Minimal symplectic atlases of Hermitian symmetric spaces
aa r X i v : . [ m a t h . S G ] M a r MINIMAL SYMPLECTIC ATLASES OF HERMITIANSYMMETRIC SPACES
ROBERTO MOSSA AND GIOVANNI PLACINI
Abstract.
In this paper we estimate the minimal number of Darboux chartsneeded to cover a Hermitian symmetric space of compact type M in termsof the degree of their embeddings in C P N . The proof is based on the recentwork of Y. B. Rudyak and F. Schlenk [20] and on the symplectic geometrytool developed by the first author in collaboration with A. Loi and F. Zuddas[14]. As application we compute this number for a large class of Hermitiansymmetric spaces of compact type. Introduction and statements of the main results
Consider the open ball of radius r , B n ( r ) = { ( x, y ) ∈ R n | n X j =1 x j + y j < r } in the standard symplectic space ( R n , ω ), where ω = P nj =1 dx j ∧ dy j . In [20] Y.B. Rudyak and F. Schlenk introduced the invariant S B ( M, ω ) for a closed symplecticmanifold (
M, ω ) of dimension 2 n defined by: S B ( M, ω ) := min { k | M = B ∪ · · · ∪ B k } , where B j is the image of a Darboux chart ϕ ( B n ( r j )) ⊂ M . This is the minimalnumber of symplectic charts needed to cover ( M, ω ). The problem of estimating thisnumber is closely related to two other problems, namely computing the Gromovwidth c G ( M, ω ) and the Lusternik-Schnirelmann category cat( M ) of M . Whilethe latter can be often computed or estimated very well, computing the former isan open and delicate matter. The Gromov width of a 2 n -dimensional symplecticmanifold ( M, ω ), introduced in [7], is defined as c G ( M, ω ) = sup (cid:8) πr (cid:12)(cid:12) ∃ ϕ : (cid:0) B n ( r ) , ω (cid:1) → ( M, ω ) (cid:9) Mathematics Subject Classification.
Key words and phrases.
Minimal symplectic atlases; Darboux chart; Gromov width; Hermitiansymmetric spaces of compact type.The first author was supported by Prin 2010/11 – Variet`a reali e complesse: geometria, topologiae analisi armonica – Italy. where ϕ is a symplectic embedding.By Darboux’s theorem c G ( M, ω ) is a positive number or ∞ . Computations andestimates of the Gromov width for various examples can be found in [2, 3, 4, 5, 7,8, 10, 11, 14, 15, 16, 17, 18, 19, 21, 24].We adopt the following notation from [14]. Notation:
From now on we shall use the shortening HSSCT to denote a Hermitiansymmetric space of compact type. Further, throughout the paper we shall denote by ω F S the canonical symplectic (K¨ahler) form on an irreducible HSSCT normalized sothat ω F S ( B ) ∈ {− π, π } when B is a generator of H ( M, Z ) , and by A the generatorfor which ω F S ( A ) = π . The following theorem and its two corollaries are the main results of this paper.
Theorem 1.
Let ( M, ω
F S ) be a n -dimensional HSSCT and let f : M ֒ → C P N beany holomorphic isometric immersion of M in C P N endowed with the Fubini–Studyform ω . Then(i) If deg( f ) ≥ n , then S B ( M, ω
F S ) = deg( f ) + 1 (ii) If deg( f ) < n , then max { n + 1 , deg( f ) + 1 } ≤ S B ( M, ω
F S ) ≤ n + 1 . As holomorphic isometric immersion f : M ֒ → C P N we can take, for example,the coherent states map described in Section 1.1. In particular when M is thecomplex Grassmannian one can take f equal to the Pl¨ucker embedding. We recallthe definition of degree of a holomorphic immersion in Section 2.1, while in Section2.2 we compute it for all irreducible HSSCT.The proof of Theorem 1 is based on the results obtained by Y. B. Rudyak andF. Schlenk in [20] about minimal atlases for compact symplectic manifolds togetherwith the explicit computation of the Gromov width given by the first author incollaboration with A. Loi and F. Zuddas in [14] and the properties of the symplecticduality map introduced by A. J. Di Scala and A. Loi in [6] which, in particular,give us a symplectic embedding of the noncompact dual (Ω , ω ) of ( M, ω
F S ) into(
M, ω
F S ).Using the explicit computation of the volume of a classical domain (Ω , ω ) givenby L. K. Hua in [9], we are able to prove the following corollary, which extends thecomputation of S B for the Grassmannians given in [20] to any classical irreducibleHSSCT. Before stating the corollary, we recall that a classical irreducible HSSCTis one of the following quotients of compacts Lie groups:I k,s = SU ( s ) /S ( U ( k ) × U ( s − k )) , II s = SO (2 s ) /U ( s ) , III s = Sp ( s ) /U ( s ) , INIMAL SYMPLECTIC ATLASES OF HERMITIAN SYMMETRIC SPACES 3 IV s = SO ( s + 2) /SO ( s ) × SO (2) . Corollary 2.
Let ( M, ω
F S ) be a classical irreducible HSSCT of dimension n .Then we have: S B (I k,s ) = deg( f ) + 1 , for ( k = 2 and s ≥ or k ≥ S B (II s ) = deg( f ) + 1 , for s ≥ S B (III s ) = deg( f ) + 1 , for s ≥ n + 1 ≤ S B (IV s ) ≤ n + 1 , for s ≥ . Otherwise, we have max { n + 1 , deg( f ) + 1 } ≤ S B ( M, ω
F S ) ≤ n + 1 . In the rank one case (i.e. M = C P n ), we can set f equal to the identity map,so that deg( f ) = 1. On the other hand, [20, Corollary 5.8] tells us that S B ( C P n , ω F S ) = n + 1 . The second corollary is a straightforward consequence of Theorem 1:
Corollary 3.
Let ( M × M , ω F S ) be a product of HSSCT of dimension n . If M × M is different from C P × C P n − and C P × C P , then S B ( M × M , ω F S ) = deg( f ) + 1 , where f : M × M ֒ → C P N is any holomorphic isometric immersion. Otherwise,we have max { n + 1 , deg( f ) + 1 } ≤ S B ( M, ω
F S ) ≤ n + 1 . Acknowledgments.
The authors would like to thank Professor Andrea Loi for hishelp and various stimulating discussions and Professor Felix Schlenk for his interestin our work and his valuable comments.1.1.
The coherent states map.
It is well know that an HSSCT M is a simply con-nected K¨ahler–Einstein manifold with strictly positive scalar curvature. Thereforethe integrality of ω FS π implies the existence of a polarizing holomorphic hermitianline bundle ( L, h ) on M such that c ( L ) = [ ω FS π ] and the Ricci curvature of h sat-isfies Ric( h ) = ω FS π (where Ric( h ) = − i π ∂∂ log ( h ( σ, σ )) in a local trivialization σ : U ⊂ M → L ). Consider the space H ( L ) consisting of global holomorphicsections s of L which are bounded with respect to h s, s i = k s k = Z M h ( s ( x ) , s ( x )) ω n n ! . R. MOSSA AND G. PLACINI As H ( L ) = { } , given an orthonormal basis { s , . . . , s N } ⊂ H ( L ) (with respect h· , ·i ), it is well defined the coherent states map , given by f : M → C P N f ( x ) = [ s ( x ) : · · · : s N ( x )] . The Fubini–Study form ω of C P N (normalized so that ω ( B ) ∈ {− π, π } , when B isa generator of H ( C P N , Z )) is given by ω = i ∂∂ log N X j =0 | Z j | , it follows that f ∗ ω = i ∂∂ log N X j =0 | s j ( x ) | = i ∂∂ log P Nj =0 h ( s j ( x ) , s j ( x )) h ( σ ( x ) , σ ( x )) ! = − i ∂∂ log ( h ( σ ( x ) , σ ( x ))) + i ∂∂ log N X j =0 h ( s j ( x ) , s j ( x )) π Ric( h ) + i ∂∂ log ǫ ( x ) = ω F S + i ∂∂ log ǫ ( x ) , where ǫ : M → R is the so called ǫ -function defined by ǫ ( x ) = N X j =0 h ( s j ( x ) , s j ( x )) , one can prove that the ǫ -function (see e.g. [12, Theorem 4.3]) is invariant withrespect the action of the group of holomorphic isometric transformation of ( M, ω
F S )which act transitively on M . Therefore the ǫ -function is constant and we concludethat f ∗ ω = ω F S . Proofs of Theorem 1, Corollary 2 and Corollary 3
Consider the following lower bound for S B ( M, ω ) given byΓ(
M, ω ) := $ V ol ( M, ω ) n ! c G ( M, ω ) n % + 1 , where ⌊ x ⌋ denote the maximal integer smaller than or equal to x . The followingtheorem summarizes the results about minimal atlases obtained in [20] that we needin the proof of Theorem 1. Theorem A (Rudyak–Schlenk [20]) . Let ( M, ω ) be a compact connected n -dimensionalsymplectic manifold.i) If Γ( M, ω ) ≥ n + 1 , then S B ( M, ω ) = Γ(
M, ω ) .ii) If Γ( M, ω ) < n + 1 then max { n + 1 , deg( f ) + 1 } ≤ S B ( M, ω ) ≤ n + 1 . INIMAL SYMPLECTIC ATLASES OF HERMITIAN SYMMETRIC SPACES 5
Proof of Theorem 1.
We start recalling the definition of the degree of anholomorphic immersion f : M → C P N . Suppose that dim( M ) = 2 n < N , bySard’s Theorem there exists a point q / ∈ f ( M ). Up to unitary transformation of C P N we can suppose q to be the point of coordinates [1 , , . . . , p k : C P k \ { q } → C P k − , p k ([ Z , . . . , Z k ]) = [ Z , . . . , Z k ] and define themap F : M → C P n by F = ˜ p ◦ f , where ˜ p = p n +1 ◦ · · · ◦ p N . The degree deg( f ) of f is by definition the degree deg( F ) of the map F , which is the integer number suchthat F ∗ [ M ] = deg( F )[ C P n ] ∈ H n ( C P n , Z ) . (2)What we need about deg( f ) is summarized in the following Lemma: Lemma 4. (W. Wirtinger [23], M. Barros, A. Ros, [1])
The degree deg( f ) is apositive integer such that Vol( M ) = deg( f )Vol( C P n ) , (3) where deg( f ) = 1 iff M is totally geodesic and deg( f ) = 2 iff f is congruent to thestandard embedding of the quadric. The proof follows from Theorem A once one observes that the volume of any n -dimensional projective variety X , with holomorphic embedding f : X ֒ → C P N ,is given by Vol( X, ω
F S ) = deg( f )Vol( C P n , ω F S ) , (4)Vol( C P n ) = π n n ! and that the Gromov width of any HSSCT (see [14]) is given by c G ( M, ω
F S ) = π .2.2. Proof of Corollary 2.
Consider (Ω , ω ), the noncompact dual of ( M, ω
F S ).In [6, Theorem 1.1] it is proved the existence of a global symplectomorphismΦ : (Ω , ω ) → ( M \ Cut p ( M ) , ω F S )where Cut p ( M ) is the cut locus of ( M, ω
F S ) with respect to a fixed point p ∈ M (see also [13]). Thus Vol( M, ω
F S ) = Vol(Ω , ω ). On the other hand the explicitexpression of the volume Vol(Ω , ω ) can be found in L. K. Hua [9] and by (4) weare able to write the expression of deg( f ) associated to any classical HSSCT, asfollows.Let I k,s be a HSSCT of type I, namely the Grassmannian of k -planes in C s .Notice that the dimension is 2 n = 2( s − k ) k and that rank(I k,s ) = k . We have thatdeg( f k,s ) = Vol(I k,s , ω F S )Vol( C P ( s − k ) k , ω F S ) == 1! 2! . . . ( s − k − . . . ( k − s − k ) k )!1! 2! . . . ( s − . (5) R. MOSSA AND G. PLACINI
The case I k,s was already done by Rudyak–Schlenk [20] and we obtain(1) by [20,Corollary 5.10]. Moreover they prove that S B (I , ) ∈ { , } S B (I , ) ∈ { , , , } . Let II s be an irreducible HSSCT of the second type. The complex dimension isgiven by n s = ( s − s . We have,deg( f II s ) = s ( s − . . . (2 s − s − s ! . . . (2 s − . In order to apply Theorem 1 we need to study whendeg( f II s ) n s ≥ s = 6 and that deg( f IIs ) n s < deg( f IIs+1 ) n s +1 for any s ≥ s be an irreducible HSSCT of the third type. The complex dimension isgiven by n s = ( s +1) s . We have,deg( f III s ) = s ( s + 1)2 ! 2! 4! . . . (2 s − s ! ( s + 1)! ( s + 2)! . . . (2 s − . Arguing as before we see that deg( f IIIs ) n s ≥ s ≥ s be an irreducible HSSCT of the fourth type (namely the complexquadric). Assume s > s = 1 or s = 2 we have respectively IV = C P orIV = C P × C P ). By Lemma 4, deg( f ) = 2. As n = s ≥
3, the result follows by( ii ) of Theorem 1.2.3. Proof of Corollary 3.
Let ω F S and ω F S be the Fubini-Study forms asso-ciated to M and M . Since the associated volume form satisfies (with abuse ofnotation) v ω FS = v ω FS ∧ v ω FS , we have Vol( M × M ) = Vol( M )Vol( M ). By (4)we get: deg( f ) = ( n + n )! n ! n ! deg( f ) deg( f ) , where n j is the complex dimension of M j , j = 1 , f , f and f are holomorphicisometric immersions of M × M , M and M . In order to apply ( i ) of Theorem 1,we have to check when deg( f ) deg( f ) ( n + n − n ! n ! ≥ . (6)First notice that when deg( f ) ≥ f ) ≥
2, since ( n + n − n ! n ! ≥
1, theinequality (6) is satisfied.
INIMAL SYMPLECTIC ATLASES OF HERMITIAN SYMMETRIC SPACES 7
Assume now that deg( f ) = deg( f ) = 1. By Lemma 4, f , f are totally geo-desic, this force M and M to have rank 1, that is M = C P n and M = C P n .Moreover it is easy to see that (6) is satisfied if and only if n ≥ n ≥ n ≥ n ≥
3. The proof is complete.
Remark 5.
When M = C P × C P n − , C P × C P we are not able to compute S B ( M, ω
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Dipartimento di Matematica e Informatica, Universit`a degli studi di Cagliari(Italy)
E-mail address : [email protected] (G. Placini) Dipartimento di Matematica, Universit`a di Pisa (Italy)
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