Singular Levi-flat hypersurfaces and codimension one foliations

Singular Levi-flat hypersurfaces and codimension one foliations

MARCOBRUNELLA

Abstract. We study Levi-flat real analytic hypersurfaces with singularities. We prove that the Levi foliation on the regular part of the hypersurface can be holo- morphically extended, in a suitable sense, to neighbourhoods of singular points.

Mathematics Subject Classification (2000):32V25 (primary); 32S65, 32C05 (secondary).

1.

Introduction

LetX be a complex manifold of dimensionn. A smooth real hypersurfaceM ⊂ X is said to beLevi-flatif the codimension one distribution

T^CM =T M∩J(T M)⊂T M

is integrable, in Frobenius’ sense. It follows that M is smoothly foliated by im- mersed complex manifolds of dimensionn−1. We shall denote by

F

Msuch aLevi foliation.

If Mis moreover real analytic, then its local structure is very well understood:

according to E. Cartan (see for instance [1, Section 1.7]), around each p ∈ M we can find local holomorphic coordinates z₁, . . . ,z_n such that M = {mz₁ = 0}, and consequently the leaves of

F

^{M are {z1 = c}, c ∈}

R

. In particular, the Levi foliation

F

^M extends on a neighbourhood of pto a codimension one holomorphic foliation, with leaves {z₁ = c}, c ∈

C

. It is then easy to see that these local extensions glue together, giving a codimension one holomorphic foliation

F

^on some neighbourhood ofM.

In this paper we are interested in finding a similar structure in the case ofsingu- larLevi-flat hypersurfaces, a subject sistematically initiated by Burns and Gong [2].

Let us firstly recall some terminology.

A closed subset M ⊂ X isreal analyticif it is locally defined by the vanish- ing of a (finite) collection of real analytic functions. A real analytic subset M is irreducible if it cannot be expressed as M = M₁∪ M₂, with bothM₁andM₂real Received January 30, 2007; accepted in revised form November 21, 2007.

analytic and different from M. Any real analytic subset can be decomposed (on relatively compact open subsets) into a finite collection of irreducible components.

An irreducible real analytic subset M has a well defined dimension dim_RM, and it can be decomposed as a disjoint union M = M_reg∪ M_sing, where: (i) M_reg is nonempty and open in M, and it is formed by those points ofM around which M is a smooth real analytic submanifold of X of dimension dim_RM; (ii) M_sing is a real analytic subset, all of whose irreducible components have dimension strictly smaller than dim_RM. Remark, however, that M_regmay fail to be dense in M, as in the well known Whitney umbrella.

When dim_RM = dim_RX −1= 2n−1, or more generally each irreducible component ofMhas dimension 2n−1, we shall callMareal analytic hypersur- face. In that case, we shall say thatM isLevi-flatifM_regis Levi-flat in the usual sense (Frobenius’ integrability).

If M ⊂ X is a Levi-flat real analytic hypersurface then we have onM_regthe Levi foliation

F

Mreg. A natural question is: does this foliation extend holomorphi- cally on some neighbourhood of theclosure M_reg? Of course, we need here to work, at least, withsingularcodimension one holomorphic foliations; see for instance [5]

for the basic dictionary. Let us see some examples.

Example 1.1. Take a nonconstant holomorphic function f : X →

C

^{and setM =} {m f =0}. ThenMis Levi-flat andM_singis the set of critical points of f lying on M. Leaves of the Levi foliation onM_reg are given by{f = c},c∈

R

. Of course, this Levi foliation can be extended to a singular holomorphic foliation on the full X, generated by the kernel ofd f. More complicated examples can be obtained by takingM = {F◦ f =0}for some real analytic functionF:

C

^→

R

, or by starting with a meromorphic f [2].

Example 1.2. In X =

C

²with coordinatesz = x+i y,w =s+i t, consider the irreducible real analytic hypersurfaceMdefined by

M = {t²=4(y²+s)y²}.

We have M_sing= {t = y=0}, a totally real plane in

C

². Note thatM_reg∩M_sing= {t = y = 0,s ≥ 0}, a proper subset of M_sing. This hypersurface is Levi-flat: on M_reg, the leaves of

F

Mreg are the complex curves

L_c = {w=(z+c)²,mz =0}, c∈

R

^.

For everyc ∈

R

the closureL_c = {w= (z+c)²}cuts M_singalong the real curve {s = (x+c)²}. Thus, each point of M_sing∩ {s > 0}belongs to two closuresL_c1 and L_c2, whereas each point of M_sing∩ {s < 0}belongs to no one. The foliation

F

^Mreg extends holomorphically to a neighbourhood of M_reg: just choose a square root ofwon

C

^{2\ {t =0,s ≥0}}(which is a neighbourhood ofM_reg), and then take the holomorphic foliation generated there bydw = 2√

wd z. However, it is clear that

F

Mreg cannot be extended to neighbourhoods of points ofM_reg∩M_sing, due to some “ramification” along M_sing(or, more precisely, along the real line{y = t =

s = 0} ⊂ M_sing: the other points are not seriously problematic, for around them M is the union of two smooth Levi-flat hypersurfaces intersecting transversely). In spite of this, we could say that

F

^Mreg can be extended in some weak or multiform sense, as solution of the implicit differential equation(^d_{d z}^w)²=4w.

Our main result is an extrapolation of this last example. The philosophy which is behind is that, even if sometimes it is impossible to extend

F

Mreg, it is always possible to extend it in a “microlocal” sense,i.e. after lifting to the cotangent bun- dle. In other words,

F

^Mreg can always be extended as a (transcendental) implicit differential equation, or web. This will be better explained in Section 2 below. For the moment, we state our result in the form of a “resolution theorem” for singular Levi-flat hypersurfaces.

Theorem 1.3. Let X be a complex manifold of dimension n, and let M ⊂ X be a Levi-flat real analytic hypersurface. Then there exist a complex manifold Y of dimension n, a Levi-flat real analytic hypersurface N ⊂ Y , a(singular)codimen- sion one holomorphic foliation

F

on Y extending

F

^Nreg, and a holomorphic map π :Y →X , such that for some open N₀⊂ N:

(i) π|N0: N₀→M_regis an isomorphism;

(ii) π|_N0: N₀→M_regis a proper map.

In the Example 1.2 above, we may take Y =

C

^{2, N} a real hyperplane, and π a quadratic map which sends complex lines inNto the curvesL_c,c∈

R

. Remark that here N₀is not equal toN_reg, it is only a (open and dense) part of it. Unfortunately, during this procedure we loose the isolated part ofM_sing. Remark thatM_reg, as well as N₀, is not a real analytic subset, it is onlysubanalytic. It could be interesting to generalise the above theorem, in some way, to the subanalytic setting, but our method is strictly inside the real analytic world.

The theorem above can be seen as a first step toward a resolution procedure for singularities of Levi-flat hypersurfaces: the mapπ is a sort of blow-up of M_reg\ M_reg, allowing afterwards the extension of the Levi foliation. The second step, which for the moment requiresn ≤ 3, consists in applying the Desingularisation Theorem of Seidenberg (n = 2) or Cano (n = 3) [3, 5] to the foliation

F

^{. The} third, and easy, step is the analysis of Levi-flat real analytic hypersurfaces which are tangent to a simple singularity of codimension one foliation. We leave the details to the interested reader.

The proof of Theorem 1.3 is based on quite elementary considerations con- cerning the complexification(s) of real analytic subsets (not only hypersurfaces) in complex manifolds, which in some sense stem from the seminal work of Diederich and Fornaess [6] and which are also exploited in [2].

2.

Lifting to the cotangent bundle

In this section we give a more precise statement of our main result, and we reduce its proof to a general statement on the intrinsic complexification of certain real analytic

subsets.

Consider, as before, a Levi-flat real analytic hypersurface M in a complex manifold X, dim_CX = n. Let P T^∗X be the projectivised cotangent bundle of X:

a

C

^Pn−1-bundle over X, whose fibre P T_x^∗Xoverx ∈X will be thought as the set of complex hyperplanes inT_xX. Denote byπ the projectionP T^∗X → X.

The regular part M_reg of M can be lifted to P T^∗X: just take, for every x ∈ M_reg, the complex hyperplane

T_x^CM_reg =T_xM_reg∩J(T_xM_reg)⊂T_xX. Call

M_reg ⊂ P T^∗X

this lifting of M_reg. Remark that it is no more a hypersurface: its (real) dimension 2n−1 is half of the real dimension ofP T^∗X. However, it is still “Levi-flat”, in a sense which will be more precisely specified below.

Take now a pointyin the closureM_reg , projecting onXto a pointx ∈M_reg. Lemma 2.1. There exists, in a neighbourhood U_y ⊂ P T^∗X of y, a real analytic subset N_y of dimension2n−1containing M_reg ∩U_y.

(The notation suggests thatU_yandN_yshould be more properly considered asgerms at y. Similarly for the proposition below.)

Proof. We choose local coordinatesz₁, . . . ,z_n, w1, . . . , wn−1 around y such that z₁, . . . ,z_n are coordinates aroundx andw1, . . . , wn−1 corresponds to the hyper- planed z_n+_n−1

j=1wjd z_j =0. Take a real analytic equation f ofMaroundx, with d f =0 onM_reg. Then the real analytic subset defined aroundyby the equations

f =0 , wj = ∂f

∂z_j ∂f

∂z_n , j =1, . . . ,n−1

containsM_reg as an open subset. Indeed,T^CM_regis the Kernel of∂f. This analytic subset may have dimension larger than 2n−1 (for instance, it contains the full fibres over M_sing), but using a stratification of it we see that it contains a real analytic subset N_y (a stratum) of dimension 2n−1 and which still containsM_reg .

We choose such a N_y as the minimal one (as a germ at y), by taking inter- sections. We have M_reg ∩U_y ⊂(N_y)reg, but the inclusion may be strict, as in the Example 1.2 of the Introduction. Also, N_y may have several irreducible compo- nents, but each one contains a part of M_reg ∩U_y, by minimality.

The crucial fact is now the following.

Proposition 2.2. There exists, in a neighbourhood V_y ⊂ U_y of y, a complex ana- lytic subset Y_y of(complex)dimension n containing N_y∩V_y.

Let us deduce Theorem 1.3 from this proposition.

First of all, N_y ∩ V_y is a real analytic hypersurface inY_y, and it is Levi-flat because each irreducible component contains a Levi-flat piece [2, Lemma 2.2]. On Yy, we have a natural codimension one holomorphic (singular) distribution of hy- perplanes, given by the restriction of the canonical contact structure of P T^∗X. At points of M_reg , this codimension one distribution coincides (by tautology of the contact form) with T^CM_reg. In other words and more precisely, around points of M_reg ∩V_ythe projectionπ : Y_y → Xis a biholomorphism, sendingM_reg to M_reg, thusY_yappears as the graph of a local holomorphic section ofP T^∗Xextending the real analytic section M_reg x → y = T_x^CM_reg ∈ M_reg . This local holomorphic section is a local hyperplane distribution on X. When lifted toY_y by the sameπ, this distribution becomes a distribution onY_y coinciding with the restriction of the contact structure.

Now, this distribution is integrable, because it is integrable on a real hypersur- face, hence it defines a codimension one (singular) foliation

F

y onY_y. Clearly, this foliation

F

yextends the Levi foliation of(N_y)reg∩V_y, because so is forM_reg ∩V_y and each irreducible component ofN_y∩V_y contains a portion ofM_reg ∩V_y.

These local constructions are sufficiently canonical to be patched together, when y varies on M_reg : ifY_y1 ⊂ V_y1 and Y_y2 ⊂ V_y2 are as above, with M_reg ∩ (V_y1∩V_y2) = ∅, thenY_y1∩(V_y1∩V_y2)andY_y2∩(V_y1∩V_y2)have some common irreducible components containing M_reg ∩(V_y1∩V_y2), and soY_y1andY_y2 can be glued by identifying those components. In this way, we obtain an analytic space Y of dimensionn, with a Levi-flat hypersurface N and a holomorphic foliation

F

^, enjoying all the properties stated in Theorem 1.3; the mapπ : Y → Xis deduced from the projection of P T^∗Xto X. Finally, to get a smoothY we just replace the possibly singular Y with a resolutionY of it, over which N and

F

can be lifted.

Note that on Y the foliation

F

is locally defined by a holomorphic 1-form, the restriction to Y of the contact form on P T^∗X; therefore onYthe lifted foliation will be also locally defined by a holomorphic 1-form, obtained by pulling-back the contact form under the resolution map.

Let us reformulate this proof from a slightly different point of view.

On some neighbourhoodX₀⊂ XofM_regwe have a holomorphic foliation

F

⁰ extending the Levi foliation

F

^Mreg. The graph of this foliation is a complex manifold Y₀ ⊂ P T^∗Xwhich projects bihlomorphically to X₀byπ. ThisY₀contains M_reg , as a Levi-flat hypersurface. The main point is then to extendY₀to a neighbourhood of M_reg , and this is the content of Proposition 2.2. In this way, an extension prob- lem for foliations (or, more appropriately, for webs) has been transformed into an extension problem for complex analytic subspaces.

Remark 2.3. The analytic subsetY_ycan be seen as an implicit differential equation on X, around x. But there are two quite different situations. Ifπ : Y_y → X is a finite map, then (by Weierstrass)Y_y has an equation which ispolynomialin the vertical variablesw1, . . . , wn−1, so that the associated implicit differential equation is “algebraic in the derivatives”. In some sense, even if there is a ramification we could say that this situation is “not-too-singular”, from the differential equation point of view. If on the contrary π : Y_y → X is not finite over x, we don’t

know if such an algebraicity in the derivatives persists,i.e.ifY_ycan be analytically continued to a neighbourhood of the full fibre overx.

Example 2.4. Consider the subsetM ⊂

C

^{2defined by}

M =

λ∈S¹

{z₂=λz₁+ϕ(λ)z²₁}

whereϕ :

S

¹ →

C

is a real analytic function. Outside the origin (and close to it) M is real analytic, smooth, and Levi-flat. ThusM\ {0}can be lifted toP T^∗

C

^{2 =}

C

^{2 ×}

C

^P1, and the closure of this lifting is a smooth real analytic threefold N which cuts the fibre over 0 along a smooth circle. The cooking recipe to obtain the complex surfaceY containingN is the following. We differentiate the complex curves contained inM,

d z₂=(λ+2ϕ(λ)z₁)d z₁,

and then we “eliminate” λfrom the two equationsz₂ = λz₁+ϕ(λ)z²₁ and ^{d z}_{d z}²

1 =

λ+2ϕ(λ)z₁, thus obtaining an analytic relation between z₁, z₂ andw1 = −^{d z}_{d z}²₁. It is quite clear that such an equation will be algebraic (inw1) if and only ifϕ is algebraic, i.e. ϕ(λ) =

j=−a_jλ^j. However, we believe that this algebraicity condition on ϕ should be also necessary and sufficient for the real analyticity of M at the origin. Thus, if M is real analytic at 0 then we would obtain an implicit differential equation which is algebraic in the derivatives, and the question of the previous Remark remains unanswered. Note, however, that whateverϕis, the sub- set N is everywhere analytic; hence our method, which is based on the analyticity ofN and not ofM, is unfortunately rather useless for these type of subtle problems.

Proposition 2.2 is a particular case of a more general fact, for which it is con- venient to introduce some more terminology.

Let Z be a complex manifold of dimensionm. Let N ⊂ Z be a real analytic subset of dimension 2n−1. We shall say that N is aLevi-flat real analytic sub- setif its regular part is foliated by complex submanifolds of (complex) dimension n−1. As in the hypersurface case, we shall denote by

F

N_reg thisLevi foliation on N_reg; it is real analytic and of real codimension one (in N_reg). This definition can be reformulated in the language of CR geometry [1, Chapter I]: N_regis a CR submanifold of CR dimensionn−1 (or CR codimension one) whose complex tan- gent bundleT^CN_reg = T N_reg∩ J(T N_reg)is integrable. It is easy to see, as in the hypersurface case [2, Lemma 2.2], that it is sufficient to check the integrability only on some open nonempty subsetN₀⊂ N_reg.

As in the hypersurface case, the local structure of N_reg is well understood [1, Section 1.8]: around every z ∈ N_reg there are local holomorphic coordinates z₁, . . . ,z_monZ such that

N_reg= {mz₁=0,z_m−n =. . .=z_m =0}.

In particular, aroundzthere is a canonically defined complex submanifold of Z, of dimensionn, which contains N_reg:

Y₀= {z_m−n=. . .=z_m=0}.

ThisY₀ is calledintrinsic complexificationof N_reg (aroundz). It is the smallest complex submanifold containing N_reg. Remark that N_reg is a (smooth) Levi-flat hypersurfaceinY₀, and the Levi foliation

F

^Nreg can be canonically extended to a holomorphic codimension one foliation

F

^Y0 onY₀. This is the situation that we have already met in the proof of Theorem 1.3.

Now the basic result, from which Proposition 2.2 follows, states that this in- trinsic complexification can be analytically prolonged around points of the closure N_reg:

Theorem 2.5. Let Z be a complex manifold of dimension m, and let N ⊂ Z be a Levi-flat real analytic subset of dimension 2n −1. Then, for every z ∈ N_reg, there exists a neighbourhood V ⊂ Z of z and a complex analytic subset Y ⊂V of dimension n which contains N_reg∩V

The proof will be given in the next section, and it is based on the following idea. There exists a second type of complexification of N, which could be called extrinsic complexification [4]. It lives in a larger space, not inside Z, but it has the advantage that it can be defined around every point of N, not only N_reg. The intrinsic complexification appears as a “projection” of the extrinsic one, and from the analytic extendability to singular points of the latter we shall deduce the analytic extendability of the former. In a vague sense, this is related to [6, Appendix].

It is however important to realize that this result holds because the CR codi- mension ofN_regis one,and no more. Here is an example showing that an analogous statement for the higher CR codimensional case may fail. This is related to the fact that, whereas a holomorphic map from a complex curve is always “locally proper”, the same is no more true for maps from higher dimensional complex manifolds, due to the phenomenon of “contraction of divisors”.

Example 2.6. We work in

C

³, with coordinatesz₁,z₂,z₃. LetS⊂

C

³be the real analytic surface defined by

z₂=(ez₁)z₁ z₃=e^ez1z₁

Note that S is smooth, being the graph over thez₁-axis of a smooth function F :

C

^z1 →

C

^z2,z₃. At the point 0 ∈ Swe haveT₀^CS = T₀S, a complex line, whereas at any other point p∈ S, p =0, we haveT_p^CS= {0}. Indeed, 0∈

C

^z1is the only point where the differential ofFis

C

-linear. Thus,S₀= S\{0}is a CR submanifold of CR dimension 0 and CR codimension 2. The “Levi foliation” is here the foliation by points.

AlongS₀we have the intrinsic complexificationY₀ ⊃ S₀, which is a complex surface. However, this complex surface cannot be extended around the point 0. To see this, observe that over{z₁ =0}the surfaceY₀has equation

z₃=e^z2/z1z₁

which has an essential singularity atz₁=0. More geometrically, if we blow up

C

³ at the origin,

C

^3→b

C

³, then the closure ofb⁻¹(Y₀)has a trace on the exceptional

divisorb⁻¹(0)

C

^P2which contains the complex curve with equationw3 =e^w2, in affine coordinates w2 = z₂/z₁,w3 = z₃/z₁. The transcendency of this curve implies thatY₀cannot be holomorphically extended at 0.

3.

Complexification of Levi-flat subsets

We start the proof of Theorem 2.5 by recalling some facts about (extrinsic) com- plexification of real analytic subsets [4], [1, Chapter X].

LetZbe a complex manifold of dimensionm. We shall denote byZ^∗the com- plex manifoldconjugateto Z, that is obtained by replacing the complex structure J of Z with the opposite complex structure−J. If A ⊂ Z is a complex analytic subset, then thesamesubset is also a complex analytic subset of Z^∗, but with the opposite complex structure; it will be denoted by A^∗. The diagonal⊂ Z×Z^∗is a totally real submanifold. The projections ofZ×Z^∗ontoZandZ^∗will be respec- tively denoted byand^∗, and theantiholomorphicinvolutionZ×Z^∗→ Z×Z^∗, (z₁,z₂)→(z₂,z₁), will be denoted by.

LetN ⊂ Zbe a real analytic subset of dimensionk, and fix a pointz₀ ∈N_reg. Without loss of generality for our purposes, we will suppose that the germ of N at z₀is irreducible. Set

N= {(z,z)∈ Z×Z^∗|z∈ N}.

Then in some sufficiently small neighbourhood of(z₀,z₀),which we may assume equal to Z×Z^∗by restricting Z, there exists an irreducible complex analytic subset N^Cof dimensionksuch that

N^C∩=N.

This N^C is called complexificationof N at z₀. It is obtained by complexifying the real analytic equations defining N, and it can be characterized as the smallest (germ at(z₀,z₀)of) complex analytic subset containingN. It enjoys the following fundamental symmetry property:  leavesN^Cinvariant, andNis the fixed point set of|N^C.

A word of caution. A real analytic subset is not necessarily coherent [4]. This means that ifz₁ ∈ N_reg is another point, close toz₀, then we can still define the complexification N^C of N at z₁ (if the germ of N at z₁ has several irreducible components, we complexify each one), however it may happen that N^C, defined in some neighbourhood of (z₁,z₁), is smaller than the restriction of N^C to that neighbourhood. More precisely, it may happen that the germ ofN^Cat(z₁,z₁)is not the full germ of N^Cat(z₁,z₁), but only a union of certain irreducible components of it. The situation is even worse if we takez₁∈ N \N_reg: the local dimension of N atz₁is smaller thank, and thus the complexification ofN atz₁has also smaller dimension; the germ ofN^Cat(z₁,z₁)is then a proper subset of the germ ofN^C.

Suppose now that N is Levi-flat,k = 2n−1. Take a regular pointz ∈ N_reg. As we saw in the previous section, in some neighbourhood V_z ⊂ Z of z there exists a unique smooth complex submanifoldYz of dimensionn, which contains N_reg∩V_zand over which the Levi foliation extends as a holomorphic codimension one foliation

F

^z. For a good choice ofV_z, we may assume that the space of leaves Y_z

F

^{zis a disc}

D

, in which the Levi leaves correspond to points of

I

^=(−1,1)⊂

D

. Hence, we have a well defined Schwarz reflection

s_z :Y_z

F

^z→Yz

F

^z with respect to

I

^.

Provided that V_z is sufficiently small, the complexification of N ∩V_z at z is a smooth complex submanifold N_z^C ofU_z = V_z×V_z^∗ ⊂ Z× Z^∗, of dimension 2n−1, which in fact coincides with an irreducible component of N^C∩U_z. The following Lemma gives the precise relation betweenY_zandN_z^C.

Lemma 3.1. The projectioninduces a smooth holomorphic fibration of N_z^Cover Y_z, and the fibre of: N_z^C →Y_zover z ∈Y_zis the Schwarz reflection of the leaf of

F

^{zthrough z(}with the opposite complex structure).

Proof. In suitable local coordinatesz₁, . . . ,z_mwe have N_reg= {mz₁ =0,z_m−n =. . .=z_m=0}

Y_z= {z_m−n=. . .=z_m =0}.

Ifz∈Y_z, then the leaf throughzis

L_z = {z₁=a,z_m−n=. . .=z_m=0} whereais thez₁-coordinate ofz, and its Schwarz reflection is

s_z(L_z)= {z₁= ¯a,z_m−n=. . .=z_m=0}.

Local coordinates on Z×Z^∗are provided byz₁, . . . ,z_m,¯z₁, . . . ,¯z_m, and in these coordinates

N_z^C = {z₁= ¯z₁,z_m−n=. . .=z_m = ¯z_m−n=. . .= ¯z_m =0}.

We therefore see that N_z^Cprojects bytoY_z, and the fibre overzis {¯z₁=a,¯z_m−n=. . .= ¯z_m =0} =s_z(L_z)^∗.

Of course, a similar statement holds also for the second projection^∗. Thus,N_z^Cis a smooth complex hypersurface inY_z×Y_z^∗ ⊂Z×Z^∗, with a double fibration over Y_zandY_z^∗.

Note that for everyz ∈Y_zwe can also consider the fibre overzof the fullN^C, i.e.the intersectionN^C∩({z}×Z^∗). This is an analytic subset ofZ^∗which extends the leafsz(L_z)^∗ ⊂ V_z^∗. In particular, each leaf of

F

^{z has an}analytic continuation to the full Z. This fundamental fact was discovered in [6], in a different context, and it was our starting point.

Let us now see what happens at the singular pointz₀. For every p∈ N^Cconsider the vertical fibre through p

F_p^∗ = {q∈ N^C|(q)=(p)}

and define

d(p)=dim_p(F_p^∗)

where dim_p denotes the local dimension at p (if F_p^∗ is reducible at p, we take the maximal dimension among irreducible components). According to Cartan or Remmert [7, Section V.3], the function d on N^C is Zariski semicontinuous, thus equal to some constantd₀over some Zariski open and denseN˘^C ⊂N^C and strictly greater thand₀over N^C\ ˘N^C. By Lemma 3.1, we haved₀=n−1, and moreover N˘^C contains generic points ofN_reg (here we say “generic”, and not “all”, because at special points (z,z) ∈ N_reg the germ of N^C could have a second irreducible component different fromN_z^C).

Set p₀=(z₀,z₀).

Ifd(p₀)=d₀=n−1, then by the Rank theorem of Remmert [7, Section V.6]

there exists a neighbourhood U_z0 ⊂ Z × Z^∗ of p₀ such that (N^C ∩U_z0)is a complex analytic subsetY_z0 of V_z0 = (U_z0), of dimension n. The situation is not very different from the one of Lemma 3.1. And we are just in the conclusion of Theorem 2.5: obviously Y_z0 contains the previously definedY_z, for every z ∈ N_reg∩V_z0.

Hence, we shall suppose from now on thatd(p₀) =l ≥n.Thus the vertical fibre F^∗_p

0 contains an irreducible component Y^∗ ⊂ Z^∗ of dimensionl, passing through p₀. Its conjugate Y =  (Y^∗) ⊂ Z is an irreducible component of the horizontal fibre F_p0 = {q ∈ N^C | ^∗(q) = ^∗(p₀)}. In the following, we shall considerY sometimes as a (horizontal) subset ofN^C, sometimes as a subset of the baseZ.

Obviously, Y ⊂ Z is contained in(N^C). Because dimN^C = 2n−1 and d₀=n−1, the image ofN^Cbyis a countable union of local analytic subsets of dimension≤n(by iterated application of the Rank theorem). ThusY cannot have dimension larger thann, and so

dimY =n.

To complete the proof, we will show thatYcontains(N^C∩U_z0), for some neigh- bourhoodU_z0ofz₀(thus, after all,Y is equal to that projection, and soY is in fact the unique irreducible component of dimensionnof the horizontal fibre). Because Y is closed, it is sufficient to verify this inclusion for(N˘^C∩U_z0).

Note that Y ⊂ N^C is not contained in N^C \ ˘N^C, for dimensional reasons:

becauseY is horizontal of dimensionn and each vertical fibre through N^C \ ˘N^C has dimension ≥ n, N^C would have dimension≥ 2n. Thus,Y intersects N˘^C. If p=(z,z₀)is a point ofY ∩ ˘N^C, then (by the Rank theorem) the image byof a small neighbourhood ofpinN^Cis an analytic subsetY_z ⊂V_z⊂ Zof dimensionn.

The germ ofY_z atzcould have several irreducible components, but at least one of them is contained inY, becauseY∩V_z ⊂Y_zand dimY =dimY_z. By a connectivity argument (recall thatN^Cis irreducible, and thereforeN˘^Cis connected), we see that Y contains all the points arising from the projection ofN˘^C toZ, as claimed.

Let us conclude by looking again at the (counter) example of the previous section.

Example 3.2. Take again the real analytic surfaceSin

C

³with equation z₂=(ez₁)z₁

z₃=e^ez1z₁

which, outside 0, is a CR submanifold of CR dimension 0 and CR codimension 2.

Its complexification is the complex analytic surfaceS^Cin

C

^3×

C

^3∗with equation (setting¯z_j =wj)

z₂ = z₁+w1

2 z₁ , w2= z₁+w1

2 w1

z₃ =e^{z1+w2 1}z₁, w3=e^{z1+w2 1}w1.

The projection of S^C to the first factor

C

³ is not analytic: something like z₃ = e^z2/z1z₁, for z₁ = 0. The horizontal fibre through 0 ∈ S^C is the complex curve C ⊂

C

^{3given by}

z₂= 1

2z²₁ , z₃=e^12z1z₁.

It is, of course, contained in the projection ofS^Cbut not equal to. The dimensional arguments used several times in the proof of Theorem 2.5 cannot be used here.

References

[1] M. S. BAOUENDI, P. EBENFELTand L. P. ROTHSCHILD, “Real Submanifolds in Complex Space and their Mappings”, Princeton Mathematical Series 47, 1999.

[2] D. BURNSand X. GONG,Singular Levi-flat real analytic hypersurfaces, Amer. J. Math.121 (1999), 23–53.

[3] F. CANO,Reduction of the singularities of codimension one singular foliations in dimension three, Ann. of Math.160(2004), 907–1011.

[4] H. CARTAN,Vari´et´es analytiques r´eelles et vari´et´es analytiques complexes, Bull. Soc. Math.

France85(1957), 77–99.

[5] D. CERVEAU,Feuilletages holomorphes de codimension 1. R´eduction des singularit´es en petites dimensions et applications, Panor. Synth`eses8(1999), 11–47.

[6] K. DIEDERICHand J. E. FORNAESS,Pseudoconvex domains with real-analytic boundary, Ann. of Math.107(1978), 371–384.

[7] S. LOJASIEWICZ, “Introduction to Complex Analytic Geometry”, Birkh¨auser, 1991.

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