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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 manifold 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 hypersurface, 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. References [BER] M. S. Baouendi, P. Ebenfelt, L. P. Rothschild, Real submanifolds in complex space and theirmappings,PrincetonMathematical Series47(1999) [B-G] D. Burns, X. Gong, Singular Levi-flat real analytic hypersurfaces, Amer. J. 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