A stability theorem for projective CR manifolds
aa r X i v : . [ m a t h . C V ] A p r A STABILITY THEOREM FOR PROJECTIVE CR MANIFOLDS
JUDITH BRINKSCHULTE, C. DENSON HILL, AND MAURO NACINOVICHA bstract . We consider smooth deformations of the CR structure of asmooth 2-pseudoconcave compact CR submanifold M of a reduced com-plex analytic variety X outside the intersection D ∩ M with the support D of a Cartier divisor of a positive line bundle F X . We show that nearbystructures still admit projective CR embeddings. Special results are ob-tained under the additional assumptions that X is a projective space or aFano variety. C ontents
1. Introduction 22. Preliminaries and notation 42.1. CR manifolds of type ( n , k ) 42.2. Natural projection onto the canonical bundle 62.3. CR maps 72.4. q -pseudoconcavity 82.5. Deformation of CR structures 83. ¯ ∂ M -cohomology 83.1. Di ff erential presentation of the tangential CR complex 93.2. CR line bundles 103.3. Deformations and CR line bundles 124. An example 135. Vanishing results 165.1. Embedded CR manifolds 165.2. Some estimates for abstract CR manifolds 175.3. Deformation of tangential CR complexes 195.4. Estimates for deformations 205.5. Vanishing theorems 216. Proof of Theorem 1.1 226.1. The general case 226.2. Generic CR submanifolds of the projective space 236.3. Generic CR submanifolds of Fano varieties 24References 25
1. I ntroduction
In this paper we study projective CR manifolds, meaning those that havea CR embedding into some CP N . We consider a small CR deformation ofsuch an M , and investigate under what conditions the deformed CR manifoldstill has a projective CR embedding. Our main result is the following CR -embedding theorem. Theorem 1.1.
Let X be an ( n + k ) -dimensional reduced complex analyticvariety and D the support of a Cartier divisor of its positive line bundle F X . Let M be a smooth compact submanifold of real dimension n + k of X reg and { M t } t ∈ R a family of CR structures of type ( n , k ) on M , smoothly depend-ing on a real parameter t . We assume that (1) M is induced by the embedding M ֒ → X ; (2) M is -pseudoconcave; (3) the CR structures of all M t agree to infinite order on M ∩ D.Then we can find ǫ > such that, for every | t | < ǫ , M t admits a generic CRembedding into a projective complex variety X t . More precise results are obtained under the additional assumptions that X is a projective space or a Fano variety (Theorems 6.1 and 6.2).The example of § M is not su ffi cient to obtain the statement of Theorem 1.1.Precise definitions and explanation of terms used will be found in the fol-lowing sections; let us start by explaining how our main theorem relates toother results in the field of CR geometry or analysis on CR manifolds.The notion of a CR structure of type ( n , k ) on a di ff erentiable manifold M of real dimension 2 n + k arises very naturally in two, a priori di ff erent,contexts. One is quite geometric in nature, and the other is more analyti-cal, being connected with fundamental questions about partial di ff erentialequations.From the geometric perspective, a CR structure of type ( n , k ) on a smoothreal submanifold M of dimension 2 n + k of some complex manifold X ofcomplex dimension n + k is induced as the tangential part of the ambientcomplex structure on X . In particular, one has the tangential Cauchy-Riemannequations on M . In this situation M is said to be generically CR embedded in X . It is also possible that M might be embedded in a complex manifold X ofcomplex dimension greater than n + k , with the CR structure on M also beinginduced as the tangential part of the ambient complex structure in X . In thissituation the real codimension of M is greater than the CR dimension k , and M is said to be non generically CR embedded in X . This is an importantdistinction only globally, because: if M can be locally non generically CRembedded, then M can be locally generically CR embedded (see e.g. [25]). STABILITY THEOREM FOR PROJECTIVE CR MANIFOLDS 3
From the PDE perspective, one begins with (at least locally) a systemof n smooth complex vector fields defined on M , which are linearly inde-pendent over C , and satisfy a formal integrability condition. Again n is the CR dimension, and k is the CR codimension. One then asks if the givencomplex vector fields might represent (in a local basis) the tangential CR equations on M given by some (local) CR embedding of M . The criterionfor this, is that one must be able to find (locally) a maximal set of func-tionally independent characteristic coordinates, which means to find n + k independent complex valued functions that are solutions of the homoge-neous system associated to the n complex vector fields. When the vectorfields can be chosen to have real analytic coe ffi cients, then this is possible([1]). Hence locally, in the real analytic case, the geometric perspective andthe analytical PDE perspective are equivalent. The famous example of L.Nirenberg [33] showed that the two perspectives are not locally equivalentif the coe ffi cients of the vector fields are assumed to be only smooth. Whydoes this matter? From an analytical PDE viewpoint, it is too restrictive toallow only real analytic structures, one needs cut o ff functions, partitionsof unity; etc., in order to employ calculus and modern analysis. So fromthis perspective it is more natural to define what is known as an abstract CR manifold M of type ( n , k ), by reformulating the local vector field basisdescription given above into a global and invariant version, and to do sowithin the smooth category. Precise definitions and details are below.The example of Nirenberg [33] was a single complex vector field in R ,so it endows R with the structure of an abstract CR manifold of type (1 , C . In other words, the generically CR embedded and real analytic CR structure of type (1 ,
1) on the sphere S , inherited from C , becomes not even locally CR embeddable near somepoint, if it is perturbed in a smooth way to obtain a very nearby abstract CR structure of type (1 ,
1) on the same S . Here the original CR structure on S is strictly pseudoconvex, and remains so under the small perturbation.Rossi [36] constructed small real analytic deformations of the CR struc-ture on S in C , such that the resulting perturbed abstract CR structuresfail to CR embed globally into C , even though they do locally CR embedinto C . On the other hand, sticking with strictly pseudoconvex CR struc-tures of hypersurface type, i.e. type ( n , , Boutet de Monvel proved thatwhen M is compact and n >
1, then M has a CR embedding into C N forsome perhaps large N . Catlin and Lempert [15] showed that there are com-pact strictly pseudoconvex CR manifolds of hypersurface type ( n , n ≥
1, that are CR embedded into some C N , which admit small deforma-tions that are also embeddable, but their embeddings cannot be chosen closeto the original embedding. Thus these examples exhibit the phenomenon of instability of the CR embedding. On the other hand, for n =
1, Lempert[31] proved that if a compact strictly pseudoconvex M is CR embeddable J. BRINKSCHULTE, C.D. HILL, AND M. NACINOVICH into C , hence generically embedded, and if the CR structure gets perturbedby a family of CR embeddable CR structures, then the embedding is sta-ble . This means that the embedding of the perturbed structure stays closeto the unperturbed one. Additional references to other work on the strictlypseudoconvex hypersurface type case can be found in the above mentionedpapers.In the higher codimensional case ( k >
1) Polyakov [35] proved a stabil-ity theorem for compact 3-pseudoconcave CR manifolds M , generically CR embedded in a complex manifold, under some additional hypotheses,such as the vanishing of a certain cohomology group. The notion of flex-ible / inflexible, which is closely related to the notion of stability / instability,was introduced in [10], also for the higher codimension case, where theemphasis was on CR manifolds M that are not compact. A CR manifold M of type ( n , k ), generically CR embedded in C n + k , is called flexible if it ad-mits a compactly supported CR deformation whose deformed structure is nolonger CR embeddable in C n + k ; in other words, the deformation causes the CR structure to “flex out” of the space where the original manifold lies. Onthe other hand, M is inflexible if any compactly supported deformation staysin the class of manifolds globally CR embeddable into C n + k . In [10], [11] itwas proved that if M is a CR submanifold of type ( n , k ) in C n + k , and it is only2-pseudoconcave, then M is inflexible. This relates to the present paper asfollows: If a CR structure of a 2-pseudoconcave compact CR manifold of areduced complex analytic variety gets perturbed by a smooth family of CR structures that “glues to” the unperturbed structure along a Cartier divisor ofa positive line bundle, then the perturbed CR structures do not “flex out” ofthe class of globally CR embeddable CR manifolds (Theorem 1.1). More-over, the embedding is stable if the unperturbed CR manifold sits generi-cally in a projective space (Theorem 6.1).We emphasize that in this paper we discuss compact CR manifolds andwe allow higher codimension and that we are interested in global results,not local ones. 2. P reliminaries and notation CR manifolds of type ( n , k ) . Let M be a smooth manifold of real di-mension m and n , k nonnegative integers with 2 n + k = m . A CR structure oftype ( n , k ) on M is a formally integrable distribution T , M of n -dimensionalcomplex subspaces of the complexified tangent bundle C T M of M , transver-sal to T M . By this transversality condition, the real parts of vectors in T , M are the elements of the vector CR distribution H M and for each p ∈ M and v ∈ H M , p (subscript “p” means “the fiber at p”) there is a unique v ′ ∈ H M , p such that v + i v ′ ∈ T , M , p . The correspondence v → v ′ defines a bundle self mapJ M : H M → H M which yields a complex structure on the fibers of H M , asJ M = − I H M . The formal integrability condition means that(2.1) [ Γ ∞ ( M , T , M ) , Γ ∞ ( M , T , M )] ⊆ Γ ∞ ( M , T , M ) , STABILITY THEOREM FOR PROJECTIVE CR MANIFOLDS 5 which is equivalent to the vanishing of the Nijenhuis tensor on H M , i.e.J M ([ X , Y ] − [J M X , J M Y ]) = [J M X , Y ] + [ X , J M Y ] , ∀ X , Y ∈ Γ ∞ ( M , H M ) . We use the standard notationT , M = { X − i J M X | X ∈ H M } ⊂ C T M , T , M = { X + i J M X | X ∈ H M } ⊂ C T M , T ∗ , M = { φ ∈ C T ∗ M | φ ( Z ) = , ∀ Z ∈ Γ ∞ ( M , T , M ) } , T ∗ , M = { φ ∈ C T ∗ M | φ ( Z ) = , ∀ Z ∈ Γ ∞ ( M , T , M ) } , H M = { ξ ∈ T ∗ M | ξ ( X ) = , ∀ X ∈ Γ ∞ ( M , H M ) } . We haveT , M ∩ T , M = { } , T , M + T , M = C H M , T ∗ , M ∩ T ∗ , M = C H M , T ∗ , M + T ∗ , M = C T ∗ M . Deformations of CR structures were considered especially for the hyper-surface-type case k = CR hypersurfaces can be described by smooth curves { t → J t } of formally integrable complex structures on a fixed contact distribution H M of a smooth real manifold M . Di ff erently, generalized contact distributionsof higher codimension may be not locally equivalent. Therefore, to describegeneral deformations of abstract CR manifolds of type ( n , k ) with arbitrary k it will be convenient to focus our attention on their canonical bundles ,which carry comprehensive information on both their contact and partialcomplex structures, as explained below.Denote by A M the sheaf of Grassmann algebras of germs of complexvalued smooth exterior di ff erential forms on M . The subsheaf A q M of homo-geneous elements of degree q consists of the germs of smooth sections ofthe C -vector bundle Λ q ( C T ∗ M ) of complex q -covectors on M . We can identify the Grassmannian Gr C n ( M ) of n -dimensional complexsubspaces of C T M with the subbundle of the projective bundle P ( Λ n + k ( C T ∗ M ))whose representatives belong to the cone Mon ( Λ n + k ( C T ∗ M )) of degree n + k monomials of the Grassmann algebra of C T ∗ M (see e.g. [14]). Then adistribution of complex n -planes of C T M is the datum of a smooth complexline bundle K M on M , with K M ⊂ Mon ( Λ n + k ( C T ∗ M )) . Definition 2.1.
Let ( M , T , M ) be an abstract CR manifold of type ( n , k ) . Thecomplex line bundle K M corresponding to the distribution T , M of its anti-holomorphic tangent vectors is called its canonical bundle . Notation 2.1.
Let us denote by K M the subsheaf of ideals in A M generatedby the sheaf K M of germs of smooth sections of K M . We recall that, if V is a vector space, the monomials of degree q of its exterior algebra Λ ∗ V are the exterior products v ∧ · · · ∧ v q of vectors v , . . . , v q of V . J. BRINKSCHULTE, C.D. HILL, AND M. NACINOVICH
Proposition 2.1.
A CR structure of type ( n , k ) on M is completely deter-mined by the datum of a complex line bundle K M ⊂ Mon ( Λ n + k ( C T ∗ M )) suchthat the ideal sheaf K M of A M generated by the germs its smooth sectionssatisfies the following conditions: (1) K M is closed, i.e. d K M ⊆ K M ( formal integrability ); (2) { X ∈ Γ ∞ ( M , T M ) | X ⌋ K M ⊆ K M } = { } ( transversality to T M ).Indeed, when (1) and (2) are valid, K M is the canonical bundle of the CRmanifold ( M , T , M ) with (2.2) Γ ∞ ( M , T , M ) = { Z ∈ Γ ∞ ( M , C T M ) | Z ⌋ K M ⊆ K M } Proof.
In fact, (1) is equivalent to the formal integrability (2.1) and (2) tothe transversality of T , M to the real tangent distribution T M . (cid:3) By the Newlander-Nirenberg theorem ([34]) abstract CR manifolds oftype ( n ,
0) are n -dimensional complex manifolds.2.2. Natural projection onto the canonical bundle.
On a CR manifold M of type ( n , k ) we have an exact sequence of smooth bundle maps(2.3) 0 −−−−−→ C H M −−−−−→ T ∗ , M ⊕ T ∗ , M −−−−−→ C T ∗ M −−−−−→ , where C H M ∋ ξ −→ ( ξ , ξ ) ∈ T ∗ , M ⊕ T ∗ , M , T ∗ , M ⊕ T ∗ , M ∋ ( ξ , ξ ) −→ ξ − ξ ∈ C T ∗ M . A CR gauge is a bundle map(2.4) λ : C T ∗ M −→ T ∗ , M , such that ξ − λ ( ξ ) ∈ T ∗ , M , ∀ ξ ∈ C T ∗ M . Note that, in particular, λ (T ∗ , M ) ⊆ C H M and ξ − λ ( ξ ) ∈ C H M , ∀ ξ ∈ T ∗ , M . Therefore a CR gauge yields a short exact sequence inverting (2.3):0 −−−−−→ C T ∗ M ( λ / ⊕ (id − λ / −−−−−−−−−−→ T ∗ , M ⊕ T ∗ , M ( λ / − id , λ / −−−−−−−−→ C H M −−−−−→ , where id denotes the identity on C T ∗ M . There are several possible choices of λ . A CR gauge λ is said to be of thereal type if(2.5) λ ( ξ ) + λ ( ξ ) = · ξ , ∀ ξ ∈ C T ∗ M and balanced if(2.6) λ ( ξ ) = ξ , ∀ ξ ∈ H M . Existence of balanced CR gauges of the real type was proved in [32, § Proposition 2.2.
The bundle map λ of a balanced CR gauge • is a projection of C T ∗ M onto T ∗ , M ; STABILITY THEOREM FOR PROJECTIVE CR MANIFOLDS 7 • its restriction to T ∗ M is an isomorphism with a real complement ofi · H M in T ∗ , M . Let us define a linear map(2.7) λ p : Λ p ( C T ∗ M ) → Λ p (T ∗ , M )in such a way that on the monomials we have λ p ( ξ ∧ · · · ∧ ξ p ) = λ ( ξ ) ∧ · · · ∧ λ ( ξ p ) . Proposition 2.3.
The map λ n + k defines a projection of Λ n + k ( C T ∗ M ) onto K M , which is independent of the particular choice of a balanced CR gauge λ ofthe real type.Proof. Fix a point p of M and a basis( ‡ ) ξ , . . . , ξ n , τ , . . . , τ k , ζ , . . . , ζ n of C T ∗ M , p with ξ i ∈ T ∗ , M , p , for 1 ≤ i ≤ n , τ j ∈ H M for 1 ≤ j ≤ k and ζ i ∈ T ∗ , M , p for1 ≤ i ≤ n . If λ is a balanced CR gauge of the real type, then λ ( ξ i ) = ξ i for 1 ≤ i ≤ n , λ ( τ j ) = τ j for 1 ≤ j ≤ k , λ ( ζ i ) ∈ C H M for 1 ≤ i ≤ n . Hence the image by λ n + k of allmonomials of degree n + k in the elements of the basis ( ‡ ) which contain afactor ζ i contain at least k + C H M and is therefore 0 , while λ n + k ( ξ ∧ · · · ∧ ξ n ∧ τ ∧ · · · ∧ τ k ) = ξ ∧ · · · ∧ ξ n ∧ τ ∧ · · · ∧ τ k . This completes the proof. (cid:3)
Definition 2.2.
We call λ n + k the natural projection onto the canonical bun-dle.2.3. CR maps. Let ( M i , K M i ) be abstract CR manifolds of type ( n i , k i ) ( i = , f : M → M a smooth map and f ∗ : C T M → C T M the bundle map obtainedby complexifying its di ff erential. Definition 2.3.
The map f is said to be: • CR if f ∗ (T , M ) ⊆ T , M ; • a CR immersion ( embedding ) if it is a smooth immersion (embed-ding) and moreover T , M = { Z ∈ C T M | f ∗ ( Z ) ∈ T , M } . • a CR submersion if it is a smooth submersion and f ∗ (T , M ) = T , M . A CR immersion (embedding) is called generic when n + k = n + k . On an abstract CR manifold M of type ( n , k ) , we consider the Grassmannsubalgebra Ω ∗ M of Λ ∗ ( C T ∗ M ) generated by T ∗ , M : Ω ∗ M = X n + k p = Ω p M , with Ω p M = Ω ∗ M ∩ Λ p ( C T ∗ M ) . With this notation, K M = Ω n + k M . CR maps and immersion can be described byusing the bundles Ω ∗ M . Proposition 2.4.
Let M i be abstract CR manifolds of type ( n i , k i ) , for i = , and f : M → M a smooth map. Then f is J. BRINKSCHULTE, C.D. HILL, AND M. NACINOVICH • CR i ff f ∗ ( Ω M ) ⊆ Ω M ; • a CR immersion i ff f is a smooth immersion and f ∗ ( Ω n + k M ) = K M . A generic CR-immersion f : M → M is characterized by (2.8) f ∗ ( K M ) = K M . (cid:3) q -pseudoconcavity. Let q be a nonegative integer. Following [21], wesay that a CR manifold M , of type ( n , k ) , is q -pseudoconcave if, for everyp ∈ M and every characteristic covector ξ ∈ H M \{ } , the scalar Levi form L p ( ξ , · ) has at least q negative and q positive eigenvalues.We recall that L p ( ξ , v ) = ξ ([J M X , X ]) = d ˜ ξ ( v , J M ( v )) , for v ∈ H p M , with X ∈ Γ ∞ ( M , H M ) , ˜ ξ ∈ Γ ∞ ( M , H M ) and X p = v , ˜ ξ p = ξ . The CR dimension n of a q -pseudoconcave CR manifold is at least 2 q (more precise lower bounds on n , depending also on the CR codimension k , can be obtained e.g. from [7]).Various equivalent definitions of the Levi form and more of its aspectsare explained in [13].2.5. Deformation of CR structures. With the preparation of § CR deformations of a CR manifold M of general type ( n , k ) in termsof canonical bundles, allowing in this way also a deformation of the under-lying contact structure H M . Definition 2.4.
A smooth one-parameter deformation of a CR structure K M on M is a smooth map M × ( − ǫ , ǫ ) ∋ (p , t ) → [ K M t , p ] ∈ P ( Λ n + k ( C T ∗ )) suchthat, for each t ∈ ( − ǫ , ǫ ) , K M t is the canonical complex line bundle of a CR structure on M and K M = K M . Line bundles over a contractible base are trivial (see e.g. [27, Cor.3.4.8]).Hence, for a compact M , we can find a finite open covering U = { U j } todescribe the deformation by the data of smooth maps(2.9) M × ( − ǫ , ǫ ) ∋ (p , t ) → ω j (p , t ) ∈ Λ n + k C T ∗ M such that ω j , t ( · , t ) ∈ Γ ∞ ( U j , K M t ) for each t ∈ ( − ǫ , ǫ ) .
3. ¯ ∂ M - cohomology Let M be an abstract CR manifold of type ( n , k ) . The ideal sheaf I M of A M generated by the germs of smooth sections ofT ∗ , M is, according to [14], the characteristic system of T , M . Hence, formalintegrability is equivalent to d ( I M ) ⊆ I M . STABILITY THEOREM FOR PROJECTIVE CR MANIFOLDS 9
For nonnegative integers p , its exterior powers I p M are also d -closed idealsand we can consider the quotients of the de Rham complex:( ∗ ) ¯ ∂ M : I p M / I p + M −→ I p M / I p + M (we put I M = A M ). Since I p M = p > n + k , the map ( ∗ ) is nontrivial if andonly if 0 ≤ p ≤ n + k . The canonical Z -grading of A M induces a Z -grading I p M = L n q = I p , q M of I p M , with I p , q M = I p M ∩ A p + q M . Let us set Q p , ∗ M = I p M / I p + M . By passing to thequotients, the Z -gradings of the exterior powers of I M induce the Z -gradings Q p , ∗ M = M nq = Q p , q M . For each pair of integers ( p , q ) with 0 ≤ p ≤ n + k , ≤ q ≤ n the summand Q p , q M isthe sheaf of germs of smooth sections of a complex vector bundle Q p , q M on M . Then for each 0 ≤ p ≤ n + k , the map ( ∗ ) is a di ff erential complex → Q p , M ¯ ∂ M −−−−−→ Q p , M −−−−−→ · · · ¯ ∂ M −−−−−→ Q p , n − M ¯ ∂ M −−−−−→ Q p , n M → , which is called the tangential Cauchy-Riemann complex in degree p (formore details see e.g. [21]).Since I n + k + M = , the bundles Q n + k , q M are subbundles of Λ n + k + q ( C T ∗ M ) and inparticular Q n + k , M is the same as the canonical bundle K M . As we will explainbelow, it is a CR line bundle and the tangential Cauchy-Riemann complexin degree n + k is also its ¯ ∂ M -complex as a CR line bundle. This is a specialexample of a notion that was described e.g. in [22, §
7] and that we willquickly recall in the next subsection.The tangential Cauchy-Riemann complexes can be described in a waythat is suitable to deal with smooth deformations of the CR structure.Let us set T ∗ p , M = Λ p (T ∗ , M ) and denote by T ∗ p , M the sheaf of germs ofsmooth sections of T ∗ p , M . Di ff erential presentation of the tangential CR complex. Let E M bea complement of T ∗ , M in C T ∗ M . We can e.g. fix a smooth Hermitian producton the fibers of C T ∗ M and take E M = (T ∗ , M ) ⊥ . The fiber bundle E M has rank n and we obtain a bigradation of the Grassmann algebra of the complexifiedcotangent bundle of M by setting Λ ∗ ( C T ∗ M ) = X ≤ p ≤ n + k ≤ q ≤ n E p , q M , with E p , q M = Λ q ( E M ) ⊗ M T ∗ p , M . Denote by E p , q M the sheaf of germs of smooth sections of E p , q M . Lemma 3.1.
For every integer p with ≤ p ≤ n + k we have: (1) E p , q M ⊆ I p M . (2) The restriction of the projection onto the quotient defines isomor-phisms (3.1) E p , q M → Q p , q M , E p , q M → Q p , q M of fiber bundles and of their sheaves of sections. (cid:3) By using the isomorphisms of (3.1) we can rewrite the tangential Cauchy-Riemann complexes in the form0 −−−−−→ E p , M d ′′ M −−−−−→ E p , M d ′′ M −−−−−→ · · · → E p , n − M d ′′ M −−−−−→ E p , n M −−−−−→ . We get explicit representations of d ′′ M , and hence of Q p , q M and ¯ ∂ M , by choos-ing suitable coframes of C T ∗ M . Take indeed on an open subset U of M acoframe ω , . . . , ω n , η , . . . , η n + k of T ∗ M with ω , . . . , ω n in E M and η , . . . , η n + k ∈ Γ ∞ ( U ′ , T ∗ , M ) . Then there are uniquely determined smooth complex vector fields ¯ L , . . . , ¯ L n , P , . . . , P n + k such that d u = X j = ( ¯ L j u ) ω j + X n + ki = ( P i u ) η i The vector fields ¯ L h , for 1 ≤ h ≤ n , are characterized by ω j ( ¯ L h ) = δ j , h (Kronecker delta) , ≤ j ≤ n , η i ( ¯ L h ) = , ≤ i ≤ n + k , and hence are sections of T , M . By our definition, d ′′ M u = X nj = ( ¯ L j u ) ω j , ∀ u ∈ C ∞ ( U ) . For each j with 1 ≤ j ≤ n we have d ω j = X ≤ j ≤ j ≤ n c j , j j ω j ∧ ω j + ζ j , with c j , j j ∈ C ∞ ( U ) , ζ j ∈ I M ( U ) . This gives in particular d ′′ M ω j = X ≤ j ≤ j ≤ n c j , j j ω j ∧ ω j and hence, computing by recurrence on the number of factors, we get d ′′ M ( ω j ∧ ω j ∧ · · · ∧ ω j q ) = d ′′ M ( ω j ) ∧ ω j ∧ · · · ∧ ω j q + ω j ∧ d ′′ M ( ω j ∧ · · · ∧ ω j q ) , d ′′ M ( ω j ∧ · · · ∧ ω j q ∧ η i ∧ · · · ∧ η i p ) = ( d ′′ M ( ω j ∧ · · · ∧ ω j q )) ∧ η i ∧ · · · ∧ η i p . These formulas yield explicit local expressions for d ′′ M . We note that for p = ω , . . . , ω n of E M . In particular, if M is compact and { M t } | t | < ǫ a smooth deformation of its CR structure, then E M stays transversal to T ∗ , M t for small | t | and thereforethe tangential Cauchy-Riemann complex in degree 0 for M t can be definedon sections of the same vector bundles for all small t . CR line bundles. A CR line bundle on M is the datum of a smoothcomplex line bundle π : F → M , together with a CR structure on F which iscompatible with the linear structure of the fibers and for which π is a CR submersion. The compatible CR structures on F are parametrized (modulo CR equivalence) by special cohomology classes of H ( M , H , M ) , where H , M is the sheaf of germs of local cohomology classes of degree (0 ,
1) on M : In a STABILITY THEOREM FOR PROJECTIVE CR MANIFOLDS 11 local trivialization { U i , s i ) } of F , with transition functions ( γ i , j ) , the structureis described by the datum of ¯ ∂ M -closed forms φ i in Q , M ( U i ) with¯ ∂ F M s i = φ i ⊗ s i on U i and φ i − φ j = γ − i , j ¯ ∂ M γ i , j on U i ∩ U j .We can consistently define a di ff erential operator ¯ ∂ F M on smooth sections of F , with values in the smooth sections of Q , M ⊗ F , in such a way that ¯ ∂ F M ( f s i ) = ( ¯ ∂ M f ⊗ s i ) + ( f · φ i ) ⊗ s i , ∀ f ∈ C ∞ ( U i ) . Let us use the notation F M to indicate the sheaf of germs of smooth sec-tions of F M and F p , q M for the sheaf of germs of smooth sections on M of thebundle Q p , q M ⊗ F . Then we can define, in general, a di ff erential operator ¯ ∂ F M mapping sections of F p , q M into sections of F p , q + M by setting for U open in M , α ∈ Q p , q M ( U ) and σ ∈ F M ( U ) , ¯ ∂ F M ( α ⊗ σ ) = ( ¯ ∂ M α ) ⊗ σ + ( − p + q α ∧ ∂ F M σ . In this way we obtain the tangential Cauchy-Riemann complexes with coef-ficients in F :0 −−−−−→ F p , M ( U ) ¯ ∂ F M −−−−−→ F p , M ( U ) −−−−−→ · · ·· · · −−−−−→ F p , n − M ( U ) ¯ ∂ F M −−−−−→ F p , n M ( U ) −−−−−→ . We denote its cohomology groups by H p , q ( U , F M ) = ker (cid:0) ¯ ∂ F M : F p , q M ( U ) → F p , q + M ( U ) (cid:1) image (cid:0) ¯ ∂ F M : F p , q − M ( U ) → F p , q M ( U ) (cid:1) . When the CR structure on M is induced from a generic embedding intoan ( n + k )-dimensional complex manifold X , the CR line bundle K M is the re-striction to M of the holomorphic canonical line bundle K X of X . In general,the canonical bundle of an abstract CR manifold might not even be locally CR trivializable, as explained e.g. in [9, 22, 28].For a CR submanifold M of an N -dimensional projective space CP N , therestrictions O M ( −
1) and O M (1) to M of its tautological and hyperplane bun-dles are CR line bundles on M , dual to each other. By the Euler sequence,the canonical bundle K CP N of the projective space is isomorphic to the ( N + O CP N ( − N −
1) of the tautological bundle (see e.g. [19, p.146]) andtherefore its dual K ∗ CP N , being isomorphic to the ( N + O CP N ( N + O CP N (1) , which is very ample, is very ample. Inparticular, if M is a generic CR submanifold of CP N , then its anti-canonicalbundle K ∗ M is the restriction to M of a very ample line bundle. If E X is avery ample line bundle on a complex compact complex manifold X , then its This is obtained by identifying F with its vertical bundle and first defining ¯ ∂ F M on thesmooth local sections s of F by setting( † ) ¯ ∂ F M s ( X + iJ M X ) = ds ( X ) + J F ds ( J M X ) , ∀ X ∈ HM . In fact, when s is a section, the right hand side of ( † ) is vertical. first Chern class c ( E X ) is positive and the Kodaira-Nakano theorem tellsus that this property is su ffi cient to ensure that the cohomology groups H q ( X , K X ⊗ E X ) vanish for q > Fano varieties , which are characterized by the amplenessof their anti-canonical bundle K ∗ X . When X is a flag manifold of a complex semisimple Lie group G , theminimal orbits in X of its real forms G R , studied e.g. in [3, 37], are examplesof generic CR submanifolds of Fano varieties.3.3. Deformations and CR line bundles. A Cartier divisor in X is the ef-fective divisor of a section of a line bundle F X on X . A positive F X has apower which is very ample and therefore the support of its Cartier divisoris the support of the pole divisor of a global meromorphic function f on X . Let X be a complex space, f a global meromorphic function on X . Thecorresponding complex line bundle F X can be described in the followingway: Let { U i } be an open covering of X by connected open subsets with theproperty that, for each index i , there are relatively prime f ′ i , f ′′ i ∈ O X ( U i )such that f ′ i / f ′′ i = f on U i . The quotients f ′ i / f ′ j = f ′′ i / f ′′ j = g i , j define nowherevanishing holomorphic functions on U i ∩ U j , which are the transition func-tions of a holomorphic line bundle F X on the trivialization atlas { U i } . Then(3.2) D = [ i { p ∈ U i | f ′′ i (p) = } is the support of the pole divisor of f . Proposition 3.2.
Let X be a complex space, D the support of the pole divisorof a global meromorphic function f on X and F X the corresponding linebundle. Let M be a smooth CR submanifold of X and { M t } a deformationon M of the CR structure M induced on M by X such that for each t theCR structures of M t and M agree to infinite order on D ∩ M . Assume thatthe CR embedding M ֒ → X is generic. Then we can define CR line bundles F M t on each M t in such a way that { F M t } is a smooth deformation of F M , andtheir CR structures agree to infinite order on the fibers over D . Proof.
We keep the notation introduced at the beginning of the subsection.The restrictions of the transition functions g i , j define a CR line bundle on M , because g − i , j ¯ ∂ M g i , j = U i ∩ U j ∩ M . Let us consider now an element of the deformation { M t } . We note that g j , i ¯ ∂ M t g i , j = f ′′ j f ′′ i f ′′ j ¯ ∂ M t f ′′ i − f ′′ i ¯ ∂ M t f ′′ j f ′′ j = ¯ ∂ M t f ′′ i f ′′ i − ¯ ∂ M t f ′′ j f ′′ j , on U i ∩ U j ∩ M STABILITY THEOREM FOR PROJECTIVE CR MANIFOLDS 13 is the di ff erence of two smooth ¯ ∂ M t -closed forms, the first one defined on U i ∩ M , the second on U j ∩ M . Indeed φ ( t ) i = f ′′ i − · ¯ ∂ M t f ′′ i , on ( U i ∩ M ) \ D , , on U i ∩ M ∩ D , is a smooth ¯ ∂ M t -closed (0 ,
1) form on U i ∩ M , because ¯ ∂ M t f ′′ i vanishes toinfinite order on D ∩ U i ∩ M by the assumption that the CR structures of M and M t agree to infinite order on D ∩ M , while f ′′ i , which extends holomor-phically to the neighourhood U i of U i ∩ M in X , cannot vanish to infiniteorder at any point of D ∩ M . The complex smooth line bundle F M is obtained by gluing the trivialline bundles { U i × C } by the transition functions { g i , j } . Let σ i be the section U i × { } on U i . The complex structure on F M t is obtained by setting(3.3) ¯ ∂ F M t σ i = φ ( t ) i · σ i on U i for all i . Clearly it agrees to infinite order on D with that of F M . (cid:3)
4. A n example
Let m be an integer ≥ M the quadric hypersurface of CP m describedin homogeneous coordinates by M = { z ¯ z + z ¯ z = z ¯ z + · · · + z m ¯ z m } . The rational function ζ = z / z has divisor ( ζ ) = D − D with D = { z = } , D = { z = } and D ∩ D ∩ M = ∅ . Since the point [1 , , . . . ,
0] does not belong to M , the projection map π : CP m \{ [1 , , . . . , } ∋ [ z , z , . . . , z m ] −→ [ z , . . . , z m ] ∈ CP m − is well defined on M . Its image is the complement in CP m of the open ball B = { z ¯ z + · · · + z m ¯ z m < z ¯ z } . Let F be a smooth function on CP m − which is holomorphic on B and qany point of its boundary ∂ B . If we can find an open neighbourhood V of qin CP m − and a solution v to the Cauchy problem v ∈ C ∞ ( V \ ¯ B ) ∩ C ( V \ B ) , ¯ ∂ v = ¯ ∂ F , on V \ ¯ B , v = , on V ∩ ∂ B , then F ′ = F , on V ∩ B , F − v , on V \ B continues analytically F | B to the point q . There are holomorphic functions on B which can be continued smoothly,but not analytically, to all points of ∂ B . Thus we can find F ∈ C ∞ ( CP m − ) , holomorphic on B , such that F | B has no analytic continuation to any point of ∂ B and moreover, since ∂ B ∩ { z = } = ∅ , we can require that its supportdoes not intersect the hyperplane { z = } . Then α = ¯ ∂ F is a smooth (0 , CP m − with the properties: • ¯ ∂ α = α ) ⊂ CP m − \ B , supp( α ) ∩ { z = } = ∅ ; • If V is any open neighourhood of a point q ∈ ∂ B and v ∈ C ∞ ( V \ ¯ B ) ∩ C ( V \ B ) satisfies ¯ ∂ v = α on V \ ¯ B , then v is not 0 at some point of V ∩ ∂ B . Let us fix a (0 ,
1) form α on CP m − with these properties.We consider the coordinate charts ( U , ζ ) , ( U , ξ ) in CP m with U = { z , } = CP m \ D , ζ = z z , ζ = z z , . . . , ζ m = z m z , U = { z , } = CP m \ D , ξ = z z , ξ = z z , . . . , ξ m = z m z . The pullback π ∗ α vanishes to infinte order on the points of M ∩ D , whichare mapped by π onto the boundary of B . Then ζ · π ∗ α , defined on M \ D , extends to a smooth (0 ,
1) form β on M , vanishing to infinite order on M ∩ D , whose support is contained in U ∩ M . We can define a smooth CR structure M t on M by requiring that its canon-ical bundle has sections s t , = d ζ ∧ d ζ ∧ · · · ∧ d ζ m , on U ∩ M \ supp( β ), s t , = ( d ξ + t β ) ∧ d ξ ∧ · · · ∧ d ξ m , on U ∩ M . Note that d s t , = t − d ξ ξ ∧ π ∗ ( α ) + ξ π ∗ ( d α ) ! ∧ d ξ ∧ · · · ∧ d ξ m = − t d ξ ξ ∧ β ∧ d ξ ∧ · · · ∧ d ξ m = t ξ β ∧ s t , because( † ) π ∗ ( d α ) ∧ d ξ ∧ · · · ∧ d ξ m = π ∗ ( d α ∧ d ξ ∧ · · · ∧ d ξ m ) = π ∗ (( ¯ ∂ α ) ∧ d ξ ∧ · · · ∧ d ξ m ) = . Note that the form β / ξ = π ∗ α / ξ extends to a smooth form β ′ , defined on M and vanishing to infinite order on D , so that( ‡ ) d s t , = U ∩ M \ supp( β ) and d s t , = t · β ′ ∧ s t , on U ∩ M shows that the line bundles K M t define formally integrable complex valueddistributions.These CR structures agree to infinite order on M ∩ D with the CR struc-ture M on M which is induced by its embedding into CP m Their canonical bundles are isomorphic complex line bundles, since thetransition functions are in all cases g , = s t , / s t , = − ζ m + = − ξ − m − , g , = g − , on M \ ( D ∪ D ∪ supp( β )) . We note that ξ , . . . , ξ m are CR functions on M t \ D for all t . A neces-sary condition for M t being locally embeddable at a point p of D is that STABILITY THEOREM FOR PROJECTIVE CR MANIFOLDS 15 one could find an open neighbourhood U ′ of p in U ∩ M and a function u ∈ C ∞ ( U ′ ) such that d ( e u s t , ) = U ′ . This equation can be rewritten, by factoring out e u ζ and taking into account( † ) , as( ∗ ) ξ d u + t ξ π ∗ α ! ∧ d ξ ∧ · · · ∧ d ξ m = U ′ \ D . The fibers of π above points of CP m − \ ¯ B are circles and reduce to sin-gle points over ∂ B . Since the projection π is proper, by shrinking we mayassume that π − ( π ( U ′ )) = U ′ . By integrating over the fibers we define v (q) = − π i I π − (q) u ξ d ξ , for q ∈ π ( U ′ ) \ ∂ B . This is a smooth function on π ( U ′ ) \ ∂ B , which extends to a continuous func-tion, that we still denote by v , vanishing on π ( U ′ ) ∩ ∂ B . Then from ( ∗ ) weobtain that d v ∧ d ξ ∧ · · · ∧ d ξ m = t α ∧ d ξ ∧ · · · ∧ d ξ m , on π ( U ′ ) \ ¯ B . This is equivalent to ¯ ∂ v = t α , on π ( U ′ ) \ ¯ B and hence contradicts the choice of α if t , , proving that M t cannot belocally embedded at any point p ∈ D if t , . This example from [24] shows that we can construct a smooth deforma-tion { M t } of the CR structure M of the quadric M such that no M t can belocally embedded into a complex manifold at any point of D if t , . The CR structure of the M t ’s are of type ( m − ,
1) and M is 1-pseudo-concave. Hence the example shows that Theorem 1.1 cannot be valid if weonly require M being 1-pseudoconcave. Remark 4.1.
Equations ( ‡ ) is of the type of those we need to solve to con-struct projective embeddings in §
6. It can be rewritten as ¯ ∂ K M t s t , = , ¯ ∂ K M t s t , = t · β ′ s t , . We also have ¯ ∂ M t u = ¯ ∂ M u − t · ∂ u ∂ ξ β . Then we obtain¯ ∂ K M t ξ s − t , ! = − t ξ β + ξ β ′ ! s − t , = − t ξ π ∗ α s − t , . An argument similar to the one used above shows that the equation( ∗∗ ) ¯ ∂ ( w · s − t , ) = ( ¯ ∂ M w − t ∂ w ∂ ξ β − t · w β ′ ) s − t , = − t ξ π ∗ α s − t , has no smooth solution on M . Indeed ( ∗∗ ) can be rewritten in the form d w − t ξ ∂ ( ξ w ) ∂ ξ π ∗ α + t ξ π ∗ α ! d ξ ∧ · · · ∧ d ξ m = ξ and integrating on the fiber, we see that the term in themiddle vanishes, while v (q) = − π i I π − (q) ( ξ w ) d ξ yields a solution in C ∞ ( CP m − \ ¯ B ) ∩ C ( CP m − \ B ) , vanishing on ∂ B , of theequation ¯ ∂ v = α on CP m − \ ¯ B , contradicting the choice of α . This adds a further argument for the need of conditions stronger than1-pseudoconcavity for the validity of the conclusions of Theorem 1.1.5. V anishing results
We keep the notation introduced in § § Embedded CR manifolds. We have the following vanishing result:
Theorem 5.1.
Assume that M is a 2-pseudoconcave generic smooth CRsubmanifold of type ( n , k ) of a complex variety X . Let F X be a positive linebundle on X and F M its restriction to M . Then we can find a positive integer ℓ such that (5.1) H p , q ( M , F − ℓ M ) = , ∀ ℓ ≥ ℓ , ≤ p ≤ n + k , q = , . Proof.
We know from [21] that a generic 2-pseudoconcave M has a funda-mental system of tubular neighborhoods U in X that are ( n − n + k − and forwhich, for all 0 ≤ p ≤ n + k , the natural restriction maps H q ( U , Ω p ( F U )) −→ H q ( M , Ω p ( F M )) , q = , , are isomorphisms for any holomorphic line bundle F U over U .This was proved in [21, Thm.4.1] for the trivial line bundle. The argu-ments of the proof given there apply also in general, yielding the isomor-phism for arbitrary holomorphic vector bundles. The statement is then aconsequence of [5, Prop.27], because U is strongly ( n + k − -concave in thesense of Andreotti-Grauert and hence also F − U ≔ F − X (cid:12)(cid:12)(cid:12) U is strongly ( n + k − ℓ such that H p , q ( U , F − ℓ X ) = ℓ ≥ ℓ , ≤ p ≤ n + k and q = , . (cid:3) A real valued smooth function φ on an N -dimensional complex manifold X is strongly q -pseudoconvex in the sense of [5] at points where its complex Hessian has at least ( N − q + X is called strictly q -pseudoconvex if there is an exhaustionfunction φ ∈ C ∞ ( X , R ) which is strictly q -pseudoconvex outside a compact subset of X andstrictly q -pseudoconcave if there is an exhaustion function φ ∈ C ∞ ( X , R ) such that ( − φ ) isstrictly q -pseudoconvex outside a compact subset of X . STABILITY THEOREM FOR PROJECTIVE CR MANIFOLDS 17
Some estimates for abstract CR manifolds. Let M be an abstractcompact CR manifold of type ( n , k ) and F M a complex CR line bundle on M . To discuss its cohomology groups H p , q ( M , F M ) by using functional analyticmethods, it is convenient to fix a smooth Riemannian metric on M and asmooth Hermitian norm on the fibers of F M . In this way we are able, foreach real r ≥ , to define in a standard way on the smooth sections of Q p , q M ⊗ F M the Sobolev W r -norm, involving the L -norms of their fractional derivativesup to order r . The completions of F p , q ( M ) with respect to these norms areHilbert spaces, that we denote by W r ( M , Q p , q M ⊗ F M ) . We use the notation( v | v ) r and k v k r = p ( v | v ) r , for v , v , v ∈ W r ( M , Q p , q M ⊗ F M )for their Hermitian scalar products and norms (see e.g. [20]).In [21] subelliptic estimates were proved for the ¯ ∂ M complex under pseu-doconcavity conditions. Since the arguments are local, they trivially extendto forms with coe ffi cients in a complex CR line bundle. We have in partic-ular the following statement. Proposition 5.2.
Let M be a compact smooth abstract CR manifold of type ( n , k ) and F M any complex CR line bundle on M . Assume that M is -pseu-doconcave. Then, for every ≤ p ≤ n + k and every real r ≥ there is a positiveconstant c r , p > such that (5.2) c r , p · k u k r + (1 / ≤ k ¯ ∂ F M u k r , ∀ u ∈ F p , ( M ) , c r , p · k v k r + (1 / ≤ k ¯ ∂ F M v k r + k ( ¯ ∂ F M ) ∗ v k r + k v k , ∀ v ∈ F p , ( M ) . (cid:3) Let us define the subspace(5.3) N p , F ( M ) = { v ∈ F p , ( M ) | ¯ ∂ F M v = , ( ¯ ∂ F M ) ∗ v = } . Keeping the assumptions and the notation above, we have
Proposition 5.3.
Under the assumptions of Proposition 5.2: • The space N p , F ( M ) is finite dimensional and equals its weak closurewith respect to the L norm; • For every pair of real numbers r , r ′ with − (1 / ≤ r ′ < r , r ≥ , there isa constant C r , r ′ , p > such that (5.4) k v k r ′ + (1 / ≤ C r , r ′ p (cid:16) k ¯ ∂ F M v k r + k ( ¯ ∂ F M ) ∗ v k r (cid:17) , ∀ v ∈ F p , ( M ) ∩ ( N p , F ( M )) ⊥ . [The orthogonal is taken with respect to the L -scalar product on F p , ( M ) . ] • H p , ( M , F M ) ≃ N p , F ( M ) . Proof.
The first statement is a consequence of (5.2): this subelliptic es-timate for r = L and W / -Sobolev normsare equivalent on N p , F ( M ) . Then by Rellich’s theorem the unit L ball in N p , F ( M ) is compact and this implies that N p , F ( M ) is finite dimensional andthus also equals its weak closure with respect to the L norm. The second statement can be proved by contradiction: assume that thereare real numbers r , r ′ , with − (1 / ≤ r ′ < r , r ≥ , for which (5.5) is not validfor any positive constant C r , r ′ , p . Then we can find a sequence { v ν } ⊂ F p , ( M ) ∩ ( N p , F ( M )) ⊥ with k v ν k r ′ + (1 / = , k ¯ ∂ F M v k r + k ( ¯ ∂ F M ) ∗ v k r < − ν . By (5.2) this sequence is bounded in the W r + (1 / -norm and therefore has asubsequence { v ν ′ } which, weakly in W r + (1 / and hence strongly in W r ′ + (1 / , converges to a v ∞ . In particluar k v ∞ k r ′ + (1 / = v ∞ , . This limitis a weak solution of ¯ ∂ F M v ∞ = ∂ F M ) ∗ v ∞ = ∂ F M ⊕ ( ¯ ∂ F M ) ∗ following from the subellipticity estimate (5.2),is a smooth section and thus a nonzero element of N p , F ( M ) . This completes the proof of the second statement.It is clear that the elements of N p , F ( M ) represent cohomology classes in H p , ( M , F M ) . Indeed, if v ∈ N p , F and v = ¯ ∂ F M u for some u ∈ F p , ( M ) , then( ¯ ∂ F M ) ∗ ¯ ∂ F M u = = ⇒ k v k = ( v | ¯ ∂ F M u ) = (( ¯ ∂ F M ) ∗ v | u ) = v = . On the other hand, if f ∈ F p , ( M ) and ¯ ∂ F M f = , then we candecompose f into the sum f = f + f , with f ∈ N p , F ( M ) and f ∈ ( N p , F ( M )) ⊥ . We get | ( f | v ) | ≤ k f k k v k ≤ C − / , , p ( k ¯ ∂ F M v k + k ( ¯ ∂ F M ) ∗ v k ) / , ∀ v ∈ F p , ( M ) ∩ ( N p , F ( M )) ⊥ . By Riesz’ representation theorem there is a unique w ∈ F p , ( M ) ∩ ( N p , F ( M )) ⊥ ( w is smooth because of the subellipticity estimate (5.2)) such that( ¯ ∂ F M w | ¯ ∂ F M v ) + (( ¯ ∂ F M ) ∗ w | ( ¯ ∂ F M ) ∗ v ) = ( f | v ) , ∀ v ∈ F p , ( M ) ∩ ( N p , F ( M )) ⊥ . Since f is L -orthogonal to N p , F ( M ) , this equality holds true for all v in F p , ( M ) and integration by parts yields( ¯ ∂ F M ) ∗ ¯ ∂ F M w = , ¯ ∂ F M ( ¯ ∂ F M ) ∗ w = f . This shows that the orthogonal projection of ker( ¯ ∂ F M : F p , ( M ) → F p , ( M ))onto N p , F ( M ) yields, by passing to the injective quotient, an isomorphismbetween H p , ( M , F ) and N p , F ( M ) . The proof is complete. (cid:3)
Likewise, we have a similar statement for CR sections of F on M . Set N p , F ( M ) = { u ∈ F p , ( M ) | ¯ ∂ F M u = } . Proposition 5.4.
Under the assumptions of Proposition 5.2: • The space N p , F ( M ) is finite dimensional and equals its weak closurewith respect to the L norm; STABILITY THEOREM FOR PROJECTIVE CR MANIFOLDS 19 • For every pair of real numbers r , r ′ with − (1 / ≤ r ′ < r , r ≥ , there isa constant C r , r ′ , p > such that (5.5) k u k r ′ + (1 / ≤ C r , r ′ p k ¯ ∂ F M u k r , ∀ u ∈ F p , ( M ) ∩ ( N p , F ( M )) ⊥ . Deformation of tangential CR complexes. Let us consider the sit-uation described at the beginning of § M is a generic compact smooth CR submanifold of a com-plex variety X of dimension n + k and that the complex CR line bundle F M isthe pullback of a holomorphic line bundle F X on X , associated to a globalmeromorphic function on X , whose pole divisor has support D . We indi-cate by M the CR structure, of type ( n , k ) , induced on M by the embedding M ֒ → X and consider a smooth one-parameter deformation { M t } on M of this CR structure M , with the constraint:(5.6) The CR structures of M t agree to infinite order on M ∩ D . We observed in § F M can be also considered as a CR line bundle on each M t . We need to consider the di ff erent tangentialCauchy-Riemann complexes with coe ffi cients in F s f M as complexes of par-tial di ff erential operators acting on the same spaces of vector valued func-tions.We showed in § CR complex (onthe trivial line bundle) by fixing a complement E M = E , M of T , M . This yieldsisomorphisms between E p , q M =Λ q ( E M ) ∧ M Λ p (T , M ) and Q p , q M for all integers p , q with 0 ≤ p ≤ n + k , ≤ q ≤ n . We will be interested in deformations M t corresponding to small valuesof the parameter t . In particular, after shrinking, we can assume that E M istransversal to T , M t for all values of the parameter t . Then the tangential CR complexes0 −−−−−→ E , M d ′′ M t −−−−−→ E , M d ′′ M t −−−−−→ · · · → E , n − M d ′′ M t −−−−−→ E , n M −−−−−→ ff erential operators, smoothlydepending on the parameter t , which act on the germs of sections of thesame fiber bundles.To obtain an analogous presentation for positive values of p , we note thatevery τ in T , M t uniquely decomposes into the sum τ ′ + τ ′′ of a τ ′ ∈ T , M and τ ′′ ∈ E M . The correspondence τ ↔ τ ′ yields smooth isomorphisms Q p , M t ≃ Q p , M of complex vector bundles, from which we obtain also for positive p repre-sentations0 −−−−−→ E p , M d ′′ M t −−−−−→ E p , M d ′′ M t −−−−−→ · · · → E , n − M d ′′ M t −−−−−→ E p , n M −−−−−→ CR complexes on M t by linear partial di ff erential operatorsacting on the same complex vector bundles and smoothly depending on theparameter t . Finally, for the line bundle F M , we use the trivialization given by thesections σ i of § ∂ F M t . The form φ ( t ) i ∈ Q , ( U i )decomposes into the sum of a ψ ( t ) i ∈ E , ( U i ) and a form in Γ ∞ ( U i , T , M t ) . Then we can define for every 0 ≤ p ≤ n + k and 0 ≤ q < n and all values ofthe parameter t a linear partial di ff erential operators d ′′ M t F mapping germs ofsmooth sections of F M ⊗ E p , q M to germs of smooth sections of F M ⊗ E p , q + M insuch a way that d ′′ M t F ( α · σ i ) = ( d ′′ M t α + ( − p + q ψ ( t ) i ∧ α ) · σ i , ∀ i , ∀ α ∈ E p , q M ( U i ) . In this way we represent the di ff erent tangential CR complexes for formswith coe ffi cients in F M as complexes of di ff erential operators, smoothly de-pending on the parameter t , but acting on the same smooth complex vectorbundles:(5.7) 0 → E p , ( M , F M ) d ′′ M tF −−−−−→ E p , ( M , F M ) → · · ·· · · → E p , n − ( M , F M ) d ′′ M tF −−−−−→ E p , n ( M , F M ) → . Estimates for deformations.
We keep the notation and assumptionsof § Proposition 5.5.
Let M be a compact generic CR submanifold of type ( n , k ) of a complex variety X . Let F X be the holomorphic line bundle associatedto a global meromorphic function on X and D the support of its polar di-visor. We assume that M is -pseudoconcave. Let { M t } be a deformationof the CR structure induced on M = M by the embedding into X , smoothlydepending on the parameter t and with CR structures agreeing to infiniteorder on D ∩ M . Then we can find ǫ > such that for all | t | < ǫ (1) for every ≤ p ≤ n + k and every real r ≥ there is a positive constantc r , p > such that (5.8) c r , p · k u k r + (1 / ≤ k d ′′ M t F u k r + k u k , ∀ u ∈ F p , ( M ) c r , p · k v k r + (1 / ≤ k d ′′ M t F v k r + k ( d ′′ M t F ) ∗ v k r + k v k , ∀ v ∈ F p , ( M ) . (2) for every pair r ′ , r of real numbers with − (1 / ≤ r ′ < r , r ≥ and ≤ p ≤ n + k , there is a positive constant C r , r , ′ p > such that (5.9) k u k r ′ + (1 / ≤ C r , r ′ , p k d ′′ M t F u k r ∀ u ∈ F p , ( M ) ∩ ( N p , F ( M t )) ⊥ k v k r ′ + (1 / ≤ C r , r ′ , p (cid:16) k d ′′ M t F v k r + k ( d ′′ M t F ) ∗ v k r (cid:17) ∀ v ∈ F p , ( M ) ∩ ( N p , F ( M t )) ⊥ . (3) dim C ( H p , ( M t , F M )) ≤ dim C ( H p , ( M , F M )) < + ∞ and dim C ( H p , ( M t , F M )) ≤ dim C ( H p , ( M , F M )) < + ∞ for all ≤ p ≤ n + k . STABILITY THEOREM FOR PROJECTIVE CR MANIFOLDS 21
Proof.
The proof of the subelliptic estimates in [21] yield (5.8), the con-stants being uniform for | t | < ǫ because they depend upon the coe ffi cients ofthe linear di ff erential operators d ′′ M t F and their derivatives.We prove (2) and (3) in the case of forms ( q = q =
0) is analogous and even simpler.We begin by proving that, for any pair r , r ′ with ( − / < r ′ < r , r ≥ ≤ p ≤ n + k , we can find ǫ > C r , r ′ , p > k v k r ′ + (1 / ≤ C r , r ′ , p (cid:16) k d ′′ M t F v k r + k ( d ′′ M t F ) ∗ v k r (cid:17) , ∀| t | < ǫ , ∀ v ∈ E p , ( M , F M ) ∩ ( N p , F ( M )) ⊥ . We argue by contradiction: if this was false, we could find a sequence { v ν } in E p , ( M , F M ) ∩ ( N p , F ( M )) ⊥ and a sequence { t ν → } such that k v ν k r ′ + (1 / = , k d ′′ M t ν F v ν k r + k ( d ′′ M t ν F ) ∗ v ν k r < − ν . By (1), the sequence { v ν } is uniformly bounded in W r + / and then, by thecompactness of the inclusion W r + (1 / ֒ → W r ′ + (1 / , contains a subsequencewhich strongly converges in W r ′ + (1 / to a v ∞ , which on one hand should bedi ff erent from 0 because k v ∞ k r ′ + (1 / = , on the other hand should be zerobecause it belongs to N p , F ( M ) ∩ ( N p , F ( M )) ⊥ . Estimate (5.10) implies that N , p F ( M t ) ∩ ( N p , F ( M )) ⊥ = { } for | t | < ǫ . Thisyields (3), because the dimension of H , p ( M t , F ) equals that of N , p F ( M t ) . We can prove (2) by an argument similar to the one used for (5.10). (cid:3)
Vanishing theorems.
We will use Proposition 5.5 to solve d ′′ F M t u t = f , for forms f in E , ( M , F M ) , with u t satisfying uniform estimates in the So-bolev norms W r , for su ffi ciently large r , yielding by the immersion theoremscontrol upon the derivatives of u t . Theorem 5.6.
Let us keep the assumptions of Proposition 5.5. Assumemoreover that F M is a positive line bundle on X . Then we can find ǫ > such that (5.11) H p , q ( M t , F − ℓ M ) = , ∀ ℓ ≥ ℓ , ≤ p ≤ n + k , q = , , | t | < ǫ . Moreover, for every f ∈ E , ( M , F − ℓ M ) satisfying d ′′ F M t f = there is a unique u t in Γ ∞ ( M , F − ℓ ) such that, for r ′ < r and r ≥ / , (5.12) d ′′ F M t u t = f and k u t k r ′ ≤ C r ′ , r k f k r − (1 / . Proof.
The first statement is a consequence of Theorem 5.1 and Proposi-tion 5.5. These results allow us to fix ǫ in such a way that, by (5.9), k| v k| t = (cid:16) k d ′′ F M t v k + k ( d ′′ F M t ) ∗ v k (cid:17) / is a pre-Hilbertian norm on E , ( M t , F − ℓ M ) . Let S t be its completion with re-spect to this norm. Given f ∈ E , ( M , F − ℓ M ) , we can consider v → ( v | f ) as a linear continuous functional on S t and hence by Riesz representation The-orem there is a unique w ∈ S t such that( v | f ) = ( d ′′ F M t v | d ′′ F M t w ) + (( d ′′ F M t ) ∗ v | ( d ′′ F M t ) ∗ w ) , ∀ v ∈ E , ( M , F − ℓ M ) . By the subelliptic estimate (5.8), in fact w ∈ E , ( M , F − ℓ M ) , i.e. w is smooth,because we assumed that f is smooth. If f satisfies the integrability condi-tion d ′′ F M t f = , then integration by parts yields0 = ( d ′′ F M t w | d ′′ F M t f ) = (( d ′′ F M t ) ∗ d ′′ F M t w | f ) = ( d ′′ F M t ( d ′′ F M t ) ∗ d ′′ F M t w | d ′′ F M t w ) + (( d ′′ F M t ) ∗ ( d ′′ F M t ) ∗ d ′′ F M t w | ( d ′′ F M t ) ∗ w ) = (( d ′′ F M t ) ∗ d ′′ F M t w | ( d ′′ F M t ) ∗ d ′′ F M t w ) = ⇒ ( d ′′ F M t ) ∗ d ′′ F M t w = = ⇒ = (( d ′′ F M t ) ∗ d ′′ F M t w | w ) = k d ′′ F M t w k = ⇒ d ′′ F M t w = . By (5.9), for all 0 < r < / u t | u t ) = (( d ′′ F M t ) ∗ w , ( d ′′ F M t ) ∗ w ) = ( w , d ′′ F M t ( d ′′ F M t ) ∗ w ) = ( w , f ) ≤ k f k − r k w k r ≤ C ′′ r k f k − r k u t k r − ≤ C ′′ r k f k − r k u t k Hence u t = ( d ′′ F M t ) ∗ w solves d ′′ F M t u t = f and k u t k ≤ C ′′ r k f k − r , ∀ < r < / , with constants C ′′ r independent of | t | < ǫ . Since O ( M t , F M ) = { } , the solu-tion is unique and therefore we obtain the estimates in (5.12) by recurrence,using, for any pseudodi ff erential operator P on M , the identity d ′′ F M t P ( u t ) = [ d ′′ F M t , P ]( u t ) + P ( f ) . If P has order r ′ , with 0 < r ′ < / , then k P ( u t ) k ≤ C ′′ r ′ (cid:16) k [ d ′′ F M t , P ]( u t ) k − r ′ + k P ( f ) k − r ′ (cid:17) ≤ const P C ′′ r ′ ( k u t k + k f k ) , with a constant const P which only depends on P . This yields k u t k r ′ ≤ C r ′ , (1 / k f k . with constants independent of t . Repeating this argument we obtain theestimate in the statement. (cid:3)
6. P roof of T heorem The general case.
We keep the notation of § f ′′ i ) define aglobal section σ of F M , which is CR for the structure on M and is , M \ D . Fix an integer ℓ such that F ℓ X is very ample and H , ( M t , F − h M ) = h ≥ ℓ and | t | < ǫ , which is possible by Theorem 5.6. By the assumption, there aresections σ , σ , . . . , σ m ∈ O M ( X , F ℓ X ) providing a holomorphic embedding of X into CP m . STABILITY THEOREM FOR PROJECTIVE CR MANIFOLDS 23
Their restrictions ( σ . , σ , , . . . , σ m , ) to M provide a CR embedding of M into CP m . For every t and 1 ≤ i ≤ m set f i , t = d ′′ M t F − ℓ ( σ − ℓ σ i ) , on M \ D , , on D . Since the CR structure of M t agrees with that of M to infinite order on D , the f i , t ’s are smooth sections in E , ( M , F − ℓ ) that vanish to infinite order on D and satisfy the integrability condition d ′′ M t F − ℓ f i , t = . By Theorem 5.6, for | t | < ǫ , we can find u i , t ∈ Γ ∞ ( M , F − ℓ ) to solve d ′′ M t F − ℓ u i , t = f i , t , and we have k u i , τ k r ′ ≤ C r ′ , r k f i , t k r − (1 / for 0 ≤ r ′ < r , with a constant C r , r ′ independent of t . Then σ i , t = σ i − u i , t σ ℓ ∈ O M t ( M t , F ℓ M ) . By Sobolev’s embedding theorems, the sup-norms of the first derivatives ofthe u i , t are bounded by their W r ′ -norms for r ′ > (2 n + k + / . As k f i , t k r − (1 / → t → , we obtain that, for | t |≪ , the sections ( σ , t , σ , t , . . . , σ m , t ) , being asmall C -perturbation of ( σ , , σ , , . . . , σ m , ) , still provide a CR immersionof M t into CP m . The last part of the statement follows from [23], where it is shown thatthe maximum degree of transcendence of the field of CR meromorphic func-tions on M t is n + k : this implies that all pseudoconcave CR manifold of type( n , k ) having a projective embedding, can be embedded into a projectivecomplex variety of dimension n + k . Generic CR submanifolds of the projective space. The line bundleson the projective space CP n + k are parametrized (modulo equivalence) bythe integers. They are all holomorphically equivalent to integral powers ofthe line bundle Θ CP n + k , which can be described, in the covering { U i = { z i , }} by the transition functions { g i , j = z − j z i } . The support of the pole divisor of ameromorphic function f on CP n + k is the set of common zeros of a homoge-neous polynomial(6.1) D = { ℘ ( z , z , . . . , z n + k ) = } . If ℘ ( z ) has degree d , this is the zero locus of a section of Θ d CP n + k . Replacing ℘ by one of its powers if necessary, we can as well assume that d > n + k . For d ≥ , we have H q ( CP n + k , Θ d CP n + k ) = { ψ ∈ C [ z , z , . . . , z n + k ] | ψ ( λ z ) = λ d ψ ( z ) } , if q = , { } for q > . In particular, since T CP n + k ≃ Θ n + k + CP n + k , the projective space is rigid : this meansthat all complex spaces of a small holomorphic deformation of CP n + k are bi-holomorphic to CP n + k (cf. e.g. [30, I. Thm.5.1]). We will however give adirect proof of stability of generic CR embeddings into projective spaces byusing the techniques developed above. The fact that the pullback Θ d M of Θ d CP n + k is a CR line bundle on M t impliesthat also all pullbacks Θ p M of Θ p CP n + k to M , for any p ∈ Z , are CR line bundleson M t . Indeed the fact that Θ d M is a CR line bundle on M t means that wecan find an open covering of M , that we can take of the form { U i , µ } with U i , µ ⊆ U i = { z i , } and d ′′ M t -closed forms φ i , µ ∈ E , ( U i , µ ) such that z j z i ! d d ′′ M t z i z j ! d = φ i , µ − φ j , ν on U i , µ ∩ U j , ν . This is equivalent to z j z i ! d ′′ M t z i z j ! = d φ i , µ − d φ j , ν on U i , µ ∩ U j , ν , showing that Θ M , and therefore all Θ p M with p ∈ Z are CR line bundles on M t . By repeating the proof in § σ i = z i of Θ M , weobtain Theorem 6.1.
Let M be a smooth compact generic CR submanifold of type ( n , k ) of CP n + k and D the zero locus in CP n + k of a homogeneous polynomialof degree d in C [ z , z , . . . , z n + k ] . Let { M t } be a family of CR structures oftype ( n , k ) on M , smoothly depending on a real parameter t . If (1) M is induced by the embedding M ֒ → CP n + k ; (2) M is -pseudoconcave; (3) the CR structures of all M t agree to infinite order on M ∩ D . Then we can find ǫ > such that, for every | t | < ǫ , M t admits a genericCR embedding into CP n + k . (cid:3) Generic CR submanifolds of Fano varieties. We recall that a com-plex variety X is Fano if its anticanonical bundle K − X is ample. If we add tothe assumptions of Theorem 1.1 the requirement that X is Fano, then we candefine a projective embedding of M by using CR sections σ , σ , . . . , σ m of apower K − ℓ M of the pullback on M of the anticanonical bundle. Indeed, by [29],the tensor product K − ℓ X ⊗ F − k X is the dual of a positive bundle and we can ap-ply the arguments in § σ i , t = σ i − u i , τ σ k in O M t ( M t , K − ℓ M ) , which, for | t |≪ CR embeddings σ t = ( σ , t , σ , t , . . . , σ m , t )of M t into CP m . The images M ′ t = σ t ( M t ) are generic submanifolds of com-plex ( n + k )-dimensional subvarieties X t of CP m which have, by construction,an ample anticanonical bundle. Indeed we know from [23, Thm.5.2] that,since the M t are pseudoconcave, global CR meromorphic functions on M t are restrictions of global meromorphic functions on X t . Thus we have
Theorem 6.2.
Add to the assumptions of Theorem 1.1 the fact that X isFano. Then there is ǫ > such that for all | t | < ǫ the abstract CR manifold M t has a generic CR embedding into a complex Fano variety X t . STABILITY THEOREM FOR PROJECTIVE CR MANIFOLDS 25
Acknowledgements.
The first author was supported by Deutsche Forschungsge-meinschaft (DFG, German Research Foundation, grant BR 3363 / R eferences
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