smooth almost complex structures

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DOI 10.4171/JEMS/742

Jean-Pierre Demailly·Herv´e Gaussier

Algebraic embeddings of

smooth almost complex structures

Received December 10, 2014

Abstract. The goal of this work is to prove an embedding theorem for compact almost complex manifolds into complex algebraic varieties. It is shown that every almost complex structure can be realized by the transverse structure to an algebraic distribution on an affine algebraic variety, namely an algebraic subbundle of the tangent bundle. In fact, there even exist universal embedding spaces for this problem, and their dimensions grow quadratically with respect to the dimension of the almost complex manifold to embed. We give precise variation formulas for the induced almost complex structures and study the related versality conditions. At the end, we discuss the original question raised by F. Bogomolov: can one embed every compact complex manifold as aCsmooth subvariety that is transverse to an algebraic foliation on a complex projective algebraic variety?

Keywords. Deformation of complex structures, almost complex manifolds, complex projective variety, Nijenhuis tensor, transverse embedding, Nash algebraic map

1. Introduction and main results

The goal of this work is to prove an embedding theorem for compact almost complex manifolds into complex algebraic varieties. As usual, analmost complex manifoldof di- mensionnis a pair(X, JX), whereXis a real manifold of dimension 2nandJXa smooth section of End(T X)such thatJX2 = −Id; we will assume here that all data areC.

Let Z be a complex (holomorphic) manifold of complex dimension N. Such a manifold carries a natural integrable almost complex structureJZ (conversely, by the Newlander–Nirenberg theorem any integrable almost complex structure can be viewed as a holomorphic structure). Now, assume that we are given a holomorphic distributionD inT Z, i.e. a holomorphic subbundleD⊂T Z. Every fiberDx of the distribution is then invariant under JZ, i.e.JZDx ⊂ Dx for every x ∈ Z. Here, the distribution Dis not assumed to be integrable. We recall thatDis integrable in the sense of Frobenius (i.e.

stable under the Lie bracket operation) if and only if the fibersDxare the tangent spaces to leaves of a holomorphic foliation. More precisely,D is integrable if and only if the torsion operatorθofD, defined by

θ:O(D)×O(D)→O(T Z/D), (ζ, η)7→ [ζ, η]modD, (1.1) J.-P. Demailly, H. Gaussier: Universit´e Grenoble Alpes, Institut Fourier, UMR 5582 du CNRS, 100 rue des Maths, 38610 Gi`eres, France;

e-mail: jean-pierre.demailly@univ-grenoble-alpes.fr, herve.gaussier@univ-grenoble-alpes.fr Mathematics Subject Classification (2010):32Q60, 32Q40, 32G05, 53C12

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vanishes identically. As is well known,θis skew symmetric in(ζ, η)and can be viewed as a holomorphic section of the bundle32D⊗(T Z/D).

LetMbe a real submanifold ofZof classCand of real dimension 2nwithn < N. We say thatMistransverse toDif for everyx ∈Mwe have

TxM⊕Dx=TxZ. (1.2)

We could in fact assume more generally that the distributionDissingular, i.e. given by a certain saturated subsheafO(D)ofO(T Z)(“saturated” means that the quotient sheaf O(T Z)/O(D) has no torsion). ThenO(D) is actually a subbundle ofT Z outside an analytic subsetDsing ⊂Z of codimension ≥ 2, and we further assume in this case that M∩Dsing= ∅.

WhenMis transverse toD, one gets a naturalR-linear isomorphism

TxM'TxZ/Dx (1.3)

at every pointx ∈ M. SinceT Z/Dcarries a structure of holomorphic vector bundle (at least overZrDsing), the complex structureJZinduces a complex structure on the quotient and therefore, through the above isomorphism (1.3), an almost complex structureJMZ,D onM.

Moreover, when Dis a foliation (i.e. O(D) is an integrable subsheaf ofO(T Z)), thenJMZ,D is anintegrablealmost complex structure onM. Indeed, such a foliation is realized near any regular pointxas the set of fibers of a certain submersion: there exists an open neighborhoodofx inZ and a holomorphic submersionσ :  → 0 to an open subset0 ⊂Cnsuch that the fibers ofσ are the leaves ofDin. We can take to be a coordinate open set in Z centered at pointx and select coordinates such that the submersion is expressed as the first projection ' 0×00 → 0 with respect to0 ⊂ Cn,00 ⊂ CN−n, and then D,T Z/Dare identified with the trivial bundles

×({0} ×CN−n)and×Cn. The restriction

σM∩:M∩⊂→σ 0

providesMwith holomorphic coordinates onM∩, and it is clear that any other local trivialization of the foliation on a different charte= e0×e00would give coordinates that are changed by local biholomorphisms0→e0in the intersection∩, thanks toe the holomorphic character ofD. Thus we directly see in that case thatJMZ,Dcomes from a holomorphic structure onM.

More generally, we say thatf :X ,→Zis atransverse embeddingof a smooth real manifoldXin(Z,D)iff is an embedding andM =f (X)is a transverse submanifold ofZ, i.e.fTxX⊕Df (x) =Tf (x)Zfor everyx ∈X (andf (X)does not meetDsingin case there are singularities). One then gets a real isomorphismT X ' f(T Z/D)and therefore an almost complex structure onX(for this it would be enough to assume thatf is an immersion, but we will actually suppose thatf is an embedding here). We denote byJf the almost complex structuref(Jf (X)Z,D).

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In this work, we are interested in the problem of embedding a compact complex or almost complex manifoldXinto aprojective algebraicmanifoldZ,transversallyto an algebraic distributionD⊂T Z. We will also make use of the concept of Nash algebraic- ity. Recall that aNash algebraic mapg : U → V between connected open (i.e. metric open) setsU, V of algebraic manifoldsY, Zis a map whose graph is a connected com- ponent of the intersection ofU ×V with an algebraic subset ofY ×Z. We say that a holomorphic foliationF of codimensionnonU isalgebraic(resp.holomorphic,Nash algebraic) if the associated distributionY ⊃ U → Gr(T Y, n) into the Grassmannian bundle of the tangent bundle is given by an algebraic (resp. holomorphic, Nash algebraic) morphism. The following very interesting question was investigated about 20 years ago by F. Bogomolov [Bog96].

Basic Question 1.1. Given an integrable complex structureJ on a compact manifoldX, can one realizeJ, as described above, by a transverse embeddingf : X ,→ Z into a projective manifold(Z,D)equipped with an algebraic foliationD, in such a way that f (X)∩Dsing= ∅andJ =Jf?

There are indeed many examples of K¨ahler and non-K¨ahler compact complex manifolds which can be embedded in that way (the case of projective ones being of course triv- ial): tori, Hopf and Calabi–Eckman manifolds, and more generally all manifolds given by the LVMB construction (see Section2). Strong indications exist that every compact complex manifold should be embeddable as a smooth submanifold transverse to an al- gebraic foliation on a complex projective variety (see Section5). We prove here that the corresponding statement in the almost complex category actually holds – provided that non-integrable distributions are considered rather than foliations. In fact, there are even

“universal solutions” to this problem.

Theorem 1.2. For all integersn≥1andk≥4n, there exists a complex affine algebraic manifoldZn,kof dimensionN =2k+2(k2+n(k−n))possessing a real structure(i.e. an anti-holomorphic algebraic involution)and an algebraic distributionDn,k ⊂T Zn,k of codimensionn, for which every compactn-dimensional almost complex manifold(X, J ) admits an embeddingf : X ,→ Zn,kR , transverse toDn,k and contained in the real part ofZn,k, such thatJ =Jf.

Remark 1.3. To constructf we first embedX differentiably into Rk,k ≥ 4n, by the Whitney embedding theorem [Whi44], or its generalization due to [Ton74]. Once the embedding of the underlying differentiable manifold has been fixed, the transverse em- beddingf depends in a simple algebraic way on the almost complex structureJ given onX, as one can see from our construction (see Section 4).

The choicek=4nyields the explicit embedding dimensionN =38n2+8n(we will see that a quadratic boundN =O(n2)is optimal, but the above explicit value could perhaps be improved). SinceZ = Zn,k andD = Dn,k are algebraic and Z is affine, one can further compactifyZ to a complex projective manifoldZ, and extendDto a saturated subsheafDofT Z. In general such distributionsDwill acquire singularities at infinity,

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and it is unclear whether one can achieve such embeddings withDnon-singular onZ, if at all possible.

Next, we consider the case of a compact almost complex symplectic manifold (X, J, ω)where the symplectic form ωis assumed to be J-compatible, i.e.Jω = ω andω(ξ, J ξ ) >0. By a theorem of Tischler [Tis77], at least under the assumption that the de Rham cohomology class{ω}is integral, we know that there exists a smooth embed- dingg: X ,→CPs such thatω=gωFSis the pull-back of the standard Fubini–Study metricωFSonCPs. A natural problem is whether the symplectic structure can be accom- modated simultaneously with the almost complex structure by a transverse embedding.

Let us introduce the following definition.

Definition 1.4. Let(Z,D)be a complex manifold equipped with a holomorphic distribu- tion. We say that a closed semipositive(1,1)-formβonZis atransverse K¨ahler structure if the kernel ofβis contained inD, in other terms, ifβinduces a K¨ahler form on any germ of complex submanifold transverse toD.

Using an effective version of Tischler’s theorem stated by Gromov [Gro86], we prove:

Theorem 1.5. For all integersn, b≥1andk≥2n+1, there exists a complex projective algebraic manifoldZn,b,k of dimensionN = 2bk(2bk+1)+2n(2bk−n)), equipped with a real structure and an algebraic distributionDn,b,k⊂T Zn,b,kof codimensionn, for which every compactn-dimensional almost complex symplectic manifold(X, J, ω)with second Betti numberb2≤band aJ-compatible symplectic formωadmits an embedding f : X ,→ ZRn,b,ktransverse toDn,b,k and contained in the real part ofZn,k, such that J =Jf andω=fβfor some transverse K¨ahler structureβon(Zn,b,k,Dn,b,k).

In Section5, we discuss Bogomolov’s conjecture for the integrable case. We first prove the following weakened version, which can be seen as a form of “algebraic embedding”

for arbitrary compact complex manifolds.

Theorem 1.6. For all integersn≥1andk≥4n, let(Zn,k,Dn,k)be the affine algebraic manifold equipped with the algebraic distributionDn,k ⊂ T Zn,k introduced in Theo- rem1.2. Then, for every compactn-dimensional(integrable)complex manifold(X, J ), there exists an embeddingf :X ,→ Zn,kR transverse toDn,k, contained in the real part ofZn,k, such that

(i) J =Jf and∂Jf is injective;

(ii) Im(∂Jf )is contained in the isotropic locusIDn,kof the torsion operatorθofDn,k, the intrinsically defined algebraic locus in the Grassmannian bundleGr(Dn,k, n)→Zn,k

of complexn-dimensional subspaces inDn,k, consisting of those subspaces Ssuch thatθ|S×S =0.

The inclusion condition (ii) Im(∂Jf ) ⊂IDn,k is in fact necessary and sufficient for the integrability ofJf.

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In Section6, we investigate the original Bogomolov conjecture and “reduce it” to a statement concerning approximations of holomorphic foliations. The flavor of the state- ment is that holomorphic objects (functions, sections of algebraic bundles, etc.) defined on a polynomially convex open set ofCncan always be approximated by polynomials or algebraic sections. Our hope is that this might also be true for the approximation of holo- morphic foliations by Nash algebraic ones. In fact, we obtain the following conditional statement.

Proposition 1.7. Assume that holomorphic foliations can be approximated by Nash al- gebraic foliations uniformly on compact subsets of any polynomially convex open subset ofCN. Then every compact complex manifold can be approximated by compact complex manifolds that are embeddable in the sense of Bogomolov in foliated projective manifolds.

In the last Section7, we briefly discuss a “categorical” viewpoint in which the above questions have a nice interpretation.

2. Transverse embeddings to foliations

We consider the situation described above, whereZis a complexN-dimensional manifold equipped with a holomorphic distribution D. More precisely, let X be a compact real manifold of classCand of real dimension 2nwithn < N. We assume that there is an embeddingf :X ,→Zthat is transverse toD, namelyf (X)∩Dsing= ∅and

fTxX⊕Df (x)=Tf (x)Z (2.1)

at every pointx ∈ X. HereDf (x)denotes the fiber atf (x)of the distributionD. As ex- plained in Section 1, this induces anR-linear isomorphismf:T X→f(T Z/D), and from the complex structures ofT ZandDwe get an almost complex structurefJf (X)Z,D onT Xwhich we will simply denote byJf here. Next, we briefly investigate the effect of isotopies.

Definition 2.1. An isotopy of smooth transverse embeddings of X into (Z,D) is by definition a family ft : X → Z of embeddings for t ∈ [0,1] such that the map F (x, t )=ft(x)is smooth onX× [0,1]andftis transverse toDfor everyt∈ [0,1]. We then get a smooth variationJft of almost complex structures onX. WhenDis in- tegrable (i.e. a holomorphic foliation), these structures are integrable and we have the following simple but remarkable fact.

Proposition 2.2. LetZ be a compact complex manifold equipped with a holomorphic foliationDand let ft : X → Z,t ∈ [0,1], be an isotopy of transverse embeddings of a compact smooth real manifold. Then all complex structures(X, Jft)are biholomor- phic to(X, Jf0)through a smooth variation of diffeomorphisms inDiff0(X), the identity component of the groupDiff(X)of diffeomorphisms ofX.

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Proof. By an easy connectedness argument, it is enough to produce a smooth variation of biholomorphismsψt,t0 : (X, Jft0)→ (X, Jft)whent is close tot0, and then extend these to allt, t0∈ [0,1]by the chain rule. Letx ∈X. Thanks to the local triviality of the foliation atz0 =ft0(x)∈ ZrDsing,Dis locally nearx the family of fibers of a holo- morphic submersionσ :Z ⊃→ 0 ⊂Cndefined on a neighborhoodofz0. Then σ◦ft :X⊃ft−1()→0is by definition a local biholomorphism from(X, Jft)to0 (endowed with the standard complex structure ofCn). Now,ψt,t0 =(σ◦ft)−1◦(σ◦ft0) defines a local biholomorphism from(X, Jft

0)to(X, Jft)on a small neighborhood ofx, and these local biholomorphisms glue together to a global one whenxandvary (this biholomorphism consists of “following the leaf ofD” from positionft0(X)to position ft(X)of the embedding). Clearlyψt,t0depends smoothly ontand satisfies the chain rule

ψt,t0◦ψt0,t1t,t1. ut

Therefore when D is a foliation, to any triple (Z,D, α) whereα is an isotopy class of transverse embeddings X→Z, one can attach a point in the Teichm¨uller space Jint(X)/Diff0(X)of integrable almost complex structures modulo biholomorphisms dif- feotopic to identity. The question raised by Bogomolov can then be stated more precisely:

Question 2.3. For any compact complex manifold (X, J ), does there exist a triple (Z,D, X, α)formed by a smooth complex projective varietyZ, an algebraic foliationD onZ and an isotopy classαof transverse embeddingsX → Z, such thatJ = Jf for somef ∈α?

The isotopy class of embeddings X → Z in a triple(Z,D, α) provides some sort of

“algebraicization” of a compact complex manifold, in the following sense:

Lemma 2.4. There is an atlas of X such that the transition functions are solutions of algebraic linear equations(rather than plain algebraic functions, as would be the case for usual algebraic varieties). In this setting, the isotopy classesαare just “topological classes” belonging to a discrete countable set.

This set can be infinite as one already sees for real linear embeddings of a real even- dimensional torusX=(R/Z)2ninto a complex torusZ=CN/3equipped with a linear foliationD.

Proof of Lemma2.4. We first coverZrDsingby a countable family of coordinate open sets ν ' 0ν ×00ν such that the first projections σν : ν → 0ν ⊂ Cn define the foliation. We assume here that0ν and00ν are balls of sufficiently small radius, so that all fibers z0 ×00ν are geodesically convex with respect to a given hermitian metric on the ambient manifoldZ, and the geodesic segment joining any two points in those fibers is unique (of course, we mean here geodesics relative to the fibers—standard results of differential geometry guarantee that sufficiently small coordinate balls will satisfy this property). Then any non-empty intersectionT

zj0 ×00ν

j of the fibers from various co- ordinate sets is still connected and geodesically convex. We further enlarge the fam- ily with all smaller balls whose centers have coordinates in Q[i] and radii in Q+, so that arbitrarily fine coverings can be extracted from the family. A transverse embedding

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f :X→Zis characterized by its imageM=f (X)up to right composition with an ele- mentψ∈Diff(X), and thus, modulo isotopy, up to an element in the countable mapping class group Diff(X)/Diff0(X). The imageM=f (X)is itself given by a finite collection of graphs of mapsgν :0ν →00ν that glue together, for a certain finite subfamily of co- ordinate sets(ν)ν∈Iextracted from the initial countable family. However, any two such transverse submanifolds(Mk)k=0,1and associated collections of graphs(gk,ν)defined on the same finite subsetI are isotopic: to see this, assume e.g.I = {1, . . . , s}and fix even smaller products of ballseν 'e0ν×e00ν bν still coveringM0andM1, and a cut-off functionθν(z0)equal to 1 on e0ν and with support in 0ν. Then we construct isotopies (Ft,k)t∈[0,1]:M0→Mt,kstep by step, fork=1, . . . , s, by taking inductively graphs of maps(Gt,k,ν)t∈[0,1], k=1,...,s, ν∈I such that

Mt,1 given by

(Gt,1,1(z0)=γ t θ1(z0);g0,1(z0), g1,1(z0)

on01, Gt,1,ν(z0)=g0,ν(z0) onσν νr(Supp(θ1)×e001)

, ν6=1, Mt,k given by

(Gt,k,k(z0)=γ t θk(z0);Gt,k−1,k(z0), Gt,k−1,k(z0)

on 0k, Gt,k,ν(z0)=Gt,k−1,ν(z0) onσν ν r(Supp(θk)×e00k)

, ν6=k,

whereγ (t;a00, b00)denotes the geodesic segment betweena00andb00in each fiberz0×00ν. By construction, we haveM0,k=M0andM1,k∩Uk =M1∩UkonUk=e1∪ · · · ∪ek, thusft :=Ft,s :M0→ Mt is a transverse isotopy betweenM0andM1. Therefore, we have at most as many isotopy classes as the cardinality of the mapping class group, times the cardinality of the set of finite subsets of a countable set, which is still countable. ut Of course, whenDis non-integrable, the almost complex structureJft will in general vary under isotopies. One of the goals of the next sections is to investigate this phenomenon, but in this section we further study some integrable examples.

Example 2.5 (Complex tori). LetX =R2n/Z2nbe an even-dimensional real torus and Z=CN/3a complex torus where3 'Z2N is a lattice ofCN,N > n. Any complex vector subspaceD ⊂CN of codimensionndefines a linear foliation onZ (which may or may not have closed leaves, but forDgeneric, the leaves are everywhere dense). Let f : X → Z be a linear embedding transverse to D. Here, there are countably many distinct isotopy classes of such linear embeddings, in fact up to a translation, f is in- duced by anR-linear mapu : R2n → CN that sends the standard basis(e1, . . . , e2n) ofZ2n to a unimodular system of 2nZ-linearly independent vectors(ε1, . . . , ε2n)of3.

Such(ε1, . . . , ε2n)can be chosen to generate any 2n-dimensionalQ-vector subspaceVε

of3⊗Q 'Q2N, thus the permitted directions forVεare dense, and for most of them f is indeed transverse toD. For a transverse linear embedding, we get anR-linear iso- morphismu˜ : R2n → CN/D, and the complex structureJf onX is precisely the one induced by that isomorphism by pulling back the standard complex structure on the quo- tient. For N ≥ 2n, we claim that all possible translation invariant complex structures onXare obtained. In fact, we can then choose the lattice vector imagesε1, . . . , ε2nto be C-linearly independent, so that the mapu :Z2n → 3,ej 7→εj, extends to an injection v : C2n →CN. Once this is done, the isotopy class of embedding is determined, and a

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translation invariant complex structureJ onX is given by a direct sum decomposition C2n = S⊕S with dimCS = n(andS the complex conjugate ofS). What we need is that the compositionv˜ :C2n→CN→CN/Ddefines aC-linear isomorphism ofSonto v(S)˜ ⊂CN/Dandv(S)˜ = {0}, i.e.D ⊃ v(S)andD∩v(S)= {0}. The solutions are obtained by takingD=v(S)⊕H, whereHis any complementary subspace ofv(S⊕S) inCN (thus the choice ofDis unique ifN =2n, and parametrized by an affine chart of a GrassmannianG(N−n, N−2n)ifN >2n). Of course, we can take hereZto be an Abelian variety—even a simple Abelian variety if we wish.

Example 2.6 (LVMB manifolds). We refer to L´opez de Medrano–Verjovsky [LoV97], Meersseman [Mer00] and Bosio [Bos01] for the original constructions, and sketch here the more general definition given in [Bos01] (or rather an equivalent one, with very minor changes of notation). Letm ≥ 1 andN ≥ 2mbe integers, and letE = Em,N+1 be a non-empty set of subsets of cardinality 2m+1 of{0,1, . . . , N}. ForJ ∈ E, defineUJ to be the open set of points [z0 : . . . : zN] ∈ CPN such that zj 6= 0 forj ∈ J and UE=S

J∈EUJ. Then, consider the action ofCmonUE given by w· [z0:. . .:zN] = [e`0(w)z0:. . .:e`N(w)zN]

where`j ∈(Cm)are complex linear formsCm→C, 0≤j ≤N. Then Bosio [Bos01, Th´eor`eme 1.4] proves that the space of orbitsX =UE/Cmis a compact complex mani- fold of dimensionn=N −mif and only if the following two combinatorial conditions are met:

(i) for anyJ1, J2 ∈E, the convex envelopes in(Cm)of{`j}j∈J

1 and{`j}j∈J

2 overlap on some non-empty open set;

(ii) for allJ ∈Eandk∈ {0, . . . , N}, there existsk0∈J such that(J r{k0})∪ {k} ∈E.

The above action can be described in terms ofmpairwise commuting Killing vector fields of the action of PGL(N+1,C)onCPN, given by

ζj =

N

X

k=0

λj kzk

∂zk

, λj k= ∂`k

∂wj

, 1≤j ≤m.

These vector fields generate a foliationF of dimensionmonCPN that is non-singular overUE. Under the more restrictive condition defining LVM manifolds, it follows from [Mer00] thatXcan be embedded as a smooth compact real analytic submanifoldMinUE that is transverse toF; such a submanifoldMis realized as the transverse intersection of hermitian quadricsP

0≤k≤Nλj,k|zk|2=0, 1≤ j ≤ m(this actually yields 2mreal con- ditions by taking real and imaginary parts). In the more general case of LVMB manifolds, Bosio has observed thatXcan also be embedded smoothly inUE ⊂CPN (see [Bos01, Prop. 2.3 and discussion thereafter] and also [BoM06, Part III, Section 12]).

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3. Deformation of transverse embeddings

Letf :X→(Z,D)be a transverse embedding. ThenJf :=f(Jf (X)Z,D)defines an almost complex structure onX. In this section we give sufficient conditions on the embeddingf that ensure that small deformations ofJf, in a suitable space of almost complex structures onX, are given asJf˜wheref˜are small deformations off in a suitable space of trans- verse embeddings ofXinto(Z,D). Since the implicit function theorem will be needed, we have to introduce various spaces ofCr mappings. For anyr ∈ [1,∞], we consider the group Diffr(X)of diffeomorphisms ofXof classCr, and the subgroup Diffr0(X)of diffeomorphisms diffeotopic to identity. Whenr = s +γ is not an integer,s = brc, thenCr denotes the H¨older space of maps of class Cs with derivatives of order s that are H¨older continuous with exponentγ. Similarly, we consider the spaceCr(X, Z)ofCr mappingsX→Zequipped with Cr convergence topology (of course, in Diffr(X), the topology also requires convergence of sequencesfν−1). IfZ is Stein, there exists a bi- holomorphism8:T Z→Z×Z from a neighborhood of the zero section ofT Z to a neighborhood of the diagonal inZ×Z, such that8(z,0)=zanddζ8(z, ζ )=0=Id onTzZ. WhenZis embedded inCN

0 for someN0, such a map can be obtained by taking 8(z, ζ ) = ρ(z+ζ ), whereρ is a local holomorphic retraction CN

0 → Z andTzZ is

identified to a vector subspace ofCN

0. In general (i.e. whenZis not necessarily Stein), one can still find aCor even real analytic map 8satisfying the same conditions, by taking e.g.8(z, ζ ) = (z,expz(ζ )), where exp is the Riemannian exponential map of a real analytic hermitian metric onZ; actually, we will not need8to be holomorphic in what follows.

Lemma 3.1. Forr ∈ [1,∞[,Cr(X, Z)is a Banach manifold whose tangent space at a pointf : X → Z isCr(X, fT Z), andDiffr0(X)is a “Banach Lie group” with “Lie algebra”Cr(X, T X)[the quotes meaning that the composition law is not real analytic as one would expect, but merely continuous and differentiable atIdX, though the underlying manifold is indeed a Banach manifold].

Let us also point out that if the composition ofCr maps is merelyCr2 for 0< r <1, it is actually aCr map forr≥1.

Proof of Lemma3.1. The use of the map8allows us to parametrize small deformations of the embeddingf asf (x)e =8(f (x), u(x))[or equivalentlyu(x)=8−1(f (x),f (x))e ], whereuis a smooth sufficiently small section offT Z. This parametrization is one-to- one, andfeisCr if and only ifuisCr (providedf is). The argument is similar, and very

well known indeed, for Diffr0(X). ut

Now, letJr(X) denote the space of almost complex structures of classCr onX. For 1 ≤r <∞, this is a Banach manifold whose tangent space at a pointJ is the space of sectionsh∈Cr(X,End

C(T X))satisfyingJ◦h+h◦J =0 (that is,C-conjugate-linear endomorphisms ofT X). There is a natural right action of Diffr0(X)onJr−1(X)defined by

(J, ψ )7→ψJ, ψJ (x)=dψ (x)−1◦J (ψ (x))◦dψ (x).

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As is well-known and as a standard calculation shows, the differential ofψ 7→ ψJ at ψ=IdXis closely related to the∂J operator

J :Cr(X, T X)→Cr−1(X, 30,1T X⊗T X1,0)=Cr−1(X,End

C(T X)), namely it is given byv7→J◦dv−dv◦J =2J ∂Jvifv∈Cr(X, T X)is the infinitesimal variation ofψ.

Let0r(X, Z,D)be the space ofCr embeddings ofXintoZthat are transverse toD.

Transversality is an open condition, so0r(X, Z,D)is an open subset inCr(X, Z). Now, Diffr0(X)acts on0r(X, Z,D)through the natural right action

0r(X, Z,D)×Diffr0(X)→0r(X, Z,D), (f, ψ )7→f ◦ψ.

We wish to consider the differential of this action at(f, ψ ),ψ =IdX, with respect to the tangent space isomorphisms of Lemma 3.1. This is just the addition law in the bundlefT Z:

Cr(X, fT Z)×Cr(X, T X)→Cr−1(X, fT Z), (u, v)7→u+fv.

A difficulty occurring here is the loss of regularity from Cr toCr−1 coming from the differentiations off andv. To overcome this difficulty, we have to introduce a slightly smaller space of transverse embeddings.

Definition 3.2. Forr∈ [1,∞] ∪ {ω}we consider the space e0r(X, Z,D)⊂0r(X, Z,D)⊂Cr(X, Z)

of transverse embeddingsf :X→Zsuch thatf is of classCr together with all “trans- verse” derivativesh·df, whereh runs over conormal holomorphic 1-forms with val- ues in(T Z/D). When r = ∞ orr = ω (real analytic case), we sete0r(X, Z,D) = 0r(X, Z,D).

Thene0r(X, Z,D)satisfies the following conditions:

Proposition 3.3. For1≤r <∞:

(i) the groupDiffr+10 (X)acts on the right one0r(X, Z,D);

(ii) the spacee0r(X, Z,D)is a Banach manifold whose tangent space at a pointf : X→ZisCr(X, fD)⊕Cr+1(X, T X).

Proof. Part (i) is clear since Diffr+10 (X)acts one0r(X, Z,D)through the natural right action

e0r(X, Z,D)×Diffr+10 (X)→e0r(X, Z,D), (f, ψ )7→f ◦ψ.

For (ii), pickf ∈e0r(X, Z,D),u ∈ Cr(X, fD)andv ∈ Cr+1(X, T X). The flow ofvyields a family of diffeomorphismsψt ∈Diffr+10 (X)withψ0=IdXandψ˙t|t=0=v (in what follows, all derivativesdtd|t=0will be indicated by a dot). Now, fixu˜ ∈Cr(Z,D) such thatu= ˜u◦f, by extending theCr vector fieldfufromf (X)toZ. The extension mappingu 7→ ˜ucan be chosen to be a continuous linear map of Banach spaces, using

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e.g. a retraction from a tubular neighborhood of theCr submanifoldf (X) ⊂ Z. Let ft be the flow ofu˜ starting at f0 = f, i.e. dtdft = ˜u(ft). Let (ej)1≤j≤N be a local holomorphic frame ofT Zsuch that(ej)n+1≤j≤Nis a holomorphic frame ofD,(ej)its dual frame and∇the unique local holomorphic connection ofT Zsuch that∇ej =0. For j =1, . . . , n, we find

d

dt(ej◦dft)=ej(ft)◦ ∇dft

dt =ej(ft)◦ ∇(u(f˜ t))=ej(ft)◦(∇ ˜u)(ft)·dft. However, if we writeu˜ = P

n+1≤k≤Nkek we see that the composition vanishes since ejek =0. Therefore dtd(ej◦dft)=0 andej◦dft =ej(f )◦df ∈Cr(X). This shows thatft ∈e0r(X, Z,D)for allt, and by definition we havef˙t = ˜u◦f =u. Now, if we definegt =ft◦ψt, we findgt ∈ e0r(X, Z,D)by (i), andg˙t =u+fvsinceψ˙t =v.

The mapping(u, v)7→g1=(ft◦ψt)|t=1defines a local “linearization” ofe0r(X, Z,D)

nearf. ut

We may now consider the differential of this action at(f, ψ ), wheref ∈ e0r(X, Z,D) andψ =IdX. If we restrictuto be inCr(X, fD), we actually get an isomorphism of Banach spaces

Cr(X, fD)×Cr(X, T X)→Cr(X, fT Z), (u, v)7→u+fv, (3.1) by the transversality condition. In fact, we can (non-canonically) define one0r(X, Z,D) a “lifting”

8(f,):Cr(X, fD)→ ˜0r(X, Z,D), u7→8(f, u)

on a small neighborhood of the zero section, and the differential of8(f,)at 0 is given by the inclusionCr(X, fD) ,→ Cr(X, fT Z). Modulo composition with elements of Diffr+10 (X)close to identity (i.e. in the quotient spacee0r(X, Z,D)/Diffr0+1(X)), small deformations offare parametrized by8(f, u)whereuis a small section ofCr(X, fD).

The first variation offdepends only on the differential of8along the zero section ofT Z, so it is actually independent of the choice of our map8. We can think of small variations offasf+u, at least if we are working in local coordinates(z1, . . . , zN)∈CNonZ, and we can assume thatDz⊂TzZ =CN; the use of a map8like those already considered is however needed to make the arguments global.

Let us summarize these observations as follows.

Lemma 3.4. For1 ≤r <∞, the quotient spacee0r(X, Z,D)/Diffr0+1(X)is a Banach manifold whose tangent space atf can be identified withCr(X, fD)via the differential of the composition

Cr(X, fD)−−−→ ˜8(f,) 0r(X, Z,D)→ ˜0r(X, Z,D)/Diffr+10 (X)

at0, where the first arrow is given byu 7→8(f, u)and the second arrow is the natural

map to the quotient. ut

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Our next goal is to computeJf and the differentialdJf off 7→ Jf whenf varies in the above Banach manifolde0r(X, Z,D). Near a pointz0∈Zwe can pick holomorphic coordinatesz=(z1, . . . , zN)centered atz0such thatDz0 =Span(∂/∂zj)n+1≤j≤N. Then we have

Dz=Span ∂

∂zj

+ X

1≤i≤n

aij(z) ∂

∂zi

n+1≤j≤N

, aij(z0)=0. (3.2)

In other words,Dz is the set of vectors of the form(a(z)η, η) ∈ Cn×CN−n, where a(z) = (aij(z))is a holomorphic map into the space L(CN−n,Cn)of n×(N −n) matrices. A trivial calculation shows that the vector fieldsej(z) =

∂zj +P

iaij(z)∂z

i

have brackets equal to [ej, ek] = X

1≤i≤n

∂aik

∂zj

(z0)−∂aij

∂zk

(z0) ∂

∂zi

atz0, n+1≤j, k≤N;

in other words, the torsion tensorθis given by θ (z0)= X

1≤i≤n, n+1≤j,k≤N

θij k(z0) dzj∧dzk⊗ ∂

∂zi,

θij k(z0)= 1 2

∂aik

∂zj (z0)−∂aij

∂zk (z0)

.

(3.3)

We now take a pointx0 ∈ X and apply this toz0 =f (x0) ∈ M =f (X)⊂ Z. In the coordinatesz=(z1, . . . , zN)chosen as above, we haveTz0M⊕Span(∂/∂zj)n+1≤j≤N = Tz0Z, so we can representMin the coordinatesz=(z0, z00)∈Cn×CN−nlocally as a graphz00=g(z0)in a small polydisc0×00centered atz0, and usez0=(z1, . . . , zn)∈0 as local (non-holomorphic !) coordinates onM. Hereg:0→00isCr+1differentiable andg(z00)=z000. The embeddingf :X→Zis itself obtained as the composition with a certain localCr diffeomorphismϕ :X⊃V →0⊂Cn, i.e.

f =F◦ϕonV , ϕ:V 3x 7→z0=ϕ(x)∈0⊂Cn, F :03z07→(z0, g(z0))∈Z.

With respect to the(z0, z00)coordinates, we get anR-linear isomorphism dF (z0):Cn→TF (z0)M⊂TF (z0)Z'Cn×CN−n,

ζ 7→(ζ, dg(z0)·ζ )=(ζ, ∂g(z0)·ζ+∂g(z0)·ζ ).

Here∂gis defined with respect to the standard complex structure ofCn 3 z0and has a priori no intrinsic meaning. The almost complex structureJf can be explicitly defined by Jf(x)=dϕ(x)−1◦JF(ϕ(x))◦dϕ(x), (3.4) whereJF is the almost complex structure onMdefined by the embeddingF :M ⊂Z, expressed in coordinates asz07→(z0, g(z0)). By construction we get

JF(z0)=dF (z0)−1◦πZ,D,M(F (z0))◦JZ(F (z0))◦dF (z0) (3.5)

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whereJZ is the complex structure onZ andπZ,D,M(z) : TzZ → TzMis theR-linear projection toTzM alongDzat a pointz ∈ M. Since these formulas depend on the first derivatives of F, we see that Jf is at least of class Cr−1 on X and JF is at least of classCr−1onM. We will see in Proposition3.5thatJf is in fact of classCr onX for f ∈e0r(X, Z,D). Using the identificationsTF (z0)M'Cn,TzZ'CNgiven by the above choice of coordinates, we simply haveJZη=iηonT Zsince the(zj)are holomorphic, and we get therefore

JZ(F (z0))◦dF (z0)·ζ =idF (z0)·ζ =i(ζ, dg(z0)·ζ )= iζ, ∂g(z0)·iζ −∂g(z0)·iζ

=(iζ, dg(z0)·iζ )−2(0, ∂g(z0)·iζ ).

By definition ofz7→a(z), we have(a(z)η, η)∈Dzfor everyη∈CN−n, and so πZ,D,M(z)(0, η)=πZ,D,M(z) (0, η)−(a(z)η, η)

= −πZ,D,M(z)(a(z)η,0).

We take hereη=∂g(z0)·iζ. As(iζ, dg(z0)·iζ )∈TF (z0)Malready, we find πZ,D,M(F (z0))◦JZ(F (z0))◦dF (z0)·ζ

=(iζ, dg(z0)·iζ )+2πZ,D,M(F (z0)) a(F (z0))∂g(z0)·iζ,0 . From (3.5), we get in this way

JF(z0)·ζ =iζ−2dF (z0)−1◦πZ,D,M(F (z0)) ia(F (z0))∂g(z0)·ζ,0

. (3.6)

In particular, sincea(z0)=0, we simply haveJF(z00)·ζ =iζ.

We want to evaluate the variation of the almost complex structureJf when the embed- dingft =Ft◦ϕtvaries with respect to some parametert∈ [0,1]. Letw∈Cr(X, fT Z) be a given infinitesimal variation of ft and w = u+fv, u ∈ Cr(X, fD), v ∈ Cr+1(X, T X)its direct sum decomposition. With respect to the trivialization ofDgiven by our local holomorphic frame(ej(z)), we can write in local coordinates

u(ϕ−1(z0))= a(F (z0))·η(z0), η(z0)

∈DF (z0)

for some sectionz07→η(z0)∈CN−n. Therefore

u(ϕ−1(z0))= 0, η(z0)−dg(z0)·a(F (z0))·η(z0)

+F a(F (z0))·η(z0) where the first term is “vertical” and the second one belongs toTF (z0)M. We then get a slightly different decompositionwe:=w◦ϕ−1=

eu+Fev∈Cr(0, FT Z)where eu(z0)= 0, η(z0)−dg(z0)·a(F (z0))·η(z0)

∈ {0} ×CN−n,

ev(z0)=ϕv(z0)+a(F (z0))·η(z0)∈Cn. This allows us to perturbf =F ◦ϕasft =Ft◦ϕt with





X3x 7→z0t(x)=ϕ(x)+tev(ϕ(x))∈Cn, Cn3z07→Ft(z0)=(z0, gt(z0))∈Z,

gt(z0)=g(z0)+teu(z0)=g(z0)+t η(z0)−dg(z0)·a(F (z0))·η(z0) ,

(3.7)

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in such a way that f˙t = d

dt(ft)|t=0 = w. We replacef, g, F, M byft, gt, Ft, Mt

in (3.6) and compute the derivative for t = 0 andz0 = z00. Sincea(z0) = 0, the only non-zero term is the one involving the derivative of the mapt 7→ a(Ft(z0)). We have F˙t(z00)=(0, η(z00))=u(x0)whereη(z00)∈CN−n, thusJ˙Ft can be expressed atz00 as

Ft(z00)·ζ := d

dt(JFt(z00)·ζ )|t=0

= −2dF (z00)−1◦πZ,D,M(z00) ida(z0)(u(x0))·∂g(z00)·ζ,0 .

Now, if we setλ=ida(z0)(u(x0))∂g(z00)·ζ, asDz0 = {0} ×CN−nin our coordinates, we immediately get

πZ,D,M(z00)(λ,0)=(λ, dg(z00)·λ)=dF (z00)·λ, so dF (z00)−1◦πZ,D,M(z00)(λ,0)=λ.

Therefore, we obtain the very simple expression

Ft(z00)= −2ida(z0)(u(x0))·∂g(z00)∈End

C(Cn) (3.8)

whereda(z0)(ξ ) ∈ L(CN−n,Cn)is the derivative of the matrix functionz 7→ a(z)at z=z0in the directionξ ∈ CN, and∂g(z00)is viewed as an element ofL

C(Cn,CN−n).

What we want is the derivative ofJft =dϕt−1◦JFtt)◦dϕt atx0fort =0. Writingϕ

as an abbreviation fordϕ, we find, fort=0,

ft = −ϕ−1◦dϕ˙t◦ϕ−1 ◦JF(ϕ)◦ϕ−1◦JF(ϕ)◦dϕ˙t−1 ◦ ˙JFt(ϕ)◦ϕ

=2JfJf−1ϕ˙t)+ϕ−1◦ ˙JFt(ϕ)◦ϕ, (3.9) where the first term on the right hand side comes from the identity−ds◦Jf+Jf ◦ds = 2JfJfswiths=ϕ−1ϕ˙t ∈Cr(X, T X)andds =ϕ−1dϕ˙t. Our choicesev=ϕv+a◦F·η andϕt =ϕ+tev◦ϕyield

ϕ˙t =

ev◦ϕ=ϕv+a◦f ·η◦ϕ, so ϕ−1ϕ˙t =v+ϕ−1(a◦f ·η◦ϕ).

If we recall thata(z0)=0 andη(ϕ(x0))=η(z00)=pr2u(x0), we get, atx =x0,

Jf−1ϕ˙t)(x0)=∂Jfv(x0)+ϕ−1 da(z0)(∂Jff (x0))·pr2u(x0)

. (3.10)

By construction,ϕ=dϕis compatible with the almost complex structures(X, Jf)and (Cn, JF). A combination of (3.8), (3.9) and (3.10) yields

ft(x0)=2JfJfv(x0)

−1 2ida(z0)(∂Jff (x0))·pr2u(x0)−2ida(z0)(u(x0))·∂g(z00)◦ϕ .

As∂Jff (x0)=(∂JFF )(z00)◦dϕ(x0)=(0, ∂g(z00))◦ϕandf=F◦ϕ, we get J˙ft(x0)=f−1F 2ida(z0)(∂Jff (x0))·pr2u(x0)−2ida(z0)(u(x0))·pr2Jff (x0)

+2JfJfv(x0).

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By (3.3), the torsion tensorθ (z0): Dz0 ×Dz0 → Tz0Z/Dz0 ' FTz0M = fTx0X is given by

θ (η, λ)= X

1≤i≤n, n+1≤j,k≤N

∂aik

∂zj (z0)−∂aij

∂zk(z0)

ηjλk

∂zi

=da(z0)(η)·λ−da(z0)(λ)·η.

Since our pointx0 ∈Xwas arbitrary andJ˙ft(x0)is the value of the differentialdJf(w) atx0, we finally get the global formula

dJf(w)=2Jf f−1θ (∂Jff, u)+∂Jfv (observe that∂Jff ∈L

C(T X, fT Z)actually takes values infD, so taking a projec- tion tofDis not needed). We conclude:

Proposition 3.5. Letr∈ [1,∞] ∪ {ω}.

(i) The natural mapf 7→Jf sendse0r(X, Z,D)intoJr(X).

(ii) The differential of the natural map

e0r(X, Z,D)→Jr(X), f 7→Jf,

along every infinitesimal variationw=u+fv:X→fT Z=fD⊕fT Xof f is given by

dJf(w)=2Jf f−1θ (∂Jff, u)+∂Jfv

whereθ:D×D→T Z/Dis the torsion tensor of the holomorphic distributionD, and ∂f = ∂Jff, ∂v = ∂Jfv are computed with respect to the almost complex structure(X, Jf).

(iii) The differentialdJf off 7→Jf one0r(X, Z,D)is a continuous morphism Cr(X, fD)⊕Cr+1(X, T X)→Cr(X,End

C(T X)), (u, v)7→2i(θ (∂f, u)+∂v).

Ifr= ∞orr=ωthen we replacer+1 byrin (iii).

Proof. Parts (i) and (ii) are clear, as it can be easily seen that∂f depends only on the transversal part ofdf by the very definition ofJf and of∂f =1

2(df +JZ◦df ◦Jf).

Part (iii) is a trivial consequence of the general variation formula. ut Our goal now is to understand under which conditionsf 7→Jf can be a local submersion frome0r(X, Z,D)toJr(X). If we do not take into account the quotient by the action of Diffr+10 onJr(X), we obtain a more demanding condition. For that stronger requirement, we see that a sufficient condition is that the continuous linear map

Cr(X, fD)→Cr(X,End

C(T X)), u7→2iθ (∂f, u), (3.11) be surjective.

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Theorem 3.6. Fixr∈ [1,∞] ∪ {ω}(again,ωmeans real analyticity here). Let(Z,D)be a complex manifold equipped with a holomorphic distribution, and letf ∈e0r(X, Z,D) be a transverse embedding with respect toD. Assume thatf and the torsion tensorθof Dsatisfy the following additional conditions:

(i) f is a totally real embedding, i.e.∂f (x)∈ End

C(TxX, Tf (x)Z)is injective at every pointx ∈X;

(ii) for everyx ∈Xand everyη∈End

C(T X), there exists a vectorλ∈Df (x)such that θ (∂f (x)·ξ, λ)=η(ξ ) for allξ ∈T X.

Then there is a neighborhoodU off ine0r(X, Z,D)and a neighborhood V ofJf in Jr(X)such thatU →V,f 7→Jf, is a submersion.

Proof. This is an easy consequence of the implicit function theorem in the Banach space situationr <∞. Let8be the real analytic mapT Z→Z×Zconsidered in Section3, and let

9f :Cr(X, fD)→e0r(X, Z,D), u7→8(f, fu).

By definitionf = 9f(0)and9f defines the infinite-dimensional manifold structure on e0r(X, Z,D) by identifying a neighborhood of 0 in the topological vector space Cr(X, fD)with a neighborhood off ine0r(X, Z,D), and providing in this way a “co- ordinate chart”. As we have seen in (3.11), the differentialu7→dJf(u)is given by

u7→Lf(u)=2iθ (∂f, u) whereLf ∈ Cr(X,Hom(fD,End

C(T X)))is by our assumption (ii) a surjective mor- phism of bundles of finite rank. The kernelK:=KerLf is aCr subbundle offD, thus we can select aCrsubbundleEoffDsuch that

fD=K⊕E.

(This can be seen by a partition of unity argument forr 6=ω; in the real analytic case, one can instead complexify the real analytic objects and apply a Steinness argument together with Cartan’s Theorem B to obtain a splitting). The differential of the composition

u7→g=9f(u), g7→Jg,

is precisely the restriction ofLf =dJf to sectionsu∈Cr(X,E)⊂Cr(X, fD), which is by construction a bundle isomorphism fromCr(X,E)ontoCr(X,End

C(T X)). Hence for r <∞,u7→g=9f(u)7→J9f(u)is aCr-diffeomorphism from a neighborhoodWrE(0) of the zero section ofCr(X,E)onto a neighborhoodVr ofJf ∈Jr(X), and sog7→Jg is aCr-diffeomorphism fromUrE := 9(WrE(0))ontoVr. This argument does not quite work forr= ∞orr=ω, since we do not have Banach spaces. Nevertheless, forr= ∞, we can apply the result for a given finiter0and considerr0 ∈ [r0,∞[arbitrarily large.

Then, by applying a local diffeomorphism argument inCr0at all nearby pointsg=9f(u) (and by using the injectivity onUrE

0), we see that the map UrE0 :=9f(WrE

0(0)∩Cr0(X,E))→Vr0 ∩Jr0(X), g7→Jg,

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