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ISSN 2074-1871 Уфимский математический журнал. Том 9. № 3 (2017). С. 172-185.
УДК 517.55
LEVI-FLAT WORLD: A SURVEY OF LOCAL THEORY
A. SUKHOV
Abstract. This expository paper concerns local properties of Levi-flat real analytic manifolds with singularities. Levi-flat manifolds arise naturally in Complex Geometry and Foliation Theory. In many cases (global) compact Levi-flat manifolds without singularities do not exist. These global obstructions make natural the study of Levi-flat objects with singularities because they always exist. The present expository paper deals with some recent results on local geometry of Levi-flat singularities. One of the main questions concerns an extension of the Levi foliation as a holomorphic foliation to a full neighborhood of singularity. It turns out that in general such extension does not exist. Nevertheless, the Levi foliation always extends as a holomorphic web (a foliation with branching) near a non-dicritical singularity. We also present an efficient criterion characterizing these singularities.
Keywords: CR structure, Levi-flat manifold. Mathematics Subject Classification: 37F75,34M,32S,32D
1. Introduction
This expository paper paper concerns local properties of real analytic Levi-flat manifolds with singularities. Such manifolds arise naturally in the theory of holomorphic foliations and differential equations, in particular, in the study of minimal sets for foliations. They were studied recently by several authors from different points of view (see, e.g., [1, 2, 3, 4, 5, 6, 11]). In the present paper I discuss recent progress achieved in the series of our joint papers [18, 16, 17] with S. Pinchuk and E. Shafikov, Of course, they always must be considered as co-authors of the present work.
This paper is written for the special issue of Ufa Mathematical Journal dedicated to the 100th anniversary of A.F. Leontiev, the foundator of the theory function research school in Ufa. I dedicate this work to the memory of this remarkable mathematician.
2. Real analytic Levi-flat hypersurfaces in cn
2.1. Real analytic sets and their complexification. Let Q С Rra be a domain. A real analytic set Г С Q is a closed set locally defined as a zero locus of a finite collection of real analytic functions. In fact, we can always take just one function to locally define any real analytic set. We say that Г is irreducible in Q if it cannot be represented as the union Г = Г U Г2 of two real analytic sets Г.,- in Q with Гj \ (Г П Г2) = 0, j = 1, 2, (this is the geometric irredueibilitv). In the present paper we always deal with germs of real analytic sets (without mentioning this
Г
if there exists a point q E Г such that near q the set Г is a real analytic submanifold of dimension n — 1. For a real hypersurface Г we call such q a regular point. The union of all regular points form a regular locus denoted bv Г*. Its complement Ysing := Г \ Г* is called the Г
A. Sukhov, Levi-flat world: a survey of local theory.
© Sukhov A. 2017.
Поступила 19 июня 2017 г.
The author is partially supported by Labex CEMPI..
point in semianalytic or subanalytic geometry, where a similar notion is less restrictive and a real analytic set is allowed to be a submanifold of some dimension near a regular point. By our definition, the points of a hvpersurfaee r, where r is a submanifold of dimension smaller than n — 1, belong to the singular locus. For that reason, r* may not be dense in T, this can happen even if r is irreducible (so-called umbrellas). Note that Tsing is a closed semianalytic subset of r (possibly empty) of real dimension at most n — 2,
In local questions we are interested in the geometry of a real hvpersurfaee r in an arbitrarily small neighbourhood of a given point a G r, i.e., of the germ at a of r. If the germ is irreducible at a, we may consider a sufficiently small open neighbourhood U of a and a representative of the germ which is irreducible at a, see [14] for details. In what follows we will not distinguish between the germ of r at a given point a and its particular representative in a suitable neighbourhood of a.
Let r c R™ be the germ of a real analytic set at the origin. By rC we denote the complexifieation of r, i.e., a complex analytic germ at the origin in C™ = R + zR™, z = x + iy, with the property that each holomorphie function vanishing on r necessarily vanishes on rC, Equivalentlv, rC is the smallest complex analytic germ in Cn that contains r. It is well known that the dimension of r is equal to the complex dimension of rC and that the germ of rC is irreducible at zero whenever the germ of r is irreducible, see Narasimhan [14] for further details and proofs. Also, given a real analytic germ J^j|>0 aj xJ, aj G R, x G Rra, we define its complexification to be the complex analytic germ YhaJ Z'J■
While the complexification of the germ of a real analytic set is canonical and is independent of the choice of the defining function, the next lemma gives a convenient way of constructing the complexification of a real analytic hvpersurfaee using a suitably chosen defining function. We will need the following notion of a minimal defining function for a complex hvpersurfaee. Given a complex hvpersurfaee A = {z G Q : f (z) = 0} in a domain Q c Cn, / is called minimal if for every open subset U C Q and any function g holomorphie on U and such that g = 0 on A n U, there exists a function h holomorphie in U such that g = h/. If f is a minimal defining function, then the singular locus of A coincides with the set f = df = 0, Locally, any irreducible complex hvpersurfaee admits a minimal defining function, see Chirka [7],
Lemma 2.1. Let r C Rra be an irreducible germ of a real analytic hypersurface at the origin. Then there exists a defining function p(x) of the germ of r at the origin such that its complexification p(z) is a minimal defining function of the complexification rC.
2.2. Levi-flat hypersurfaces. Let z = (zi,...,zn), Zj = Xj + iyj, be the standard coordinates in Cn. Let r be an irreducible germ of a real analytic hvpersurfaee at the origin defined by a function p provided by Lemma 2,1, In a (connected) sufficiently small neighbourhood of the origin Q c Cn, the hypersurface r is a closed irreducible real analytic subset of Q of dimension 2n — 1,
For q G r* ^^^^^^^r ^te ^^^^^orphic tangent space Hq (r) := Tq (r) n JTq (r). The Levi form of r is a Hermitian quadratic form defined on Hq (r) by
Lq (v) = Y1 (q)Vk
k,j
with v G Hq (r) A real analytic hvpersurfaee r is called Levi-flat if its Levi form vanishes on Hq(r) for every regular point g of r. By the classical result of Elie Cartan, for every point q G r* ^^^re ^^^^^s a local biholomorphie change of coordinates centred at q such that in the new coordinates r in some neighbour hood U of q = 0 has the fo rm {z G U : zn + zn = 0}, Hence, r n U is locally foliated by complex hvperplanes {zn = c, c G i R}. This foliation is called the Levi foliation of r*, and will be denoted by We denote by Cq the leaf of the Levi foliation through q. Note that by definition it is a connected complex hvpersurfaee closed in r*.
Let 0 G r , We choose the neighbourhood Q of the origin in the form of a polydisc A(e) = {z G Cn : Izj| < e] of radius e > 0, Then for e small enough, the function p admits the Taylor expansion convergent in U:
p(z, z) = ^ CuzJzJ, Cu G C, I,J G Nra (1)
J-J
t-ij * ij
The coefficients cij satisfy the condition
cij = cJ]:, (2)
because p is a real-valued function. Note that in local questions we may further shrink Q as needed.
By Lemma 2.1, the choice of the defining function p guarantees that the complexifieation of (the germ of) r is given by
rC = {(z,w) G Cn x C : p(z,w) = 0]. (3)
The hvpersurface r lifts canonically to rC as
r = rC n{w = z]. In what follows we denote by rC the singular locus of rC.
2.3. Segre Varieties. Our key tool is the family of Segre varieties associated with a real analytic hvpersurface r, For w G A(e) consider a complex analytic hvpersurface given by
Qw = {z G A(e) : p(z,w) = 0]. (4)
It is called the Segre variety of the point w. This definition uses the defining function p of r in a neighbourhood of the origin which appears in (3). We will always consider the case where the germ of r at the origin is irreducible and everywhere through the paper we use a defining function provided by Lemma 2.1 in a neighbourhood of the origin (the same convention is used in [18]). In general the Segre varieties Qw also depend on the ehoice of e (some irreducible components of Qw may disappear when we shrink e). Throughout the paper we consider only the Segre varieties Qw defined by means of the complexifieation at the origin. The reader should keep this in mind. Also note that if 0 is a regular point of T, then the notion of the Segre variety Qw is independent of the choice of a defining function p with non-vanishing gradient when w is close enough to the origin.
The following properties of Segre varieties are immediate.
Lemma 2.2. Let r be a germ of an irreducible real analytic hypersurface in Cn, n > 1. Then
(a) z G Qz if and only if z G T,
(b) z G Qw if and only if w G Qz.
We also recall the property of local biholomorphic invarianee of some distinguished components of the Segre varieties near regular points. Since here we are working near a singularity, we state this property in detail using the notation introduced above. Consider a regular point a G r* n A(e) and fix a > 0 small enough with respeet to e. Consider any function pa real analytic on the polvdise A(a,a) = {Izj — aj| < a,j = 1,...,n] such that r n A(a, a) = p-1(0) and the gradient of pa does not vanish on A(a, a;). Then for w G A(a, a) we can define the Segre variety aQw ("the Segre variety with respect to the regular point a") as
aQw = {z G A(a,a) : pa(z,w) = 0],
(we use the Taylor series of p„ at a to define the complexifieation). For a small enough, aQw is a connected nonsingular complex submanifold of dimension n — 1 in A(a,a). This definition is independent of the choice of the local defining function pa satisfying the above properties.
We have the inclusion aQw C Qw. Note that in general Qw can have irreducible components in A(e) which do not contain aQw.
Lemma 2.3. (Invariance property) Let r, r' he irreducible germs of real analytic hyper surf aces, a G r*, a' G (r')*, and A(a,a), A(a',a') he small polydiscs. Let f : A(a,a) ^ A(a', a') he a holomorphic 'map such that f (r n A(a,a)) C r' n A(a' ,a') and f (a) = a'. Then
f (aQ w) c Qf (w)
for all w G A(a, a) close enough to a. In particular, if f : A(a,a) ^ A(a' ,a!) is biholomorphic, then f (aQw) = a Q'f(w). Here aQw and a Q'fare the Segre varieties associated with r and r' and the points a and a' respectively.
For the proof see for instance, [8]. As a simple consequence of Lemma 2,2 we have
Corollary 2.4. Let r C Cn be an irreducible germ at the origin of a real analytic Levi-flat hypersurface. Let a G r*. Then the following holds:
(a) There exists a unique irreducible component Sa of Qa containing the leaf Ca. This is also a unique complex hypersurface through a which is contained in r.
(b) For every a,b G r* one has b G Sa ^^ Sa = Sb.
(c) Suppose that a G r* and Ca touches a point q G r such that dimC Qq = n — 1 (the point q may be singular). Then Qq contains Sa as an irreducible component.
The proof is contained in [18]. Again, we emphasize that Corollary 2,4 concerns the "global" Segre varieties, i.e., those defined by (4) using the eomplexifieation at the origin,
2.4. Characterization of dicritical singularities for Levi-flat hypersurfaces. Let r
be an irreducible germ of a real analytic Levi-flat hypersurface in Cn at 0 G r*. Fix a local defining function p chosen by Lemma 2.1 so that the eomplexifieation rC is an irreducible germ of a complex hypersurface in C2ra given as the zero locus of the eomplexifieation of p. As already mentioned above, all Segre varieties which we consider are defined by means of this eomplexifieation at the origin.
Fix also e > 0 small enough; all considerations are in the polvdise A(e) centred at the origin, A point q G r* n A(e) is called a dicritical singularitv if q belongs to the closure of infinitely many geometrically different leaves Ca. Singular points in r* which are not dicritical are called nondicritical.
A singular point q is called Segre degenerate if dim Qq = n.
Lemma 2.5. Let r be a real analytic Levi-flat hypersurface. Then dicritical singular points form a complex analytic subset of r of complex dimension at most n — 2, in particular, it is a discrete set if n = 2. If r is algebraic, then the set of dicritical singularities is also complex algebraic.
We recall that the Segre degenerate singular points form a complex analytic subset of A(e) of complex dimension at most n — 2, in particular, it is a discrete set if n = 2, For the proof see [11, 18],
Theorem 2.6. Let r = p-1(0) be an irreducible germ at the origin of a real analytic Levi-flat hypersurface in Cn and 0 G r*. Then 0 is a dicritical point if and only if it is Segre degenerate.
This result is obtained in [16],
3. Singular webs
In this section we define singular holomorphic webs and outline the connection between webs and differential equations. This connection is transparent in dimension two, so we will discuss this case separately. For a comprehensive treatment of singular webs see, e.g., [15].
3.1. Webs in C2. Recall that the germ of a holomorphie codimension one foliation T in Cra, n > 2, can be given by the germ of a holomorphie 1-form u G A1(U) satisfying the Frobenius integrabilitv condition u A du = 0. The leaves of T are then complex hypersurfaces L that are tangent to ker u. The foliation T is -singular if the set Tsng = {z : u(z) = 0] is nonempty and of eodimension at least 2,
In dimension 2 the integrabilitv eondition for u always holds, and the above definition of a (nonsingular) foliation can be interpreted in the following way: for a suitably chosen open set U and coordinate svstem in C2 the foliation T is given by a holomorphie first order ODE
S = F ("1,"2) ^
with respect to unknown function z2 = z2(z1). The leaves of the foliation T are then the graphs of solutions of the ODE, This interpretation admits a far reaching generalization which we now describe. Our considerations are local and should be understood on the level of germs, but to simplify the discussion we will work with appropriate representatives of the germs.
Let U1, U2 be domains in C containing the origin. Set U = U1 x U2 C C2, and consider a holomorphie function $ on U x C, It defines a holomorphie ordinary differential equation on U x C,
$(Z1,Z2,P)=0 (6)
with z = (z1,z2) G U and p = ^ G C. This is an equation for the unknown function z2 = z2(z1); in other words, we view z1 and z2 as the independent and the dependent variables respectively. For d G N a singular holomorphie d-web W in U is defined by equation (6) where $ is of the form
d
$(z,p) = £ (z)f. (7)
i=0
In general, there are d families of solutions of (6) (with $(z,p) as in (7)), which are either unrelated to each other or may fit together along some complex curves (branching), The graphs of solutions are called the leaves of W.
Example 3.1. Consider the ODE of the form p2 = 4z2 in C2, Its solutions form a complex one-dimensional family of curves Lc = {z2 = (z1 + c)2], c G C. For every point b = (b1, b2) G C2 with b2 = 0 there exist exactly two curves passing through this point, namely, L_biand L_bl(we can take an arbitrary braneh of These curves meet at b transversely. But any point (b1, 0) is contained only in one curve L_bl of the family o
If d = 1, then (6) becomes resolved with respect to the derivative, so 1-webs simply coincide with holomorphie foliations (possibly singular). If (7) factors into distinct, linear in p terms, i.e., $(z,p) = ndj=1(p — fj(z)), where fj(z) ^re holomorphie functions, then each ODE p = fj(z) defines a holomorphie foliation Tj. If the leaves of Tj intersect in general position (resp, pairwise transversely) then the union of Tj is called a smooth (resp. quasi-smooth) holomorphie d-web. Thus, our definition of a singular d-web is a proper generalization of smooth webs. From this point of view one can consider singular d-webs as a "branched"version of their smooth counterparts,
3.2. Webs in Cra, n > 2. The definition of a d-web (singular or smooth) via differential equations does not have a simple generalization to higher dimensions. There are several equivalent definitions in the literature. We will use a more geometric one that is more suitable for our purposes.
We denote by PT* := PT*Cn the projectivization of the cotangent bundle of Cn with the natural projection n : PT,* ^ Cra, A local trivialization of PT,* is isomorphic to U x G(1,n),
where U C Cn is an open set and G(1,n) = CPn—1 is the Grassmanian space of linear complex one dimensional subspaees in Cn. The space PT* has the canonical structure of a contact manifold, which can be described (using coordinates) as follows. Let z = (z1,...,zn) be the coordinates in Cn and (p1,... ,pn) be the fibre coordinates corresponding to the basis of differentials dz1,..., dzn. We may view [f>1,... ,pn] as homogeneous eoordinates on G(1,n). Then in the affine chart {pn = 0} in nonhomogeneous coordinates pj = pj/pn, j = 1,... ,n — 1, the 1-form
n— 1
rq = dzn + ^ Pj dzj (8)
j=1
is a local contact form. Considering all affine charts {pj = 0} we obtain a global contact structure.
Let U be a domain in Cn. Consider a complex purely n-dimensional analvtie subset W in n—1(U) C PT*. Suppose that the following conditions hold:
(a) the image under n of every irreducible component of W has dimension n;
(b) a generic fibre of n intersects W in d regular (smooth) points and at every such point q the differential dn(q) : TqW M Cn is surjective;
(c) the restriction of the contact form on the regular part of W is Frobenius integrable. So rqlw = 0 defines the foliation T-^ of the regular part of W. (The leaves of the foliation Tw are called Legendrian submanifolds.)
Under these assumptions we define a -singular d-web W in U as a triple (W,n, Tw), A leaf of the web W is a component of the projection of a leaf of Tw into U. Note that at a generic point z G U a ^b (W,-n, Tw) ^^toes in U Mar z exactly d families of smooth foliations.
We need first to interpret a first order PDE as a subvarietv of a 1-jet bundle. Recall that two smooth functions and 02 have the same fc-jet at a source point x° G Cn if |01(x) — 02(x)| = o(lx — x°lk), In other words, this simply means that their Taylor expansions of order fc at x0 coincide. The equivalence classes with respect to this relation are called fc-jets at x0.
Let U C Cn— 1 be a domain. Consider J 1(U, C), the space of 1-jets of holomorphic functions f : U ^ C We can view such functions as sections of the trivial line bundle U x C ^ U. Then J1 (U, C) can be viewed as a vector bundle
-k : J 1(U, C) ^ U x C (9)
of rank (n — 1). Let z' = (z1,..., zn—1) ^e ^te coordinates on U C Cn—\ zn be the coordinate in the target space, and let pj denote the partial derivatives of zn with respect to Zj. Then (z,p) = (z1,..., zn,p1,... ,pn—1) form the coordinate system on J 1(U, C), Note that dim J 1(U, C) = 2n — 1. The sp ace J 1(U, C) admits the structure of a contact manifold with the contact form
n—1
6 = dzn — ^^ pj dzj. (10)
j=1
Given a local section f : U ^ C, let j 1f : U ^ J 1(U, C) j 1f : z M- jlzf denote the corresponding section of the 1-jet bundle. Then a section F : U M J 1(U, C) locally coincides with j1f for some section f : U M C if and onlv if F annihilates 6. Now observe that the map l : (z,p) M (z, —p) in the chosen coordinate systems is a biholomorphism whose pullbaek sends ^ to 0 in (8), i.e., l : J 1(U, C) M PT* is a contactomorphism. Using the map i we may view the projectivized cotangent bundle PT,* as a compactification of the 1-jet bundle. Alternatively, we may eompaetifv J 1(U, C) in the variables p, that is, we compactify every fibre C™—1 to CPn—1. Since the dependence of the form 9 is linear in p, the eompaetified bundle will be a contact complex manifold.
Any first order holomorphie PDE of the form
$ (--—Si--it-,)=0 ^
with respect to the unknown function zn = zn(z1,..., zn_1) defines a complex hvpersurface in J 1(U, C) given by the equation $(z,p) = 0, Any solution zn = f (z1,..., zn_1) of (11) admits prolongation to J 1(U, C), i.e., defines there an (n — 1)-dimensional submanifold Sf given by
|zra = f(z1,...,zn_1), Pj = ^(z1,...,zn_1), j = 1,...,n — 1J .
Hence, solutions of this differential equation can be identified with holomorphie sections Sf of annihilated by the contact form 6. As an example, for equation (5), the corresponding hvpersurface W C J 1(U, C), U C C, is simply the graph of a holomorphie function p = F(z). It is foliated by graphs of solutions, which are integral curves of the distribution defined by 9, and the corresponding foliation T in C2 is obtained by the biholomorphic projection : W ^ C2Z.
Suppose now that we have several differential equations of the form (11) such that the intersection of the corresponding hvpersurfaees is a complex analytic subset W of J 1(U, C) of pure dimension n. For example, we can have n — 1 equations in general position. Suppose further that the eompaetifieation of W in the projectivized cotangent bundle PT,*^ still forms a complex subvarietv of the same dimension. This is the case, for example, if all $(z,p) are polynomial with respect to p with coefficients holomorphie in z, Then W satisfies the definition of a singular web given in the previous subsection. Note that we need to consider eompaetifieation of W only if the projection in (9) has fibres of positive dimension, since otherwise the projection from J 1(U, C) gives the same web in U C Cn.
Also note that for n = 2 both definitions of a singular web agree. Indeed, given a differential equation (6), (7), the function $(z,p) is polvnomial in p and thus it can be projectivized to define a hvpersurface in PT2*U. This gives the hvpersurfaee in PT2*U that has the required properties. Conversely, let U be a neighbourhood of the origin in C2, let W be a complex hvpersurface in PT2* U with the surjective projection n : W ^ U. Without loss of generality assume that W is irreducible. If ■n_1 is discrete, then by the Weierstrass preparation theorem, in a sufficiently small neighbourhood U of the origin W can be represented by a Weierstrass pseudo-polynomial in p, and we obtain the definition of the web given in Section 3,1, Suppose that dim ^_1(0) = 1, Let t : C2popi) \ {0] ^ CP1 ^e ^te natural projection given by t(p0,p1) = [p0,p1]- Let
f = (Id, t) : U x (C2 \ {0]) ^ U x CP1.
Then the set W = ;f_1(W) is analytic in U x (C2 \{0]) of dimension 3, The set U x {0]
is removable, and so we may assume that W is analvtic in U x C2, In a neighbourhood
of (0, 0) it can be given by an equation $(z,p) = 0, But since its image is complex analytic in U x CP\ ^te fonction 0 is in fact a homogenous polynomials in p. This shows that in a neighbourhood of the origin in U, the hvpersurface W can be given by an equation which is polynomial in variable p, and we again recover the definition of Section 3,1,
We also need a related notion of a multi-valued meromorphic first integral. Let X and Y be two complex manifolds and nx : X x Y ^ X and ny : X x Y ^ Y be the natural projections, A d-valued meromorphic correspondence between X and Y is a complex analytic subset Z C X x Y such that the restriction -nx IZ is a proper surjective generieallv d-to-one map. Hence, nY°is defined generieallv on X (i.e., outside a proper complex analytic subset in X), and can be viewed as a d-valued map. In what follows we denote a meromorphic correspondence by a triple (Z; X, Y) equipped with the canonical projections:
A multiple-valued meromorphic first integral of a singular d-web W in U is a d-valued meromorphic correspondence (Z; U, CP) such that level sets nx ° n—1(c), c G CP are invariant subsets of W, i.e., they consist of the leaves of W.
Definition 3.2. Let r be a real analytic Levi-flat hvpersurface in a domain Q c Cn. We say that a holomorphie d-web W in Q is the extension of the Levi foliation of r* if every leaf of the Levi foliation is a leaf of W.
Although in this definition we do not require W to be irreducible, we suppose that at least one leaf of every component of W meets T*. Clearly, under this condition the singular web extending the Levi foliation is unique.
4. Extension of the Levi foliation
Global or local extension of the Levi foliation to the ambient space is an important question,
see, e.g., [2, 3, 5, 9] for recent results in this direction. Brunella [2] gave an example of a
C2
neighbourhood of the origin as a singular web, but not as a foliation, see Example 4.3. The following result is obtained in [18].
Theorem 4.1. Let r C Q be an irreducible Levi-flat real analytic hypersurface in a domain Q C Cra, n > 2, and 0 G r*. Assume that at least one of the following conditions holds:
(a) 0 G r is not a dicritical singularity.
(b) r is a real algebraic hypersurface.
Then there exist a neighbourhood U of the origin and a singular holomorphie d-web W in U such that W extends the Levi foliation C. Furthermore, W admits a multiple-valued meromorphic first integral in U.
We note that under some additional assumptions on the singular locus of T, part (a) of our result was obtained recently by Fernandez-Perez [9]. Our approach is rather constructive, especially under condition (b) in Theorem 4.1. In many cases one can write down explicitly the d-web that gives the extension of the Levi foliation. The key point of our approach lies in the connection between singular webs and first order analytic partial differential equations, although we do not claim any particular originality here. Presently, the most commonly used definition of webs is through the geometry the projeetivized cotangent bundle. We reconstruct the connection between geometry of singular webs and analytic PDEs through eompaetifieation of the 1-jet bundle of functions on Cn—1 and its identification with the projeetivized cotangent bundle of Cra, see Section 3 for details.
The proof of Theorem 4.1 (in the Case (a)) is based on the following idea. The leaves of the Levi foliation can be identified with the components of the Segre varieties associated with r. It is possible to find a complex line parametrizing all Segre varieties of r. While for a general real analytic hypersurface r in Cn, the corresponding family of Segre varieties is n-dimensional, Levi-flat hvpersurfaees can be characterized as those whose Segre family is one-dimensional, and ultimately this is the reason why the Levi foliation admits extension to the ambient space. Essentially, a suitably chosen one-dimensional family of Segre varieties is the meromorphic (perhaps, multiple-valued) first integral. Its graph can be described by a system of n — 1 first order PDEs. This system defines an n-dimensional complex analytic subvarietv of the 1-jet
bundle of holomorphie functions on Cra_1, This subvariety can then be compactified in the projectivized cotangent bundle of Cn, which gives the singular d-web. The key statement is the following
Proposition 4.2. Under the assumptions of Theorem J^.i, for a sufficiently small neighbourhood Q of the origin there exists a complex line A C Cn with the following properties:
(i) A n Qo = {0];
(ii) A C rsng;
(iii) For every q G r* n Q, there exists a point w G A such that Cq C Qw.
The existence of such A should be compared to the transversal of the Levi foliation in the smooth case: if r is given by {Re zn = 0], then the complex line {z1 = ■ ■ ■ = zn_1 = 0] intersects all Segre varieties, and can be used as a local parametrization both of the Levi foliation and its extension. The complex one-parameter family of the Segre varieties constructed in Proposition 4,2 can be viewed as a holomorphie web. The equations of this web can be explieitelv constructed using tools of the local complex analytic geometry (on jet bundles). As an illustration consider the following example.
Example 4.3. This example, discovered by M, Brunella [2], shows that in general the Levi foliation of a Levi-flat hvpersurfaee admits extension to a neighbourhood of a singular point only as a web, not as a singular foliation. Consider the Levi-flat hvpersurfaee
r = {z G C2 : yl = 4(y2 + X2)y2]. (12)
The singular locus of r is the set {y1 = y2 = 0]. Its subset given by {y1 = y2 = 0, x2 < 0] is a "stick i.e., it does not belong to the closure of smooth points of r. The Segre varieties of r are given by
Qw = {z G C2 : (Z2 — W2)2 + (Z1 — W1)4 — 2(z2 + W2)(Z1 — W1)2 = 0]. (13)
We see that Q0 = {zl + z4 — 2z2z2 = 0], and the origin is a nondieritieal singularity. Following the algorithm in the proof of Theorem 4,1 we choose A(t) to be given by w1 = 0 w2 = Then (13) becomes
(Z2 — t)2 + z4 — 2^2(^2 + t) = 0. After differentiation with respect to zi, and using the notation p = ^ we obtain
2(Z2 — t)p + 4zf — 4*1 (Z2 + t) — 2z\p = 0.
Direct calculation shows that the resultant of the two polynomials in t above vanishes (after dropping irrelevant factors) when
p2 = 4*2. (14)
This is the 2-web that extends the Levi foliation of r*. Its behaviour is described in Example 3,1, Note that the exceptional set {z2 = 0] interseets r along the 1 ine {z2 = y1 = 0] C rsrafl. By inspection of solutions of (14) we see that a first integral of r can be taken to be
f (Z1,Z2) = Z1 ± ^Z~2,
where f is as a multiple-valued 1 — 2 map. In fact, one can immediately verify that
the closure of the smooth points of r is given by
{z G C2 : Im (Z1 ± yfiD = 0] = {Im (Z1 + ^2)] U {Im (Z1 — y/^)].
o
5. Levi-flat subsets, Segre varieties and Segre envelopes
A real analytic Levi-flat set M in CN is a real analytic set such that its regular part is a Levi-flat CE manifold of hvpersurfaee type. An important special case (closely related to the theory of holomorphie foliations) arises when M is a hvpersurfaee. The main question here concerns an extension of the Levi foliation of the regular part of M as a (singular) holomorphie foliation (or, more generally, a singular holomorphie web) to a full neighbourhood of a singular point. The existence of such an extension allows one to use the holomorphie resolution theorems in order to study local geometry of singular Levi-flat hvpersurfaees.
In this section we provide relevant background material on real analytic Levi-flat sets (of higher codimension) and their Segre varieties. This is quite similar to the special case of hvpersurfaees considered in previous sections,
5.1. Real and complex analytic sets. Let Q be a domain in CN. We denote by z = (z1,...,zN) the standard complex coordinates, A closed subset M C Q is called a real
Q
(resp, holomorphie) functions.
For a real analytic M this means that for every point q G Q there exist a neighbourhood U of q and real analytic vector function p = (p1,..., pk) : U M Rfc such that
M n U = p-1(0) = [z G U : pj (z,z) = 0, j = 1,...,k}. (15)
In fact, one can reduce the situation to the case k = 1 by considering the defining function p\ + ... + p\. Without loss of generality assume q = 0 and choose a neighbourhood U in (15) in the form of a polvdise A(e) = [z G CN : Izj | < e} of radius e > 0. Then, for e small enough, the (vector-valued) function p admits the Taylor expansion (1) convergent in U. Of course, the (Cfc-valued) coefficients cjj also satisfy the condition (2), since p is a real-valued (Revalued) function.
An analytic subset M is called irreducible (as a germ) if its germ at 0 g M can not be represented as a union M = M1 U M2 where Mj are analytic germs at 0 different from the germ of M. In what follows we always use this notion of irredueibilitv, A set M can be decomposed into a disjoint union M = Mreg U Msing, the regular and the singular part respectively. Notice that here we change the notation with respect to the case of a hvpersurfaee. The regular part Mreg is a nonempty and open subset of M. In the real analytic case we adopt the following convention: M is a real analytic submanifold of maximal dimension in a neighbourhood of every point of Mreg, This dimension is called the dimension of M and is denoted by dim M. The set Msing is a real semianalytic subset of Q of dimension < dim M. Unlike complex analytic sets, for a real analytic M, the set Msing may contain manifolds of smaller dimension which are not in the closure of Mreg, as seen in the classical example of the Whitney umbrella. Therefore, in general Mreg is not dense in M.
Recall that the dimension of a complex analytic set A at a point a G A is defined as
dimtt A := lim dim^ A,
and that the the function z M- dim^ A is upper semicontinuous. Suppose that A is an irreducible complex analytic subset of a domain Q and let F : A M X be a holomorphie mapping into some complex manifold X. The local dimension of F at a point z G Q is defined as dim^ F = dim A — dim^ F-1(F(z)) and the dimension of F is set to be dim F = dim^ F.
Note that the identity dim^ F = dim F holds on a Zariski open subset of A, and that dim F coincides with the rank of the map F.
5.2. Complexiflcation and Segre varieties. Let M be (the germ of) an irreducible real analytic subset of CN defined by (1). Denote by J the standard complex strueture of CN and
consider the opposite structure — J, Consider the space C. := (Ct; , J) x (C^, —J) and the diagonal
A = {(z,w) G C2,N : z = w} . The set M can be lifted to C™ as the real analytic subset
M := {(z,z) G C2N : z G M} .
There exists a unique irreducible complex analytic subset Mc in ClN of complex dimension equal to the real dimension of M such that M = Mc n A (see [14]). The set Mc is called the complexification of M. The antiholomorphic involution
t : C^ M C2W, t : (z,w) M (w,z)
leaves Mc invariant and M is the set of fixed points of t\mc.
The complexification Mc is equipped with two canonical holomorphie projections ■nz : (z,w) M z and nw : (z,w) M w. We always suppose by convention that the domain of these projections is Mc. The tripie (Mc,nz,tw) is represented by the following diagram
M c
(CN ,J) (CN, —J)
which leads to the central notion of the present paper in full generality. The Segre variety of a point w G CN is defined as
Qw := frz ° = {z G CN : (z,w) G Mc) .
When M is a hypersurface defined by (15) (with k =1) this definition coincides with the usual definition
Qw = {z : p(z,w) = 0] . The following properties of Segre varieties are well-known for hvpersurfaees.
Proposition 5.1. Let M he any real analytic subset of a domain Q. Then
(a) z G Qz ^^ z G M.
(b) z G Qw ^^ w G Qz,
(c) (invariance property) Let M1} M2 be real analytic subsets in CN and CK respectively, p G (MJ^, q G (M2)reg, and U1 3 p, U2 3 q be small neighbourhoods such, that Mj n Uj is a CR manifold. Let also f : U1 M U2 be a holomorphie map such that f (M1 n^1) C M2nU2. Then
f (Q ) C Qf(w)
for all w close to p. In particular, if f : U1 M U2 is biholomorphic, then f (Q^) = Q2f(w)-Here Ql and Q2^) are Segre varieties associated with M1 and M2 respectively.
5.3. Levi-flat sets. We say that an irreducible real analytic set M C Cn+m is Levi-flat if dim M = 2n — 1 and Mrefl is locally foliated by complex manifolds of complex dimension n — 1. In particular, Mreg is a CR manifold of hypersurface type. The most known case arises when m = 0, i.e., when M is a Levi-flat hypersurface in Cra,
We use the notation z" = (zn+1,..., zn+m), and similarly for the w variable. It follows from the Frobenius theorem and the implicit function theorem that for every point q G Mreg there exist an open neighbourhood U and a local biholomorphic change of coordinates F : (U,q) M (U', 0) such that F(M) has the form
{z G U' : zn + ~zn = 0, z" = 0]. (16)
The subspace F(M) is foliated by complex affine subspaees Lc = [zn = ic, z" = 0, c G R}, which gives a foliation of Mreg n U by complex submanifolds F-1(LC). This defines a foliation on Mreg which is called the Levi foliation and denoted by C. Every leaf of C is tangent to the complex tangent space of Mreg. The complex affine subspaees
[zn = c^z" = 0} , c G C (17)
in local coordinates given by (16) are precisely the Segre varieties of M for every complex c. Thus, the Levi foliation is closely related to Segre varieties.
For M defined by (16) its Segre varieties (17) fill the complex subspace z" = 0 of Cn+m. In particular, if w is not in this subspace, then Qw is empty. We need to study some general properties of projections f^d nw.
Let n be one of the projections -kz or tw. Following the discussion in the previous subsection we introduce the dimension of n by setting dimn = max^,w)eMc dim^,™) If M is irreducible, then so is Mc (see [14, p.92]). Hence, (Mc)reg is a connected complex manifold of dimension 2n — 1. Then the equality dim^,™) n = dim n holds on a Zariski open set
MC := Mc \ E C (Mc)reg, (18)
where E is a complex analytic subset of dimension < 2n — 1. Here dim n coincides with the rank of n. Furthermore, dim(^l(MC)sing) < dimn.
Lemma 5.2. (a) We have dimnz = n. (b) The image nz(Mc) is contained in the (at most) countable union of complex analytic sets of dimension < n.
Proof. (a) The image of nz near a regular point of M is swept out by the Segre varieties (17). Hence it coincides with the subspace [z" = 0}.
(b) This is a consequence of (a). □
Of course, the projection nw has similar properties. Therefore, we have the following
Lemma 5.3. For every w, one of the following holds:
(a) Qw is empty.
(b) Qw is a complex analytic subset of dimension n — 1.
(c) Qw is a complex analytic subset of dimension n. The case (b) holds for nw(M<C).
A singular point q G M is called Segre degenerate if dim Qq = n. Note that the set of Segre degenerate points is contained in a complex analytic subset of dimension n — 2. The proof is quite similar to [18] (where this claim is established for hypersurfaces) so we skip the proof.
Let q G Mreg. Denote by Cq the leaf of the Levi foliation through q. Note that by definition this is a connected complex submanifold of complex dimension n — 1 closed in Mreg. Denote by M* C Mreg the image of M n M< under the projection n, where M< is defined as in (18). This set coincides with Mreg \ A for some proper complex analytic subset A.
Finally, let M be an irreducible real analytic Levi-flat subset of dimension 2n — 1 in a domain Q in Cn+m. Consider the set
S(M) = [z G Q : z G Qw for some w} .
We call S(M) the Segre envelope of M. Thus, we simply have
5 (M) = n(Mc).
This notion will play a crucial role in our approach. It follows from (16) and (17) that near every regular point of M the Segre envelope of M is a complex submanifold of dimension n containing Mreg. One of our goals is to describe the Segre envelope near a singular point of M.
Let M be a real analytic Levi-flat subset of dimension 2n — 1 in Cn+m. A singular point q E M is called dicritical if q belongs to infinitely many geometrically different leaves Ca. Singular points which are not dicritical are called nondicritical. We have the following efficient criterion obtained in [17].
Theorem 5.4. Let M be an irreducible real analytic Levi-flat subset of dimension 2n — 1 in a domain П С Cn+m. Then the point 0 E Mreg is dicritical if and only if dimC Q0 = n, that is, the origin is a Segre degenerate point.
This generalizes the case of hypersurfaces treated in Theorem 2.6.
The connection between holomorphic webs and the Levi foliation is described by the following proposition [17] generalizing Proposition 4.2 :
Proposition 5.5. Let M be an irreducible real analytic Levi-flat subset of dimension 2n-1 in a domain П С Cn+m. Assume that 0 E Mreg is a nondicritical singularity for M. For a sufficiently small neighbourhood П of the origin there exists a complex linear map L : Cm+l ^ cn+m with the following properties:
(i) L(Cm+1) n Qo = {0};
(ii) the 1-dimensional real analytic set 7 = L(Cm+1) П M is not contained in Msing
(iii) For every q E Mreg n П, there exists a point w E 7 such that Cq С Qw.
The first consequence is the following theorem obtained by Brunella [2].
Corollary 5.6. Let M be an irreducible real analytic Levi-flat subset in Cn+m .The Segre envelope S(M) is an irreducible complex analytic subset of dimension n containing Mreg.
Proof. We have S(M) = ж(МC). There are two cases.
Case 1. dim Q0 = n — 1. Then the desired result follows from Proposition 5.5 and the Remmert Rank theorem.
Case 2. dimQ0 = n. By Theorem 5.4 for every regular point a of M we have 0 E Qa, hence a E Q(0). □
The main application of Corollary 5.6 is the following result due to Brunella [2]:
Corollary 5.7. Let M be an irreducible real analytic Levi-flat hypersurface in a complex manifold V of dimension n. Then there exist a complex manifold X of dimension n, a real analytic Levi-flat hypersurface N in X, a holomorphic foliation T in X extending the Levi foliation of N, and a holomorphic map ж : X ^ V such that for some Zariski open subset U С X one has:
(i) ж : N П U ^ Mreg is an embedding;
(ii) ж : N П U ^ Mreg is a proper map.
The main idea of the proof is to consider the holomorphic tangent bundle H(M) of M. This is a real analytic Levi-flat subset of the projectivization of the cotangent bundle of V, see Section 3. The second step is to apply the Hironaka desingularization theorem to the Segre envelope of H(M).
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Alexandre Sukhov,
Université de Lille (Sciences et Technologies), U.F.R. de Mathematiques, 59655 Villeneuve d'Ascq, Cedex, France E-mail: [email protected]
