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 aC^{∞}smooth
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, J_{X}), whereXis a real manifold of dimension 2nandJ_{X}a smooth
section of End(T X)such thatJ_{X}^{2} = −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 structureJ_{Z} (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 fibersD_{x}are 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

vanishes identically. As is well known,θis skew symmetric in(ζ, η)and can be viewed
as a holomorphic section of the bundle3^{2}D^{∗}⊗(T Z/D).

LetMbe a real submanifold ofZof classC^{∞}and of real dimension 2nwithn < N.
We say thatMistransverse toDif for everyx ∈Mwe have

T_{x}M⊕D_{x}=T_{x}Z. (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 subsetD_{sing} ⊂Z of codimension ≥ 2, and we further assume in this case that
M∩D_{sing}= ∅.

WhenMis transverse toD, one gets a naturalR-linear isomorphism

T_{x}M'T_{x}Z/D_{x} (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 structureJ_{M}^{Z,D}
onM.

Moreover, when Dis a foliation (i.e. O(D) is an integrable subsheaf ofO(T Z)),
thenJ_{M}^{Z,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 subset^{0} ⊂C^{n}such 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
to^{0} ⊂ C^{n},^{00} ⊂ C^{N−n}, and then D,T Z/Dare identified with the trivial bundles

×({0} ×C^{N−n})and×C^{n}. 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= e^{0}×e^{00}would give coordinates
that are changed by local biholomorphisms^{0}→e^{0}in the intersection∩, thanks toe
the holomorphic character ofD. Thus we directly see in that case thatJ_{M}^{Z,D}comes 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.f∗T_{x}X⊕D_{f (x)} =T_{f (x)}Zfor everyx ∈X (andf (X)does not meetD_{sing}in
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
byJ_{f} the almost complex structuref^{∗}(J_{f (X)}^{Z,D}).

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 =J_{f}?

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
manifoldZ_{n,k}of dimensionN =2k+2(k^{2}+n(k−n))possessing a real structure(i.e. an
anti-holomorphic algebraic involution)and an algebraic distributionD_{n,k} ⊂T Z_{n,k} of
codimensionn, for which every compactn-dimensional almost complex manifold(X, J )
admits an embeddingf : X ,→ Z_{n,k}^{R} , transverse toD_{n,k} and contained in the real part
ofZ_{n,k}, such thatJ =J_{f}.

Remark 1.3. To constructf we first embedX differentiably into R^{k},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 =38n^{2}+8n(we will see
that a quadratic boundN =O(n^{2})is optimal, but the above explicit value could perhaps
be improved). SinceZ = Z_{n,k} andD = D_{n,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,

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.J^{∗}ω = 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 ,→CP^{s} such thatω=g^{∗}ω_{FS}is the pull-back of the standard Fubini–Study
metricω_{FS}onCP^{s}. 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 manifoldZ_{n,b,k} of dimensionN = 2bk(2bk+1)+2n(2bk−n)), equipped
with a real structure and an algebraic distributionD_{n,b,k}⊂T Z_{n,b,k}of codimensionn, for
which every compactn-dimensional almost complex symplectic manifold(X, J, ω)with
second Betti numberb_{2}≤band aJ-compatible symplectic formωadmits an embedding
f : X ,→ Z^{R}_{n,b,k}transverse toD_{n,b,k} and contained in the real part ofZ_{n,k}, such that
J =J_{f} andω=f^{∗}βfor some transverse K¨ahler structureβon(Z_{n,b,k},D_{n,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,D_{n,k})be the affine algebraic
manifold equipped with the algebraic distributionD_{n,k} ⊂ T Zn,k introduced in Theo-
rem1.2. Then, for every compactn-dimensional(integrable)complex manifold(X, J ),
there exists an embeddingf :X ,→ Z_{n,k}^{R} transverse toD_{n,k}, contained in the real part
ofZ_{n,k}, such that

(i) J =J_{f} and∂_{J}f is injective;

(ii) Im(∂Jf )is contained in the isotropic locusI_{D}_{n,k}of the torsion operatorθofDn,k, the
intrinsically defined algebraic locus in the Grassmannian bundleGr(Dn,k, n)→Zn,k

of complexn-dimensional subspaces inD_{n,k}, consisting of those subspaces Ssuch
thatθ|S×S =0.

The inclusion condition (ii) Im(∂_{J}f ) ⊂I_{D}_{n,k} is in fact necessary and sufficient for the
integrability ofJ_{f}.

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 ofC^{n}can 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
ofC^{N}. 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 classC^{∞}and of real dimension 2nwithn < N. We assume that there is an
embeddingf :X ,→Zthat is transverse toD, namelyf (X)∩D_{sing}= ∅and

f∗T_{x}X⊕D_{f (x)}=T_{f (x)}Z (2.1)

at every pointx ∈ X. HereD_{f (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 structuref^{∗}J_{f (X)}^{Z,D}
onT Xwhich we will simply denote byJ_{f} 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 variationJf_{t} 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 f_{t} : X → Z,t ∈ [0,1], be an isotopy of transverse embeddings
of a compact smooth real manifold. Then all complex structures(X, J_{f}_{t})are biholomor-
phic to(X, J_{f}_{0})through a smooth variation of diffeomorphisms inDiff_{0}(X), the identity
component of the groupDiff(X)of diffeomorphisms ofX.

Proof. By an easy connectedness argument, it is enough to produce a smooth variation
of biholomorphismsψt,t_{0} : (X, Jf_{t}_{0})→ (X, Jf_{t})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 atz_{0} =f_{t}_{0}(x)∈ ZrD_{sing},Dis locally nearx the family of fibers of a holo-
morphic submersionσ :Z ⊃→ ^{0} ⊂C^{n}defined on a neighborhoodofz_{0}. Then
σ◦f_{t} :X⊃f_{t}^{−1}()→^{0}is by definition a local biholomorphism from(X, J_{f}_{t})to^{0}
(endowed with the standard complex structure ofC^{n}). Now,ψ_{t,t}_{0} =(σ◦f_{t})^{−1}◦(σ◦f_{t}_{0})
defines a local biholomorphism from(X, J_{f}_{t}

0)to(X, J_{f}_{t})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 positionf_{t}_{0}(X)to position
f_{t}(X)of the embedding). Clearlyψ_{t,t}_{0}depends smoothly ontand satisfies the chain rule

ψ_{t,t}_{0}◦ψ_{t}_{0}_{,t}_{1} =ψ_{t,t}_{1}. 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
J^{int}(X)/Diff_{0}(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 = J_{f} 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)^{2n}into a complex torusZ=C^{N}/3equipped with a linear
foliationD.

Proof of Lemma2.4. We first coverZrD_{sing}by a countable family of coordinate open
sets ν ' ^{0}_{ν} ×^{00}_{ν} such that the first projections σν : ν → ^{0}_{ν} ⊂ C^{n} define the
foliation. We assume here that^{0}_{ν} and^{00}_{ν} are balls of sufficiently small radius, so that
all fibers z^{0} ×^{00}_{ν} 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

z_{j}^{0} ×^{00}_{ν}

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

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)/Diff_{0}(X). The imageM=f (X)is itself given by a finite collection
of graphs of mapsg_{ν} :^{0}_{ν} →^{00}_{ν} that glue together, for a certain finite subfamily of co-
ordinate sets(_{ν})_{ν}_{∈I}extracted from the initial countable family. However, any two such
transverse submanifolds(M_{k})_{k=0,1}and associated collections of graphs(g_{k,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 'e^{0}_{ν}×e^{00}_{ν} bν still coveringM0andM1, and a cut-off
functionθν(z^{0})equal to 1 on e^{0}_{ν} and with support in ^{0}_{ν}. 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

M_{t,1} given by

(G_{t,1,1}(z^{0})=γ t θ_{1}(z^{0});g_{0,1}(z^{0}), g_{1,1}(z^{0})

on^{0}_{1},
G_{t,1,ν}(z^{0})=g_{0,ν}(z^{0}) onσ_{ν} _{ν}r(Supp(θ_{1})×e^{00}_{1})

, ν6=1,
M_{t,k} given by

(Gt,k,k(z^{0})=γ t θk(z^{0});Gt,k−1,k(z^{0}), Gt,k−1,k(z^{0})

on ^{0}_{k},
Gt,k,ν(z^{0})=Gt,k−1,ν(z^{0}) onσν ν r(Supp(θk)×e^{00}_{k})

, ν6=k,

whereγ (t;a^{00}, b^{00})denotes the geodesic segment betweena^{00}andb^{00}in each fiberz^{0}×^{00}_{ν}.
By construction, we haveM0,k=M0andM1,k∩U_{k} =M1∩U_{k}onU_{k}=e1∪ · · · ∪e_{k},
thusf_{t} :=F_{t,s} :M0→ M_{t} 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 structureJf_{t} 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 =R^{2n}/Z^{2n}be an even-dimensional real torus and
Z=C^{N}/3a complex torus where3 'Z^{2N} is a lattice ofC^{N},N > n. Any complex
vector subspaceD ⊂C^{N} 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 : R^{2n} → C^{N} that sends the standard basis(e1, . . . , e2n)
ofZ^{2n} 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 'Q^{2N}, thus the permitted directions forV_{ε}are dense, and for most of them
f is indeed transverse toD. For a transverse linear embedding, we get anR-linear iso-
morphismu˜ : R^{2n} → C^{N}/D, and the complex structureJ_{f} 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 imagesε_{1}, . . . , ε_{2n}to be
C-linearly independent, so that the mapu :Z^{2n} → 3,e_{j} 7→ε_{j}, extends to an injection
v : C^{2n} →C^{N}. Once this is done, the isotopy class of embedding is determined, and a

translation invariant complex structureJ onX is given by a direct sum decomposition
C^{2n} = S⊕S with dim_{C}S = n(andS the complex conjugate ofS). What we need is
that the compositionv˜ :C^{2n}→C^{N}→C^{N}/Ddefines aC-linear isomorphism ofSonto
v(S)˜ ⊂C^{N}/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)
inC^{N} (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 = E_{m,N+1} be a
non-empty set of subsets of cardinality 2m+1 of{0,1, . . . , N}. ForJ ∈ E, defineU_{J}
to be the open set of points [z_{0} : . . . : z_{N}] ∈ CP^{N} such that z_{j} 6= 0 forj ∈ J and
U_{E}=S

J∈EU_{J}. Then, consider the action ofC^{m}onU_{E} given by
w· [z0:. . .:zN] = [e^{`}^{0}^{(w)}z0:. . .:e^{`}^{N}^{(w)}zN]

where`_{j} ∈(C^{m})^{∗}are complex linear formsC^{m}→C, 0≤j ≤N. Then Bosio [Bos01,
Th´eor`eme 1.4] proves that the space of orbitsX =U_{E}/C^{m}is 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(C^{m})^{∗}of{`_{j}}_{j∈J}

1 and{`_{j}}_{j∈J}

2 overlap on some non-empty open set;

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

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

ζ_{j} =

N

X

k=0

λ_{j k}z_{k} ∂

∂zk

, λ_{j k}= ∂`k

∂wj

, 1≤j ≤m.

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

0≤k≤Nλ_{j,k}|z_{k}|^{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 inU_{E} ⊂CP^{N} (see [Bos01,
Prop. 2.3 and discussion thereafter] and also [BoM06, Part III, Section 12]).

3. Deformation of transverse embeddings

Letf :X→(Z,D)be a transverse embedding. ThenJ_{f} :=f^{∗}(J_{f (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 ofC^{r} mappings. For anyr ∈ [1,∞], we consider
the group Diff^{r}(X)of diffeomorphisms ofXof classC^{r}, and the subgroup Diff^{r}_{0}(X)of
diffeomorphisms diffeotopic to identity. Whenr = s +γ is not an integer,s = brc,
thenC^{r} denotes the H¨older space of maps of class C^{s} with derivatives of order s that
are H¨older continuous with exponentγ. Similarly, we consider the spaceC^{r}(X, Z)ofC^{r}
mappingsX→Zequipped with C^{r} convergence topology (of course, in Diff^{r}(X), the
topology also requires convergence of sequencesf_{ν}^{−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)=zandd_{ζ}8(z, ζ )_{|ζ}_{=0}=Id
onT_{z}Z. WhenZis embedded inC^{N}

0 for someN^{0}, such a map can be obtained by taking
8(z, ζ ) = ρ(z+ζ ), whereρ is a local holomorphic retraction C^{N}

0 → Z andT_{z}Z is

identified to a vector subspace ofC^{N}

0. In general (i.e. whenZis not necessarily Stein),
one can still find aC^{∞}or even real analytic map 8satisfying the same conditions, by
taking e.g.8(z, ζ ) = (z,exp_{z}(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,∞[,C^{r}(X, Z)is a Banach manifold whose tangent space at a
pointf : X → Z isC^{r}(X, f^{∗}T Z), andDiff^{r}_{0}(X)is a “Banach Lie group” with “Lie
algebra”C^{r}(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 ofC^{r} maps is merelyC^{r}^{2} for 0< r <1, it is
actually aC^{r} 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 off^{∗}T Z. This parametrization is one-to-
one, andfeisC^{r} if and only ifuisC^{r} (providedf is). The argument is similar, and very

well known indeed, for Diff^{r}_{0}(X). ut

Now, letJ^{r}(X) denote the space of almost complex structures of classC^{r} onX. For
1 ≤r <∞, this is a Banach manifold whose tangent space at a pointJ is the space of
sectionsh∈C^{r}(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 Diff^{r}_{0}(X)onJ^{r−1}(X)defined
by

(J, ψ )7→ψ^{∗}J, ψ^{∗}J (x)=dψ (x)^{−1}◦J (ψ (x))◦dψ (x).

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

∂J :C^{r}(X, T X)→C^{r−1}(X, 3^{0,1}T X^{∗}⊗T X^{1,0})=C^{r−1}(X,End

C(T X)),
namely it is given byv7→J◦dv−dv◦J =2J ∂_{J}vifv∈C^{r}(X, T X)is the infinitesimal
variation ofψ.

Let0^{r}(X, Z,D)be the space ofC^{r} embeddings ofXintoZthat are transverse toD.

Transversality is an open condition, so0^{r}(X, Z,D)is an open subset inC^{r}(X, Z). Now,
Diff^{r}_{0}(X)acts on0^{r}(X, Z,D)through the natural right action

0^{r}(X, Z,D)×Diff^{r}_{0}(X)→0^{r}(X, Z,D), (f, ψ )7→f ◦ψ.

We wish to consider the differential of this action at(f, ψ ),ψ =Id_{X}, with respect
to the tangent space isomorphisms of Lemma 3.1. This is just the addition law in the
bundlef^{∗}T Z:

C^{r}(X, f^{∗}T Z)×C^{r}(X, T X)→C^{r}^{−1}(X, f^{∗}T Z), (u, v)7→u+f∗v.

A difficulty occurring here is the loss of regularity from C^{r} toC^{r−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
e0^{r}(X, Z,D)⊂0^{r}(X, Z,D)⊂C^{r}(X, Z)

of transverse embeddingsf :X→Zsuch thatf is of classC^{r} 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 sete0^{r}(X, Z,D) =
0^{r}(X, Z,D).

Thene0^{r}(X, Z,D)satisfies the following conditions:

Proposition 3.3. For1≤r <∞:

(i) the groupDiff^{r+1}_{0} (X)acts on the right one0^{r}(X, Z,D);

(ii) the spacee0^{r}(X, Z,D)is a Banach manifold whose tangent space at a pointf :
X→ZisC^{r}(X, f^{∗}D)⊕C^{r+1}(X, T X).

Proof. Part (i) is clear since Diff^{r+1}_{0} (X)acts one0^{r}(X, Z,D)through the natural right
action

e0^{r}(X, Z,D)×Diff^{r+1}_{0} (X)→e0^{r}(X, Z,D), (f, ψ )7→f ◦ψ.

For (ii), pickf ∈e0^{r}(X, Z,D),u ∈ C^{r}(X, f^{∗}D)andv ∈ C^{r+1}(X, T X). The flow
ofvyields a family of diffeomorphismsψ_{t} ∈Diff^{r+1}_{0} (X)withψ_{0}=Id_{X}andψ˙_{t|t=0}=v
(in what follows, all derivatives_{dt}^{d}_{|t=0}will be indicated by a dot). Now, fixu˜ ∈C^{r}(Z,D)
such thatu= ˜u◦f, by extending theC^{r} vector fieldf∗ufromf (X)toZ. The extension
mappingu 7→ ˜ucan be chosen to be a continuous linear map of Banach spaces, using

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

d

dt(e_{j}^{∗}◦df_{t})=e_{j}^{∗}(f_{t})◦ ∇dft

dt =e_{j}^{∗}(f_{t})◦ ∇(u(f˜ _{t}))=e_{j}^{∗}(f_{t})◦(∇ ˜u)(f_{t})·df_{t}.
However, if we writeu˜ = P

n+1≤k≤Nu˜_{k}e_{k} we see that the composition vanishes since
e_{j}^{∗}e_{k} =0. Therefore _{dt}^{d}(e_{j}^{∗}◦df_{t})=0 ande_{j}^{∗}◦df_{t} =e_{j}^{∗}(f )◦df ∈C^{r}(X). This shows
thatf_{t} ∈e0^{r}(X, Z,D)for allt, and by definition we havef˙_{t} = ˜u◦f =u. Now, if we
defineg_{t} =f_{t}◦ψ_{t}, we findg_{t} ∈ e0^{r}(X, Z,D)by (i), andg˙_{t} =u+f∗vsinceψ˙_{t} =v.

The mapping(u, v)7→g1=(f_{t}◦ψ_{t})|t=1defines a local “linearization” ofe0^{r}(X, Z,D)

nearf. ut

We may now consider the differential of this action at(f, ψ ), wheref ∈ e0^{r}(X, Z,D)
andψ =Id_{X}. If we restrictuto be inC^{r}(X, f^{∗}D), we actually get an isomorphism of
Banach spaces

C^{r}(X, f^{∗}D)×C^{r}(X, T X)→C^{r}(X, f^{∗}T Z), (u, v)7→u+f∗v, (3.1)
by the transversality condition. In fact, we can (non-canonically) define one0^{r}(X, Z,D)
a “lifting”

8(f,^{•}):C^{r}(X, f^{∗}D)→ ˜0^{r}(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 inclusionC^{r}(X, f^{∗}D) ,→ C^{r}(X, f^{∗}T Z). Modulo composition with elements of
Diff^{r+1}_{0} (X)close to identity (i.e. in the quotient spacee0^{r}(X, Z,D)/Diff^{r}_{0}^{+1}(X)), small
deformations offare parametrized by8(f, u)whereuis a small section ofC^{r}(X, f^{∗}D).

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)∈C^{N}onZ, and
we can assume thatD_{z}⊂TzZ =C^{N}; 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 spacee0^{r}(X, Z,D)/Diff^{r}_{0}^{+1}(X)is a Banach
manifold whose tangent space atf can be identified withC^{r}(X, f^{∗}D)via the differential
of the composition

C^{r}(X, f^{∗}D)−−−→ ˜^{8(f,}^{•}^{)} 0^{r}(X, Z,D)→ ˜0^{r}(X, Z,D)/Diff^{r+1}_{0} (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

Our next goal is to computeJf and the differentialdJf off 7→ Jf whenf varies in
the above Banach manifolde0^{r}(X, Z,D). Near a pointz0∈Zwe can pick holomorphic
coordinatesz=(z1, . . . , zN)centered atz0such thatD_{z}_{0} =Span(∂/∂zj)n+1≤j≤N. Then
we have

D_{z}=Span
∂

∂zj

+ X

1≤i≤n

a_{ij}(z) ∂

∂zi

n+1≤j≤N

, a_{ij}(z_{0})=0. (3.2)

In other words,D_{z} is the set of vectors of the form(a(z)η, η) ∈ C^{n}×C^{N−n}, where
a(z) = (a_{ij}(z))is a holomorphic map into the space L(C^{N−n},C^{n})of n×(N −n)
matrices. A trivial calculation shows that the vector fieldsej(z) = ^{∂}

∂z_{j} +P

iaij(z)_{∂z}^{∂}

i

have brackets equal to
[e_{j}, e_{k}] = X

1≤i≤n

∂aik

∂zj

(z_{0})−∂a_{ij}

∂zk

(z_{0})
∂

∂zi

atz_{0}, 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⊗ ∂

∂z_{i},

θ_{ij k}(z_{0})= 1
2

∂a_{ik}

∂z_{j} (z_{0})−∂aij

∂z_{k} (z_{0})

.

(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 haveTz_{0}M⊕Span(∂/∂z_{j})n+1≤j≤N =
T_{z}_{0}Z, so we can representMin the coordinatesz=(z^{0}, z^{00})∈C^{n}×C^{N−n}locally as a
graphz^{00}=g(z^{0})in a small polydisc^{0}×^{00}centered atz_{0}, and usez^{0}=(z_{1}, . . . , z_{n})∈^{0}
as local (non-holomorphic !) coordinates onM. Hereg:^{0}→^{00}isC^{r+1}differentiable
andg(z^{0}_{0})=z^{00}_{0}. The embeddingf :X→Zis itself obtained as the composition with a
certain localC^{r} diffeomorphismϕ :X⊃V →^{0}⊂C^{n}, i.e.

f =F◦ϕonV , ϕ:V 3x 7→z^{0}=ϕ(x)∈^{0}⊂C^{n}, F :^{0}3z^{0}7→(z^{0}, g(z^{0}))∈Z.

With respect to the(z^{0}, z^{00})coordinates, we get anR-linear isomorphism
dF (z^{0}):C^{n}→T_{F (z}^{0}_{)}M⊂T_{F (z}^{0}_{)}Z'C^{n}×C^{N−n},

ζ 7→(ζ, dg(z^{0})·ζ )=(ζ, ∂g(z^{0})·ζ+∂g(z^{0})·ζ ).

Here∂gis defined with respect to the standard complex structure ofC^{n} 3 z^{0}and 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)
whereJ_{F} is the almost complex structure onMdefined by the embeddingF :M ⊂Z,
expressed in coordinates asz^{0}7→(z^{0}, g(z^{0})). By construction we get

J_{F}(z^{0})=dF (z^{0})^{−1}◦π_{Z,D,M}(F (z^{0}))◦J_{Z}(F (z^{0}))◦dF (z^{0}) (3.5)

whereJZ is the complex structure onZ andπZ,D,M(z) : TzZ → TzMis theR-linear
projection toTzM alongD_{z}at a pointz ∈ M. Since these formulas depend on the first
derivatives of F, we see that Jf is at least of class C^{r−1} on X and JF is at least of
classC^{r−1}onM. We will see in Proposition3.5thatJ_{f} is in fact of classC^{r} onX for
f ∈e0^{r}(X, Z,D). Using the identificationsT_{F (z}^{0}_{)}M'C^{n},T_{z}Z'C^{N}given by the above
choice of coordinates, we simply haveJ_{Z}η=iηonT Zsince the(z_{j})are holomorphic,
and we get therefore

J_{Z}(F (z^{0}))◦dF (z^{0})·ζ =idF (z^{0})·ζ =i(ζ, dg(z^{0})·ζ )= iζ, ∂g(z^{0})·iζ −∂g(z^{0})·iζ

=(iζ, dg(z^{0})·iζ )−2(0, ∂g(z^{0})·iζ ).

By definition ofz7→a(z), we have(a(z)η, η)∈D_{z}for everyη∈C^{N−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(z^{0})·iζ. As(iζ, dg(z^{0})·iζ )∈T_{F (z}^{0}_{)}Malready, we find
π_{Z,D,M}(F (z^{0}))◦J_{Z}(F (z^{0}))◦dF (z^{0})·ζ

=(iζ, dg(z^{0})·iζ )+2π_{Z,D,M}(F (z^{0})) a(F (z^{0}))∂g(z^{0})·iζ,0
.
From (3.5), we get in this way

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

. (3.6)

In particular, sincea(z_{0})=0, we simply haveJ_{F}(z^{0}_{0})·ζ =iζ.

We want to evaluate the variation of the almost complex structureJ_{f} when the embed-
dingf_{t} =F_{t}◦ϕ_{t}varies with respect to some parametert∈ [0,1]. Letw∈C^{r}(X, f^{∗}T Z)
be a given infinitesimal variation of f_{t} and w = u+f∗v, u ∈ C^{r}(X, f^{∗}D), v ∈
C^{r+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}(z^{0}))= a(F (z^{0}))·η(z^{0}), η(z^{0})

∈D_{F (z}^{0}_{)}

for some sectionz^{0}7→η(z^{0})∈C^{N−n}. Therefore

u(ϕ^{−1}(z^{0}))= 0, η(z^{0})−dg(z^{0})·a(F (z^{0}))·η(z^{0})

+F∗ a(F (z^{0}))·η(z^{0})
where the first term is “vertical” and the second one belongs toT_{F (z}^{0}_{)}M. We then get a
slightly different decompositionwe:=w◦ϕ^{−1}=

eu+F∗ev∈C^{r}(^{0}, F^{∗}T Z)where
eu(z^{0})= 0, η(z^{0})−dg(z^{0})·a(F (z^{0}))·η(z^{0})

∈ {0} ×C^{N−n},

ev(z^{0})=ϕ∗v(z^{0})+a(F (z^{0}))·η(z^{0})∈C^{n}.
This allows us to perturbf =F ◦ϕasf_{t} =F_{t}◦ϕ_{t} with

X3x 7→z^{0}=ϕ_{t}(x)=ϕ(x)+tev(ϕ(x))∈C^{n},
C^{n}3z^{0}7→F_{t}(z^{0})=(z^{0}, g_{t}(z^{0}))∈Z,

g_{t}(z^{0})=g(z^{0})+teu(z^{0})=g(z^{0})+t η(z^{0})−dg(z^{0})·a(F (z^{0}))·η(z^{0})
,

(3.7)

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 andz^{0} = z^{0}_{0}. Sincea(z0) = 0, the only
non-zero term is the one involving the derivative of the mapt 7→ a(Ft(z^{0})). We have
F˙_{t}(z^{0}_{0})=(0, η(z^{0}_{0}))=u(x_{0})whereη(z^{0}_{0})∈C^{N−n}, thusJ˙_{F}_{t} can be expressed atz_{0}^{0} as

J˙_{F}_{t}(z^{0}_{0})·ζ := d

dt(J_{F}_{t}(z^{0}_{0})·ζ )_{|t=0}

= −2dF (z_{0}^{0})^{−1}◦π_{Z,D,M}(z^{0}_{0}) ida(z0)(u(x0))·∂g(z_{0}^{0})·ζ,0
.

Now, if we setλ=ida(z0)(u(x0))∂g(z^{0}_{0})·ζ, asD_{z}_{0} = {0} ×C^{N−n}in our coordinates,
we immediately get

π_{Z,D,M}(z^{0}_{0})(λ,0)=(λ, dg(z^{0}_{0})·λ)=dF (z^{0}_{0})·λ, so dF (z^{0}_{0})^{−1}◦π_{Z,D,M}(z^{0}_{0})(λ,0)=λ.

Therefore, we obtain the very simple expression

J˙F_{t}(z^{0}_{0})= −2ida(z_{0})(u(x0))·∂g(z^{0}_{0})∈End

C(C^{n}) (3.8)

whereda(z_{0})(ξ ) ∈ L(C^{N−n},C^{n})is the derivative of the matrix functionz 7→ a(z)at
z=z_{0}in the directionξ ∈ C^{N}, and∂g(z^{0}_{0})is viewed as an element ofL

C(C^{n},C^{N−n}).

What we want is the derivative ofJ_{f}_{t} =dϕ_{t}^{−1}◦J_{F}_{t}(ϕ_{t})◦dϕ_{t} atx_{0}fort =0. Writingϕ∗

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

J˙f_{t} = −ϕ_{∗}^{−1}◦dϕ˙t◦ϕ^{−1}_{∗} ◦JF(ϕ)◦ϕ∗+ϕ_{∗}^{−1}◦JF(ϕ)◦dϕ˙t+ϕ^{−1}_{∗} ◦ ˙JF_{t}(ϕ)◦ϕ∗

=2Jf∂J_{f}(ϕ_{∗}^{−1}ϕ˙t)+ϕ_{∗}^{−1}◦ ˙JF_{t}(ϕ)◦ϕ∗, (3.9)
where the first term on the right hand side comes from the identity−ds◦J_{f}+J_{f} ◦ds =
2Jf∂J_{f}swiths=ϕ_{∗}^{−1}ϕ˙t ∈C^{r}(X, T X)andds =ϕ_{∗}^{−1}dϕ˙t. Our choicesev=ϕ∗v+a◦F·η
andϕt =ϕ+tev◦ϕyield

ϕ˙_{t} =

ev◦ϕ=ϕ∗v+a◦f ·η◦ϕ, so ϕ_{∗}^{−1}ϕ˙_{t} =v+ϕ_{∗}^{−1}(a◦f ·η◦ϕ).

If we recall thata(z_{0})=0 andη(ϕ(x_{0}))=η(z^{0}_{0})=pr_{2}u(x_{0}), we get, atx =x_{0},

∂_{J}_{f}(ϕ_{∗}^{−1}ϕ˙_{t})(x_{0})=∂_{J}_{f}v(x_{0})+ϕ^{−1}_{∗} da(z_{0})(∂_{J}_{f}f (x_{0}))·pr_{2}u(x_{0})

. (3.10)

By construction,ϕ∗=dϕis compatible with the almost complex structures(X, Jf)and
(C^{n}, JF). A combination of (3.8), (3.9) and (3.10) yields

J˙_{f}_{t}(x_{0})=2J_{f} ∂_{J}_{f}v(x_{0})

+ϕ_{∗}^{−1} 2ida(z0)(∂J_{f}f (x0))·pr_{2}u(x0)−2ida(z0)(u(x0))·∂g(z^{0}_{0})◦ϕ∗
.

As∂_{J}_{f}f (x_{0})=(∂_{J}_{F}F )(z^{0}_{0})◦dϕ(x_{0})=(0, ∂g(z^{0}_{0}))◦ϕ∗andf∗=F∗◦ϕ∗, we get
J˙f_{t}(x0)=f_{∗}^{−1}F∗ 2ida(z0)(∂J_{f}f (x0))·pr_{2}u(x0)−2ida(z0)(u(x0))·pr_{2}∂J_{f}f (x0)

+2J_{f}∂_{J}_{f}v(x_{0}).

By (3.3), the torsion tensorθ (z0): D_{z}_{0} ×D_{z}_{0} → Tz_{0}Z/D_{z}_{0} ' F∗Tz_{0}M = f∗Tx_{0}X is
given by

θ (η, λ)= X

1≤i≤n, n+1≤j,k≤N

∂a_{ik}

∂z_{j} (z_{0})−∂aij

∂z_{k}(z_{0})

η_{j}λ_{k} ∂

∂z_{i}

=da(z_{0})(η)·λ−da(z_{0})(λ)·η.

Since our pointx_{0} ∈Xwas arbitrary andJ˙_{f}_{t}(x_{0})is the value of the differentialdJ_{f}(w)
atx_{0}, we finally get the global formula

dJf(w)=2J_{f} f_{∗}^{−1}θ (∂J_{f}f, u)+∂J_{f}v
(observe that∂J_{f}f ∈L

C(T X, f^{∗}T Z)actually takes values inf^{∗}D, so taking a projec-
tion tof^{∗}Dis not needed). We conclude:

Proposition 3.5. Letr∈ [1,∞] ∪ {ω}.

(i) The natural mapf 7→J_{f} sendse0^{r}(X, Z,D)intoJ^{r}(X).

(ii) The differential of the natural map

e0^{r}(X, Z,D)→J^{r}(X), f 7→J_{f},

along every infinitesimal variationw=u+f∗v:X→f^{∗}T Z=f^{∗}D⊕f∗T Xof
f is given by

dJ_{f}(w)=2J_{f} f_{∗}^{−1}θ (∂_{J}_{f}f, u)+∂_{J}_{f}v

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

(iii) The differentialdJ_{f} off 7→J_{f} one0^{r}(X, Z,D)is a continuous morphism
C^{r}(X, f^{∗}D)⊕C^{r+1}(X, T X)→C^{r}(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
frome0^{r}(X, Z,D)toJ^{r}(X). If we do not take into account the quotient by the action of
Diff^{r+1}_{0} onJ^{r}(X), we obtain a more demanding condition. For that stronger requirement,
we see that a sufficient condition is that the continuous linear map

C^{r}(X, f^{∗}D)→C^{r}(X,End

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

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 ∈e0^{r}(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∈D_{f (x)}such that
θ (∂f (x)·ξ, λ)=η(ξ ) for allξ ∈T X.

Then there is a neighborhoodU off ine0^{r}(X, Z,D)and a neighborhood V ofJf in
J^{r}(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

9_{f} :C^{r}(X, f^{∗}D)→e0^{r}(X, Z,D), u7→8(f, f∗u).

By definitionf = 9_{f}(0)and9_{f} defines the infinite-dimensional manifold structure
on e0^{r}(X, Z,D) by identifying a neighborhood of 0 in the topological vector space
C^{r}(X, f^{∗}D)with a neighborhood off ine0^{r}(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)
whereL_{f} ∈ C^{r}(X,Hom(f^{∗}D,End

C(T X)))is by our assumption (ii) a surjective mor-
phism of bundles of finite rank. The kernelK:=KerL_{f} is aC^{r} subbundle off^{∗}D, thus
we can select aC^{r}subbundleEoff^{∗}Dsuch that

f^{∗}D=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 ofL_{f} =dJ_{f} to sectionsu∈C^{r}(X,E)⊂C^{r}(X, f^{∗}D), which is
by construction a bundle isomorphism fromC^{r}(X,E)ontoC^{r}(X,End

C(T X)). Hence for
r <∞,u7→g=9_{f}(u)7→J_{9}_{f}_{(u)}is aC^{r}-diffeomorphism from a neighborhoodW_{r}^{E}(0)
of the zero section ofC^{r}(X,E)onto a neighborhoodV_{r} ofJ_{f} ∈J^{r}(X), and sog7→J_{g}
is aC^{r}-diffeomorphism fromU_{r}^{E} := 9(W_{r}^{E}(0))ontoV_{r}. 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 finiter_{0}and considerr^{0} ∈ [r_{0},∞[arbitrarily large.

Then, by applying a local diffeomorphism argument inC^{r}^{0}at all nearby pointsg=9_{f}(u)
(and by using the injectivity onU_{r}^{E}

0), we see that the map
U_{r}^{E}0 :=9_{f}(W_{r}^{E}

0(0)∩C^{r}^{0}(X,E))→V_{r}_{0} ∩J^{r}^{0}(X), g7→J_{g},