis function isthe K¨ahler potential D gp around p defined by D gp ( q ) = ˆΦ( q, ¯ q ) + ˆΦ( p, ¯ p ) − ˆΦ( p, ¯ q ) − ˆΦ( q, ¯ p ) . Example 2.4
Let g be the Euclidean metric on C N , N ≤ ∞ , namely themetric whose associated K¨ahler form is given by ω = i P Nj =1 dz j ∧ d ¯ z j .Here C ∞ is the complex Hilbert space l ( C ) consisting of sequences ( z j ) j ≥ , z j ∈ C such that P + ∞ j =1 | z j | < + ∞ . The diastasis function D g : C N → R around the origin 0 ∈ C N is given by D g ( z ) = N X j =1 | z j | . (26) Example 2.5
Let ( Z , Z , . . . , Z N ) be the homogeneous coordinates in thecomplex projective space in C P N , N ≤ ∞ , endowed with the Fubini–Studymetric g F S . Let p = [1 , , . . . , U = { Z = 0 } endowedwith coordinates ( z , . . . , z N ) , z j = Z j Z the diastasis around p reads as: D g F S p ( z ) = log(1 + N X j =1 | z j | ) . (27) Example 2.6
Let C H N = { z ∈ C N | P Nj =1 | z j | < } ⊂ C N , N ≤ ∞ be thecomplex hyperbolic space endowed with the hyperbolic metric g hyp . Thenthe diastasis around the origin is given by: D g hyp ( z ) = − log(1 − N X j =1 | z j | ) . (28)A very useful characterization of the diastasis (see below) can be obtainedas follows. Let ( z ) be a system of complex coordinates in a neighbourhoodof p where D gp is defined. Consider its power series development: D gp ( z ) = X j,k ≥ a jk ( g ) z m j ¯ z m k , (29)13here we are using the following convention: we arrange every n -tuple ofnonnegative integers as the sequence m j = ( m ,j , m ,j , . . . , m n,j ) j =0 , ,... such that m = (0 , . . . , | m j | ≤ | m j +1 | , with | m j | = P nα =1 m α,j and z m j = Q nα =1 ( z α ) m α,j . Further, we order all the m j ’s with the same | m j | using the lexicographic order.Characterization of the diastasis: Among all the potentials the diastasisis characterized by the fact that in every coordinates system ( z ) centered in p the coefficients a jk ( g ) of the expansion (29) satisfy a j ( g ) = a j ( g ) = 0 for every nonnegative integer j . Definition 2.7
A K¨ahler immersion ϕ of ( M, g ) into a complex space form ( S, G ) is said to be full if ϕ ( M ) is not contained in a proper complex totallygeodesic submanifold of ( S, G ) . Definition 2.8
Let g be a real analytic K¨ahler metric on a complex mani-fold M . The metric g is said to be resolvable of rank N if the ∞ × ∞ matrix a jk ( g ) given by (29) is positive semidefinite and of rank N . Consider thefunction e D gp − (resp. − e − D gp ) and its power series development: e D gp − X j,k ≥ b jk ( g ) z m j ¯ z m k . (30) (resp. − e − D gp = X j,k ≥ c jk ( g ) z m j ¯ z m k . ) (31) The metric g is said to be -resolvable (resp. − -resolvable) of rank N at p if the ∞ × ∞ matrix b jk ( g ) (resp. c jk ( g ) ) is positive semidefinite andof rank N . We are now in the position to state Calabi’s fundamental theorem andto prove our Theorem 1.2.
Theorem 2.9 (Calabi)
Let M be a complex manifold endowed with a realanalytic K¨ahler metric g . A neighbourhood of a point p admits a (full)K¨ahler immersion into ( C N , g ) if and only if g is resolvable of rank atmost (exactly) N at p . A neighbourhood of a point p admits a (full) K¨ahlerimmersion into ( C P N , g F S ) (resp. ( C H N , g hyp ) ) if and only if g is -resolvable (resp. − -resolvable) of rank at most (exactly) N at p . roof of Theorem 1.2 Without loss of generality we can assume Φ(0) = 0.Then, it follows by the characterization of the diastasis function that Φ isindeed the (globally defined) diastasis function for the K¨ahler metric g Φ (associated to ω Φ ) around the origin, namely Φ = D g Φ . Since Φ = D g Φ isrotation invariant, namely it depends only on | z | , . . . , | z n | , the matrices a jk ( g ), b jk ( g ) and c jk ( g ) above are diagonal, i.e. a jk ( g ) = a j δ jk , b jk ( g ) = b j δ jk , c jk ( g ) = c j δ jk , a j , b j , c j ∈ R . (32)Therefore, by Calabi’s Theorem 2.9 if ( M, g Φ ) admits a (full) K¨ahler immer-sion into ( C N , g ) (resp. ( C P N , g F S ) or ( C H N , g hyp )) then all the a ′ j s (resp.the b ′ j s or the c ′ j s ) are greater or equal than 0 and at most (exactly) N ofthem are positive. Moreover, it follows by the fact that the metric g Φ is posi-tive definite (at 0 ∈ M ) that the coefficients a k (resp. b k or c k ), k = 1 , . . . , n ,are strictly greater than zero . Hence, by using (29) (resp. (30) or (31)) with p = 0 and g = g Φ we get ∂ ˜Φ ∂x k ( x ) = a k + P ( x ) (resp. ∂ ˜Φ ∂x k ( x ) = b k + P + ( x )1+ P j b j x mj or ∂ ˜Φ ∂x k ( x ) = c k + P − ( x )1 − P j c j x mj ) where P (resp. P + or P − ) is a polynomial withnon-negative coefficients in the variables x = ( x , . . . , x n ) , x j = | z j | . Hencecondition (3) above is satisfied. The last two assertions of Theorem 1.2 areimmediately consequences of Theorem 1.1. ✷ Example 3.1 (cfr. [7]) Let x ∈ R + ∪ { + ∞} and let F : [0 , x ) → (0 , + ∞ )be a non-increasing smooth function. Consider the domain D F = { ( z , z ) ∈ C | | z | < x , | z | < F ( | z | ) } endowed with the 2-form ω F = i ∂∂ log F ( | z | ) −| z | . If the function A ( x ) = − xF ′ ( x ) F ( x ) satisfies A ′ ( x ) > x ∈ [0 , x ), then ω F is a K¨ahler formon D F and ( D F , ω F ) is called the complete Reinhardt domain associatedwith F . Notice that ω F is rotation invariant with associated real function˜ F ( x , x ) = log F ( x ) − x . We now apply Theorem 1.1 to ( D F , ω F ). We have ∂ ˜ F∂x = − F ′ ( x ) F ( x ) − x > , ∂ ˜ F∂x = 1 F ( x ) − x > , x j = | z j | , j = 1 , . D F , ω F ) admits a special symplectic immersion in( C , ω ) (and in ( C H , ω hyp )). Moreover, this immersion is a global sym-plectomorphism only when ∂ ˜ F∂x x + ∂ ˜ F∂x x = x − F ′ ( x ) x F ( x ) − x tends to infinity on the boundary of D F . For example, let F : [0 , + ∞ ) → R + given by F ( x ) = cc + x , c > F ( x ) = x ) p , p ∈ N + ). Then P i =1 ∂ ˜ F∂x i x i = x ( c + x ) + cx c ( c + x ) − x ( c + x ) (resp. P i =1 ∂ ˜ F∂x i x i = x + px (1+ x ) − p − (1+ x ) − p − x ) doesnot tend to infinity, for t → ∞ , along the curve x = t , x = εcc + t , for any ε ∈ (0 ,
1) (resp. does not tend to infinity, for t → ∞ , along the curve x = t , x = ε (1 + t ) − p , for any ε ∈ (0 , F : [0 , + ∞ ) → R + is given by F ( x ) = e − x (resp. F : [0 , → R + , F ( x ) = (1 − x ) p , p > P i =1 ∂ ˜ F∂x i x i = x + e − x x e − x − x (resp. P i =1 ∂ ˜ F∂x i x i = x + px (1 − x ) p − (1 − x ) p − x ) tends to infinity on the boundary of D F . We then recover the conclusions of Examples 3.3, 3.4, 3.5, 3.6 in [7]. Example 3.2
Let us endow C \ { } with the rotation invariant K¨ahlerform ω Φ = i ∂ ¯ ∂ Φ with associated real function˜Φ( x , x ) = a log( x + x ) + b ( x + x ) + c, a, b, c > . The metric g Φ associated to ω Φ is used in [21] (see also [16]) for the construc-tion of K¨ahler metrics of constant scalar curvature on bundles on C P n − .Since ∂ ˜Φ ∂x i = b + ax + x >
0, by Theorem 1.1 there exists a special sym-plectic immersion of ( C \ { } , ω Φ ) in ( C , ω ) (or in ( C H , ω hyp )). Example 3.3
Let us endow C \ { } with the metric ω Φ = i ∂ ¯ ∂ Φ, where˜Φ = p r + 1 + 2 log r − log( p r + 1 + 1) , r = p | z | + | z | . The metric g Φ is used in [14] for the construction of the Eguchi–Hansonmetric. A straight calculation shows that ∂ ˜Φ ∂x i = ∂ ˜Φ ∂r ∂r∂x i = (cid:20) r √ r + 1 (cid:18) − √ r + 1 + 1 (cid:19) + 2 r (cid:21) r > , so by Theorem 1.1 there exists a special symplectic immersion of( C \ { } , ω Φ ) in ( C , ω ) (or in ( C H , ω hyp )).16 emark 3.4 Notice that in the previous Example 3.3 one has ∂ ˜Φ ∂x x + ∂ ˜Φ ∂x x = r √ r + 1 (cid:18) − √ r + 1 + 1 (cid:19) + 1 > , so again by Theorem 1.1 it does not exist a special symplectic immersion of( C \ { } , ω Φ ) in ( C , ω F S ). Moreover, such an immersion does not exist forany arbitrarily small U ⊆ C \ { } endowed with the restriction of ω Φ (cfr.Remark 2.3 above). Example 3.5
In [15] Claude LeBrun constructed the following family ofK¨ahler forms on C defined by ω m = i ∂ ¯ ∂ Φ m , whereΦ m ( u, v ) = u + v + m ( u + v ) , m ≥ u and v are implicitly defined by | z | = e m ( u − v ) u, | z | = e m ( v − u ) v. For m = 0 one gets the flat metric, while for m > C having the same volume form of the flat metric ω , namely ω m ∧ ω m = ω ∧ ω . Moreover, for m >
0, these metrics are isometric (upto dilation and rescaling) to the Taub-NUT metric.Now, with the aid of Theorem 1.1, we prove that for every m the K¨ahlermanifold ( C , ω m ) admits global symplectic coordinates. Set u = U , v = V . Then ˜Φ m (the function associated to Φ m ) satisfies: ∂ ˜Φ m ∂x = ∂ ˜Φ m ∂U ∂U∂x + ∂ ˜Φ m ∂V ∂V∂x ,∂ ˜Φ m ∂x = ∂ ˜Φ m ∂U ∂U∂x + ∂ ˜Φ m ∂V ∂V∂x , where x j = | z j | , j = 1 ,
2. In order to calculate ∂U∂x j and ∂V∂x j , j = 1 ,
2, let usconsider the map G : R → R , ( U, V ) ( x = e m ( U − V ) U, x = e m ( V − U ) V )and its Jacobian matrix J G = (cid:18) (1 + 2 mU ) e m ( U − V ) − mU e m ( U − V ) − mV e m ( V − U ) (1 + 2 mV ) e m ( V − U ) (cid:19) .
17e have detJ G = 1 + 2 m ( U + V ) = 0, so J − G = J G − = 11 + 2 m ( U + V ) (cid:18) (1 + 2 mV ) e m ( V − U ) mU e m ( U − V ) mV e m ( V − U ) (1 + 2 mU ) e m ( U − V ) (cid:19) . Since J G − = ∂U∂x ∂U∂x ∂V∂x ∂V∂x ! , by a straightforward calculation we get ∂ ˜Φ m ∂x = (1 + 2 mV ) e m ( V − U ) > , ∂ ˜Φ m ∂x = (1 + 2 mU ) e m ( U − V ) > , and lim k x k→ + ∞ ( ∂ ˜Φ m ∂x x + ∂ ˜Φ m ∂x x ) = lim k x k→ + ∞ ( U + V + 4 mU V ) = + ∞ , namely (5) and (6) above respectively . Hence, by Theorem 1.1, the mapΨ : C → C , ( z , z ) (cid:16) (1 + 2 mV ) e m ( V − U ) z , (1 + 2 mU ) e m ( U − V ) z (cid:17) is a special global symplectomorphism from ( C , ω m ) into ( C , ω ). Remark 3.6
Notice that for m > C , ω m ). Indeed, the sectional curvature of ( C , g m ) (where g m is the K¨ahler metric associated to ω m ) is positive at some point since g m is Ricci-flat but not flat. Remark 3.7
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