Singular Levi-flat hypersurfaces and codimension one foliations

0.17 MB·English

by  Marco Brunella

SINGULAR LEVI-FLAT HYPERSURFACES
AND CODIMENSION ONE FOLIATIONS
7
0
0
2 MARCOBRUNELLA

n
a Abstract. We study Levi-flat real analytic hypersurfaces with singularities.
J WeprovethattheLevifoliationontheregularpartofthehypersurfacecanbe
2 holomorphically extended, in a suitable sense, to neighbourhoods of singular
2 points.

]
V
C
.
h
t
a
m
1. Introduction
[
Let X be a complex manifold of dimension n. A smooth real hypersurface

1
M X is said to be Levi-flat if the codimension one distribution
v ⊂
7 C
T M =TM J(TM) TM
0 ∩ ⊂
6
is integrable, in Frobenius’ sense. It follows that M is smoothly foliated by im-
1
mersedcomplexmanifoldsofdimensionn 1. We shalldenote by suchaLevi
0 M
− F
7 foliation.
0 If M is moreover real analytic, then its local structure is very well understood:
/ according to E. Cartan (see for instance [BER, 1.7]), around each p M we
h
§ ∈
t can find local holomorphic coordinates z1,...,zn such that M = mz1 = 0 , and
ma consequentlythe leavesofFM are{z1 =c}, c∈R. In particular,{tℑhe Levifol}iation
extends on a neighbourhood of p to a codimension one holomorphic foliation,
M
v: Fwith leaves z1 = c , c C. It is then easy to see that these local extensions glue
{ } ∈
i together,givingacodimensiononeholomorphicfoliation onsomeneighbourhood
X F
of M.
r
a Inthispaperweareinterestedinfindingasimilarstructureinthecaseofsingular
Levi-flathypersurfaces,asubjectsistematicallyinitiatedbyBurnsandGong[B-G].
Let us firstly recall some terminology.
A closed subset M X is real analytic if 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 both M1 and M2 real
∪
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 dimRM, and
it can be decomposed as a disjoint union M = M M , where: (i) M is
reg sing reg
∪
nonempty and open in M, and it is formed by those points of M around which
M is a smooth real analytic submanifold of X of dimension dimRM; (ii) Msing is
a real analytic subset, all of whose irreducible components have dimension strictly
1
SINGULAR LEVI-FLAT HYPERSURFACES 2
smaller that dimRM. Remark, however, that Mreg may fail to be dense in M, as
in the well known Whitney umbrella.
When dimRM =dimRX 1=2n 1, or more generally each irreducible com-
− −
ponentofM hasdimension2n 1,weshallcallM areal analytic hypersurface.
−
Inthatcase,weshallsaythatM isLevi-flatifM isLevi-flatinthe usualsense
reg
(Frobenius’ integrability).
IfM X isaLevi-flatrealanalytichypersurfacethenwehaveonM theLevi
reg
⊂
foliation . A natural question is: does this foliation extend holomorphically
FMreg
on some neighbourhood of the closure M ? Of course, we need here to work, at
reg
least, with singular codimension one holomorphic foliations; see for instance [Cer]
for the basic dictionary. Let us see some examples.
Example 1.1. Take a nonconstant holomorphic function f : X C and set
→
M = mf = 0 . Then M is Levi-flat and M is the set of critical points of
sing
f lying{ℑon M. L}eaves of the Levi foliation on M are given by f = c , c R.
reg
{ } ∈
Of course, this Levi foliation can be extended to a singular holomorphic foliation
on the full X, generated by the kernel of df. More complicated examples can be
obtainedby taking M = F f =0 for some realanalytic function F :C R, or
{ ◦ } →
by starting with a meromorphic f [B-G].
Example 1.2. In X = C2 with coordinates z = x+iy, w = s+it, consider the
irreducible real analytic hypersurface M defined by
2 2 2
M = t =4(y +s)y .
{ }
WehaveM = t=y =0 ,atotallyrealplaneinC2. NotethatM M =
sing reg sing
{ } ∩
t = y = 0,s 0 , a proper subset of M . This hypersurface is Levi-flat: on
sing
{ ≥ }
M , the leaves of are the complex curves
reg FMreg
L = w =(z+c)2, mz =0 , c R.
c
{ ℑ 6 } ∈
For every c R the closure L = w = (z+c)2 cuts M along the real curve
c sing
s = (x+c)∈2 . Thus, each point {of M s}> 0 belongs to two closures L
{ } sing ∩{ } c1
and L , whereas each point of M s < 0 belongs to no one. The foliation
c2 sing ∩{ }
extends holomorphically to a neighbourhood of M : just choose a square
FroMotregof w on C2 t = 0,s 0 (which is a neighbourhroegod of M ), and then
reg
\{ ≥ }
take the holomorphic foliation generated there by dw = 2√wdz. However, it is
clearthat cannotbe extended to neighbourhoodsof points ofM M ,
FMreg reg∩ sing
due to some “ramification” along M (or, more precisely, along the real line
sing
y = t = s = 0 M : the other points are not seriously problematic, for
sing
{ } ⊂
around them M is the union of two smooth Levi-flat hypersurfaces intersecting
transversely). In spite of this, we could say that can be extended in some
FMreg
weakormultiformsense,assolutionoftheimplicitdifferentialequation(dw)2 =4w.
dz
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 , it is always
FMreg
possible to extend it in a “microlocal” sense, i.e. after lifting to the cotangent
bundle. Inotherwords, canalwaysbeextendedasa(transcendental)implicit
FMreg
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.
SINGULAR LEVI-FLAT HYPERSURFACES 3
Theorem 1.1. 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
dimensionn,aLevi-flatrealanalytichypersurfaceN Y,a(singular)codimension
⊂
oneholomorphicfoliation onY extending ,andaholomorphicmapπ :Y
F FNreg →
X, such that for some open N0 N:
⊂
(i) π|N0 :N0 →Mreg is an isomorphism;
(ii) π|N0 :N0 →Mreg is a proper map.
In the Example 1.2 above, we may take Y = C2, N a real hyperplane, and π a
quadraticmapwhichsendscomplexlinesinN tothecurvesL ,c R. Remarkthat
c
∈
hereN0 isnotequaltoNreg,itisonlya(openanddense)partofit. Unfortunately,
duringthisprocedurewelosstheisolatedpartofM . RemarkthatM ,aswell
sing reg
as N0, is not a real analytic subset, it is only subanalytic. 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 Theoremabovecanbe seenasafirststeptowardaresolutionprocedurefor
singularitiesofLevi-flathypersurfaces: themapπisasortofblow-upofM M ,
reg reg
\
allowing afterwards the extension of the Levi foliation. The second step, which for
the moment requires n 3, consists in applying the Desingularisation Theorem of
≤
Seidenberg (n=2) or Cano (n=3) [Cer] [Can] to the foliation . The third, and
F
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.
TheproofofTheorem1.1isbasedonquiteelementaryconsiderationsconcerning
thecomplexification(s)ofrealanalyticsubsets(notonlyhypersurfaces)incomplex
manifolds, which in some sense stem from the seminal work of Diederich and For-
naess [D-F] and which are also exploited in [B-G].
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 hypersurfaceM in a complex man-
ifold X, dimCX = n. Let PT∗X be the projectivised cotangent bundle of X: a
CPn−1-bundle overX, whose fibre PT∗X overx X will be thought as the set of
complex hyperplanes in T X. Denote bxy π the pr∈ojection PT∗X X.
x
TheregularpartM ofM canbeliftedtoPT∗X: justtake,for→everyx M ,
reg reg
∈
the complex hyperplane
C
T M =T M J(T M ) T X.
x reg x reg∩ x reg ⊂ x
Call
′ ∗
M PT X
reg ⊂
this lifting ofM . Remarkthat it is no more a hypersurface: its (real)dimension
reg
2n 1 is half of the real dimension of PT∗X. However, it is still “Levi-flat”, in a
−
sense which will be precised below.
Take now a point y in the closure M′ , projecting on X to a point x M .
reg ∈ reg
Lemma 2.1. There exists, in a neighbourhood U PT∗X of y, a real analytic
y
subset N of dimension 2n 1 containing M′ U⊂.
y − reg∩ y
SINGULAR LEVI-FLAT HYPERSURFACES 4
(The notation suggests that U and N should be more properly considered as
y y
germs at y. Similarly for the Proposition below).
Proof. Wechooselocalcoordinatesz1,...,zn,w1,...,wn−1aroundysuchthatz1,...,zn
are coordinates around x and w1,...,wn−1 corresponds to the hyperplane dzn +
n−1w dz = 0. Take a real analytic equation f of M around x, with df = 0 on
Pj=1 j j 6
M . Then the real analytic subset defined around y by the equations
reg
∂f ∂f
f =0 , w = , j =1,...,n 1
j (cid:14)
∂z ∂z −
j n
containsM′ asanopensubset. Indeed,TCM istheKernelof∂f. Thisanalytic
reg reg
subsetmayhavedimensionlargerthan2n 1(forinstance,itcontainsthefullfibres
−
over M ), but using a stratification of it we see that it contains a real analytic
sing
subset N (a stratum) of dimension 2n 1 and which still contains M′ . (cid:3)
y − reg
WechoosesuchaN astheminimalone(asagermaty),bytakingintersections.
y
We have M′ U (N ) , but the inclusion may be strict, as in the Example
reg∩ y ⊂ y reg
1.2 of the Introduction. Also, N may have several irreducible components, but
y
each one contains a part of M′ U , by minimality.
reg∩ y
The crucial fact is now the following.
Proposition 2.1. There exists, in a neighbourhood V U of y, a complex ana-
y y
⊂
lytic subset Y of (complex) dimension n containing N V .
y y y
∩
Let us deduce Theorem 1.1 from this Proposition.
First of all, N V is a real analytic hypersurface in Y , and it is Levi-flat
y y y
∩
because each irreducible component contains a Levi-flat piece [B-G, Lemma 2.2].
On Y , we have a natural codimension one holomorphic (singular) distribution of
y
hyperplanes, given by the restriction of the canonical contact structure of PT∗X.
Atpoints ofM′ ,this codimensionone distributioncoincides (by tautologyofthe
reg
contact form) with TCM . In other words and more precisely, around points of
reg
M′ V the projection π :Y X is a biholomorphism, sending M′ to M ,
thuresgY∩ ayppearsasthegraphofya→localholomorphicsectionofPT∗X exrteegndingrtehge
y
real analytic section M x′ y′ = TCM M′ . This local holomorphic
reg ∋ 7→ x′ reg ∈ reg
section is a local hyperplane distribution on X. When lifted to Y by the same π,
y
thisdistributionbecomesadistributiononY coincidingwiththe restrictionofthe
y
contact structure.
Now, this distribution is integrable, because it is integrable on a real hypersur-
face, hence it defines a codimensionone (singular) foliation on Y . Clearly,this
y y
foliation extends the Levi foliation of (N ) V , becaFuse so is for M′ V
and eachFiryreducible component of N V cyornetgai∩nsya portion of M′ Vre.g∩ y
y∩ y reg∩ y
Theselocalconstructionsaresufficientlycanonicaltobepatchedtogether,when
yvariesonM′ : ifY V andY V areasabove,withM′ (V V )=
reg y1 ⊂ y1 y2 ⊂ y2 reg∩ y1∩ y2 6
,thenY (V V )andY (V V )havesomecommonirreduciblecompo-
n∅entsconyt1a∩ininyg1M∩ ′y2 (V y2V∩ ),ya1n∩dsyo2 Y andY canbe gluedbyidentifying
reg∩ y1∩ y2 y1 y2
thosecomponents. Inthisway,weobtainananalyticspaceY ofdimensionn,with
a Levi-flat hypersurface N and a holomorphic foliation , enjoying all the proper-
F
ties stated in Theorem 1.1; the map π : Y X is deduced from the projection of
PT∗X to X. Finally, to get a smooth Y w→e just replace the possibly singular Y
with a resolution of it, over which N and can be lifted.
F
SINGULAR LEVI-FLAT HYPERSURFACES 5
Let us reformulate this proof from a slightly different point of view.
On some neighbourhood X0 X of Mreg we have a holomorphic foliation 0
⊂ F
extendingtheLevifoliation . Thegraphofthisfoliationisacomplexmanifold
Y0 ⊂PT∗X whichprojectsFbiMhlroegmorphicallytoX0 byπ. ThisY0 containsMr′eg,as
a Levi-flat hypersurface. The main point is then to extend Y0 to a neighbourhood
of M′ , and this is the content of Proposition 2.1. In this way, an extension
reg
problemfor foliations (or,moreappropriately,for webs)has been transformedinto
an extension problem for complex analytic subspaces.
Remark2.1. TheanalyticsubsetY canbeseenasanimplicitdifferentialequation
y
on X, around x. But there are two quite different situations. If π : Y X is
y
→
a finite map, then (by Weierstrass) Y has an equation which is polynomial in the
y
vertical variables w1,...,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 X is not finite over x, we don’t know
y
→
if such an algebraicity in the derivatives persists, i.e. if Y can be analytically
y
continued to a neighbourhood of the full fibre over x.
Example 2.1. Consider the subset M C2 defined by
⊂
2
M = [ z2 =λz1+ϕ(λ)z1
{ }
λ∈S1
where ϕ : S1 C is a real analytic function. Outside the origin (and close to it)
M is real ana→lytic, smooth, and Levi-flat. Thus M 0 can be lifted to PT∗C2 =
C2 CP1, and the closure of this lifting is a sm\oo{th}real analytic threefold N
×
which cuts the fibre over 0 along a smooth circle. The cooking recipe to obtain
the complex surface Y containing N is the following. We differentiate the complex
curves contained in M,
dz2 =(λ+2ϕ(λ)z1)dz1,
and then we “eliminate” λ from the two equations z2 = λz1+ϕ(λ)z12 and ddzz21 =
λ+2ϕ(λ)z1,thusobtainingananalyticrelationbetweenz1,z2andw1 = dz2. Itis
−dz1
quiteclearthatsuchanequationwillbealgebraic(inw1)ifandonlyifϕisalgebraic,
i.e. ϕ(λ)= ℓ a λj. However,thisalgebraicityconditiononϕseemstobealso
Pj=−ℓ j
necessary and sufficient for the real analyticity of M at the origin. Thus, if M is
realanalyticat0thenweobtainanimplicitdifferentialequationwhichisalgebraic
in the derivatives, and the question of the previous Remark remains unanswered.
Note, however,that whatever ϕ is, the subset N is everywhere analytic; hence our
method, which is based on the analyticity of N and not of M, is unfortunately
rather useless for these type of subtle problems.
Proposition 2.1 is a particular case of a more generalfact, for which it is conve-
nient to introduce some more terminology.
Let Z be a complex manifold of dimension m. Let N Z be a real analytic
⊂
subsetofdimension2n 1. WeshallsaythatN isaLevi-flatrealanalyticsubset
−
ifitsregularpartisfoliatedbycomplexsubmanifoldsof(complex)dimensionn 1.
−
As in the hypersurface case, we shall denote by this Levi foliation on
FNreg
N ; it is real analytic and of real codimension one (in N ). This definition
reg reg
can be reformulated in the language of CR geometry [BER, Ch. I]: N is a
reg
CR submanifold of CR dimension n 1 (or CR codimension one) whose complex
−
SINGULAR LEVI-FLAT HYPERSURFACES 6
tangentbundleTCN =TN J(TN )isintegrable. Itiseasytosee,asinthe
reg reg reg
∩
hypersurface case [B-G, Lemma 2.2], that it is sufficient to check the integrability
only on some open nonempty subset N0 Nreg.
⊂
Asinthehypersurfacecase,thelocalstructureofN iswellunderstood[BER,
reg
1.8]: around every z Nreg there are local holomorphic coordinates z1,...,zm on
§ ∈
Z such that
Nreg = mz1 =0,zm−n =...=zm =0 .
{ℑ }
In particular,aroundz there is a canonicallydefined complex submanifold of Z, of
dimension n, which contains N :
reg
Y0 = zm−n =...=zm =0 .
{ }
ThisY0 iscalledintrinsic complexificationofNreg (aroundz). Itisthesmallest
complex submanifold containing N . Remark that N is a (smooth) Levi-flat
reg reg
hypersurface in Y0, and the Levi foliation FNreg can be canonically extended to a
holomorphic codimension one foliation FY0 on Y0. This is the situation that we
have already met in the proof of Theorem 1.1.
Nowthebasicresult,fromwhichProposition2.1follows,statesthatthisintrinsic
complexification can be analytically prolonged around points of the closure N :
reg
Theorem 2.1. 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 V
reg
∩
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 [Car]. 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 . The intrinsic complexificationappearsasa “projection”ofthe extrinsic one,
reg
andfrom the analytic extendability to singularpoints of the latter we shalldeduce
the analytic extendability of the former. In a vague sense, this is related to [D-F,
Appendix].
ItishoweverimportanttorealizethatthisresultholdsbecausetheCRcodimen-
sion of N is one, and no more. Here is an example showing that an analogous
reg
statementforthehigherCRcodimensionalcasemayfail. Thisisrelatedtothefact
that, whereasa holomorphicmapfroma complex curve is always“locallyproper”,
thesameisnomoretrueformapsfromhigherdimensionalcomplexmanifolds,due
to the phenomenon of “contraction of divisors”.
Example 2.2. We work in C3, with coordinates z1,z2,z3. Let S C3 be the real
⊂
analytic surface defined by
z2 =( ez1)z1
ℜ
z3 =eℜez1z1
Note that S is smooth, being the graph over the z1-axis of a smooth function
Fat:aCnyz1o→theCrzp2,oz3in.tApt thSe,ppoi=nt00,∈weShwaevehaTvCeST=0CS 0=.T0ISnd,eaedco,m0pleCx linies,twhheeornealys
pointwherethediffere∈ntialo6fF isC-linear. pThus,{S0}=S 0 isa∈CRz1submanifold
\{ }
ofCRdimension0andCRcodimension2. The“Levifoliation”isherethefoliation
by points.
SINGULAR LEVI-FLAT HYPERSURFACES 7
Along S0 we have the intrinsic complexification Y0 S0, which is a complex
⊃
surface. However,this complexsurface cannotbe extendedaroundthe point0. To
see this, observe that over z1 =0 the surface Y0 has equation
{ 6 }
z3 =ez2/z1z1
which has an essential singularity at z1 =0. More geometrically, if we blow up C3
at the origin, C3 b C3, then the closure of b−1(Y0) has a trace on the exceptional
divisorb−1(0)fC→P2 whichcontainsthecomplexcurvewithequationw3 =ew2,in
≃
affinecoordinatesw2 =z2/z1,w3 =z3/z1. Thetranscendencyofthiscurveimplies
that Y0 cannot be holomorphically extended at 0.
3. Complexification of Levi-flat subsets
We start the proof of Theorem 2.1 by recalling some facts about (extrinsic)
complexification of real analytic subsets [Car] [BER, Ch. X].
LetZ beacomplexmanifoldofdimensionm. WeshalldenotebyZ∗thecomplex
manifold conjugate to 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,
thenthesamesubsetisalsoacomplexa−nalyticsu⊂bsetofZ∗,butwiththeopposite
complex structure; it will be denote by A∗. The diagonal ∆ Z Z∗ is a totally
real submanifold. The projections of Z Z∗ onto Z and Z⊂∗ wil×l be respectively
denoted by Π and Π∗, and the antiholo×morphic involution Z Z∗ Z Z∗,
× → ×
(z1,z2) (z2,z1), will be denoted by .
7→
Let N Z be a real analytic subset of dimension k, and fix a point z0 Nreg.
⊂ ∈
Without loss of generality for our purposes, we will suppose that the germ of N at
z0 is irreducible. Set
∆ ∗
N = (z,z) Z Z z N .
{ ∈ × | ∈ }
Then in some sufficiently small neighbourhood of (z0,z0), which we may assume
equalto Z Z∗ by restricting Z,thereexistsanirreduciblecomplexanalyticsubset
NC of dim×ension k such that
C ∆
N ∆=N .
∩
This NC is called complexification of N at z0. It is obtained by complexifying
the real analytic equations defining N, and it can be characterized as the smallest
(germat(z0,z0)of)complexanalyticsubsetcontainingN∆. Itenjoysthefollowing
fundamental symmetry property:  leaves NC invariant, and N∆ is the fixed point
set of |NC.
Awordofcaution. Arealanalyticsubsetisnotnecessarilycoherent[Car]. This
means that if z1 Nreg is another point, close to z0, then we can still define the
∈
complexification NC of N at z1 (if the germ of N at z1 has several irreducible
g
components, we complexify each one), however it may happen that NC, defined
in some neighbourhood of (z1,z1), is smaller than the restriction ofgNC to that
neighbourhood. Moreprecisely,itmayhappenthatthegermofNCat(z1,z1)isnot
C g
the full germ of N at (z1,z1), but only a union of certain irreducible components
of it. The situation is even worst if we take z1 N Nreg: the local dimension of
∈ \
N atz1 issmallerthank,andthusthe complexificationofN atz1 hasalsosmaller
dimension; the germ of NC at (z1,z1) is then a proper subset of the germ of NC.
g
SINGULAR LEVI-FLAT HYPERSURFACES 8
Suppose now that N is Levi-flat, k = 2n 1. Take a regular point z N .
reg
− ∈
As we saw in the previous section, in some neighbourhood V Z of z there
z
⊂
exists a unique smooth complex submanifold Y of dimension n, which contains
z
N V and over which the Levi foliation extends as a holomorphic codimension
reg z
∩
one foliation . For a good choice of V , we may assume that the space of leaves
z z
Y isadisFcD,inwhichtheLevileavescorrespondtopointsofI=( 1,1) D.
z(cid:14) z
F − ⊂
Hence, we have a well defined Schwarz reflection
s :Y Y
z z(cid:14) z z(cid:14) z
F → F
with respect to I.
Provided that V is sufficiently small, the complexification of N V at z is a
z z
smooth complex submanifold NC of U =V V∗ Z Z∗, of dime∩nsion 2n 1,
which in fact coincides with anzirreduzcible zco×mpzon⊂ent o×f NC U . The follow−ing
z
Lemma gives the precise relation between Y and NC. ∩
z z
Lemma 3.1. The projection Π induces a smooth holomorphic fibration of NC over
z
Y , and the fibre of Π: NC Y over z′ Y is the Schwarz reflection of the leaf
ofz through z′ (with thezo→pposzite comple∈x stzructure).
z
F
Proof. In suitable local coordinates z1,...,zm we have
Nreg = mz1 =0,zm−n =...=zm =0
{ℑ }
Yz = zm−n =...=zm =0 .
{ }
If z′ Y , then the leaf through z′ is
z
∈
Lz′ = z1 =a,zm−n =...=zm =0
{ }
where a is the z1-coordinate of z′, and its Schwarz reflection is
sz(Lz′)= z1 =a¯,zm−n =...=zm =0 .
{ }
Local coordinates on Z Z∗ are provided by z1,,zm,z¯1,...,z¯m, and in these coor-
×
dinates
C
Nz ={z1 =z¯1,zm−n =...=zm =z¯m−n =...=z¯m =0}.
We therefore see that NC projects by Π to Y , and the fibre over z′ is
z z
∗
z¯1 =a,z¯m−n =...=z¯m =0 =sz(Lz′) .
{ }
(cid:3)
Of course, a similar statement holds also for the second projection Π∗. Thus,
NC isasmoothcomplexhypersurfaceinY Y∗ Z Z∗,withadoublefibration
ovzer Y and Y∗. z× z ⊂ ×
z z
Note that for every z′ Y we can also consider the fibre over z′ of the full NC,
z
i.e. theintersectionNC (∈z′ Z∗). ThisisananalyticsubsetofZ∗ whichextends
the leaf sz(Lz′)∗ ⊂Vz∗.∩In{pa}r×ticular, each leaf of Fz has an analytic continuation
tothe fullZ. This fundamentalfactwasdiscoveredin[D-F],inadifferentcontext,
and it was our starting point.
Let us now see what happens at the singular point z0.
For every p NC consider the vertical fibre through p
∈
∗ C
F = q N Π(q)=Π(p)
p { ∈ | }
and define
∗
d(p)=dim (F )
p p
SINGULAR LEVI-FLAT HYPERSURFACES 9
where dim denotes the local dimension at p (if F∗ is reducible at p, we take
p p
the maximal dimension among irreducible components). According to Cartan or
Remmert [Loj, V.3], the function d on NC is Zariski semicontinuous, thus equal
to some constan§t d0 over some Zariski open and dense N˘C NC and strictly
greater than d0 over NC N˘C. By Lemma 3.1, we have d0 =n⊂ 1, and moreover
N˘C contains generic poin\ts of N∆ (here we say “generic”, and−not “all”, because
reg
at special points (z,z) N∆ the germ of NC could have a second irreducible
∈ reg
component different from NC).
z
Set p0 =(z0,z0).
If d(p0)=d0 =n 1, then by the Rank Theorem of Remmert [Loj, V.6] there
exists a neighbourho−od Uz0 ⊂ Z ×Z∗ of p0 such that Π(NC ∩Uz0) is§a complex
analytic subset Y of V = Π(U ), of dimension n. The situation is not very
z0 z0 z0
differentfromtheoneofLemma3.1. Andwearejustinthe conclusionofTheorem
2.1: obviously Y contains the previously defined Y , for every z N V .
z0 z ∈ reg∩ z0
Hence, we shall suppose fromnow on that d(p0)=l n. Thus the verticalfibre
F∗ contains an irreducible component Y∗ Z∗ of dim≥ension l, passing through
p0p.0ItsconjugateY =(Y∗) Z isanirredu⊂ciblecomponentofthehorizontalfibre
Fasp0a=(h{oqri∈zoNntCal|)Πsu∗b(qse)t=ofΠN∗⊂(Cp,0)s}o.mIentitmheesfoalsloawsinugb,sewteosfhtahlelcboansseidZe.r Y sometimes
Obviously, Y Z is contained in Π(NC). Because dimNC = 2n 1 and
d0 = n 1, the i⊂mage of NC by Π is a countable union of local analytic−subsets
−
of dimension n (by iterated application of the Rank Theorem). Thus Y cannot
≤
have dimension larger than n, and so
dimY =n.
To complete the proof, we will show that Y contains Π(NC U ), for some
∩ z0
neighbourhood Uz0 of z0 (thus, after all, Y is equal to that projection, and so Y
is in fact the unique irreducible component of dimension n of the horizontalfibre).
Because Y is closed, it is sufficient to verify this inclusion for Π(N˘C U ).
NotethatY NC isnotcontainedinNC N˘C,fordimensionalrea∩sonzs0: because
Y ishorizontalo⊂fdimensionnandeachvertic\alfibrethroughNC N˘Chasdimension
n, NC would have dimension 2n. Thus, Y intersects N˘\C. If p = (z,z0)
≥is a point of Y N˘C, then (by t≥he Rank Theorem) the image by Π of a small
neighbourhood o∩f p in NC is an analytic subset Y V Z of dimension n. The
z z
⊂ ⊂
germofY atz couldhaveseveralirreduciblecomponents,butatleastoneofthem
z
is contained in Y, because Y V Y and dimY = dimY . By a connectivity
z z z
argument (recall that NC is i∩rredu⊂cible, and therefore N˘C is connected), we see
that Y contains all the points arising from the projection of N˘C to Z, as claimed.
Letusconcludebylookingagainatthe(counter)exampleofthepreviousSection.
Example 3.1. Take again the real analytic surface S in C3 with equation
z2 =( ez1)z1
ℜ
z3 =eℜez1z1
which, outside 0, is a CR submanifold of CR dimension 0 and CR codimension 2.
Its complexification is the complex analytic surface SC in C3 C3∗ with equation
×
SINGULAR LEVI-FLAT HYPERSURFACES 10
(setting z¯ =w )
j j
z1+w1 z1+w1
z2 = z1 , w2 = w1
2 2
z1+w1 z1+w1
z3 =e 2 z1 , w3 =e 2 w1
The projection of SC to the first factor C3 is not analytic: something like z3 =
ez2/z1z1, for z1 = 0. The horizontal fibre through 0 SC is the complex curve
C C3 given by6 ∈
⊂ 1
z2 = z12 , z3 =e12z1z1.
2
Itis,ofcourse,containedintheprojectionofSC butnotequalto. Thedimensional
arguments used several times in the proof of Theorem 2.1 cannot be used here.
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