singular levi-flat real analytic hypersurfaces

Singular Levi-Flat Real Analytic HypersurfacesAuthor(s): Daniel Burns and Xianghong GongSource: American Journal of Mathematics, Vol. 121, No. 1 (Feb., 1999), pp. 23-53Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/25098656 .

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SINGULAR LEVI-FLAT REAL ANALYTIC HYPERSURFACES

By Daniel Burns and Xianghong Gong

Abstract. We initiate a systematic local study of singular Levi-flat real analytic hy persurf aces, con

centrating on the simplest nontrivial case of quadratic singularities. We classify the possible tangent cones to such hypersurfaces and prove the existence and convergence of a rigid normal form in the

case of generic (Morse) singularities. We also characterize when such a hypersurface is defined by the vanishing of the real part of a holomorphic function. The main technique is to control the be

havior of the homorphic Segre varieties contained in such a hypersurface. Finally, we show that not

every such singular hypersurface can be defined by the vanishing of the real part of a holomorphic or meromorphic function, and give a necessary condition for such a hypersurface to be equivalent to an algebraic one.

1. Introduction. A well-known theorem of E. Cartan says that a real an

alytic smooth hypersurface M in Cn has no local holomorphic invariant, if M

is Levi-flat, i.e., it is foliated by smooth holomorphic hypersurfaces of Cn. In

suitable local coordinates such a hypersurface is given by 3?zi = 0. On the other

hand, if the Levi-form is nondegenerate, the invariants of M are given by the

theory of Cartan [5], Chern-Moser [6] and Tanaka [11]. In this paper we are

concerned with real analytic hypersurfaces in Cn with singularity. Such a real

analytic hypersurface M in C" is decomposed into M* and Ms, where M* is a

smooth real analytic hypersurface and Ms, the singular locus, is contained in a

proper analytic subvariety of lower dimension. We say that a real analytic hy

persurface M with singularity is Lewi-flat if its smooth locus M *

is Levi-flat. An

interesting class of Levi-flat hypersurfaces are those real analytic varieties defined

by the vanishing of the real part of a meromorphic function. Not every Levi-flat

hypersurface, however, can be realized in this way.

Singular Levi-flat real analytic sets occur naturally as invariant sets of inte

grable holomorphic Hamiltonian systems. One of the motivations of the current

paper is the relationship between the invariant sets of a Hamiltonian system and

the convergence of the normalization for the Birkhoff normal form of that system. The techniques developed in this paper have been applied by the second author

to the study of holomorphic Hamiltonian systems [8].

Manuscript received May 5, 1998; revised July 2, 1998.

Research of the first author supported in part by NSF grant DMS-9408994; Research of the second author

supported in part by NSF grant DMS-9704835 and a Rackham fellowship from the University of Michigan. American Journal of Mathematics 121 (1999), 23-53.

23



24 DANIEL BURNS AND XIANGHONG GONG

Let M be a real analytic hypersurface defined by r = 0. The Segre varieties

associated to M are the complex varieties in Cn defined by

Qw:={zeCn\r(z,w) =

0}.

Segre varieties, used first by B. Segre [10] and developed by S. M. Webster

[13], Diederich-Fornaess [7] and others, have been a powerful tool in dealing with smooth real analytic hypersurfaces. In this paper, we need to understand the

Segre variety at a singular point of a Levi-flat hypersurface. The Segre varieties at singular points are important invariants for Levi-flat

hypersurfaces, in addition to the usual invariants of such singular points such as

the tangent cones. The simplest singular Levi-flat hypersurface is the real cone

(1.1) ??l{z? + zj + -'+z2n} = 09

in which 0 is an isolated singular point. In this example the Segre variety go is

contained in the cone. However, in Cn(n > 2) another example is the complex cone

(1.2) z\Z\ -

z2z2 = 0,

for which the singular locus is C"-2, and the Segre variety go is the whole

space C". This paper will show how the Segre variety go provides the essential

information in constructing a normal form for certain Levi-flat hypersurfaces. The hypersurfaces to which most of our results apply are those with defining function r with quadratic leading terms. Fortunately, the possible tangent cones

which can occur for such hypersurfaces can be classified and understood. As in

many similar areas of study in complex analysis, we can analyze completely some

of these possible cases, but for hypersurfaces with some of the more degenerate

tangent cones our results are incomplete. The main results are as follows. We consider a real analytic hypersurface in

Cn given by

(1.3) &{z\ + zl + + z2n} +H(z,z) = 0

with

(1.4) H(z,z) = 0(\z\3l H(z,z) = H(z,z).

We have the following.

Theorem 1.1. Let M be a real analytic hypersurface defined by (1.3) and (1.4).

Assume that M is Levi-flat away from the origin ofCn. Then there is a biholomorphic

mapping defined near 0 which transforms M into the real cone (1.1).



SINGULAR LEVI-FLAT REAL ANALYTIC HYPERSURFACES 25

We next characterize Levi-flat hypersurfaces which are equivalent to the com

plex cone (1.2) as follows.

Theorem 1.2. Let M.r = 0be a Levi-flat real analytic hypersurface in Cn with

r = q(z,z) + 0(|z|3). Assume that the quadratic form q is positive definite on a

complex line and its Levi-form has rank at least 2. Then M is biholomorphically

equivalent to the complex cone (1.2).

In very general terms, a Levi-flat hypersurface, even a singular one, is a fam

ily of complex hypersurfaces dependent on one real parameter t. The difficulties

encountered in the proofs of these theorems result from needing to control care

fully the analytic properties of this parameter, in particular, extending it when

possible to a holomorphic function of complex values of t.

The contents of the paper in the various sections are as follows. In Section 2

we shall give a complete classification of Levi-flat quadrics in Cn, that is, Levi-flat

hypersurfaces defined by the vanishing of a homogeneous quadratic polynomial.

(We return to the case of a homogeneous quadratic polynomial whose zero set is

smaller dimensional in Section 3.) We shall also show that the tangent cone of

a singular Levi-flat hypersurface remains Levi-flat, if the cone is a hypersurface.

Combining Theorem 1.1 and Theorem 1.2, we shall see that a Levi-flat hyper surface in Cn, defined by a real analytic function with a nondegenerate critical

point, is biholomorphically equivalent to the real cone (1.1), or possibly to the

complex cone (1.2) when n = 2. Section 3 is devoted to the study of the Segre varieties at singular points of Levi-flat hypersurfaces. We shall prove that the

smooth locus of a real analytic Levi-flat hypersurface must be connected if the

dimension of its singular locus is not too large. In Section 3 we shall focus on

the question when a Levi-flat hypersurface is defined by the real part of a holo

morphic function. We shall prove a theorem stronger than Theorem 1.1, which

allows some degeneracy at singular points. The proof of Theorem 1.2 will be

presented in Section 4. At the end of Section 4 we summarize what we know

about irreducible Levi-flat hypersurfaces with quadratic singularities according to their possible tangent cones. In Section 5, we shall study Levi-flat hypersur faces generated by meromorphic functions, and we give a sufficient condition

for a Levi-flat real analytic hypersurface to be equivalent to an algebraic hyper surface.

Acknowledgments. Singular Levi-flat hypersurfaces have previously been stud

ied by E. Bedford [3] in relation to continuation of holomorphic functions defined

on open sets bounded by such singular hypersurfaces. His hypersurfaces are as

sumed to have singularities contained in a complex subvariety of codimension at

least two. We thank him for calling [3] to our attention.

We would also like to thank Salah Baouendi and Linda Rothschild for helpful discussions and a correction to Lemma 3.8.




2. Levi-flat quadrics. In this section we shall first give a complete classi

fication of Levi-flat quadrics. We shall also see that quadratic tangent cones of

Levi-flat hypersurfaces are Levi-flat, if the cones are of dimension 2n ? 1. The

quadratic tangent cones of smaller dimension will be discussed in section 3 by

using Segre varieties.

Recall that a germ of real analytic variety V at 0 G R* is the zero set of

a finite number of real analytic functions. A germ of irreducible real analytic

variety is always defined by an irreducible real analytic function. Throughout this paper, we shall denote by V* the set of points x G V such that near x9 V is

a real analytic submanifold of R* of dimension d = dim V. Then Vs = V \ V* is

contained in a germ of real analytic variety of dimension less than d. We shall

denote by dim Vs the dimension of the smallest germ of real analytic variety at

0 which contains Vs.

For a holomorphic function f(z), we shall define the holomorphic function

f(z) =f(z). We need the following elementary fact.

Lemma 2.1. Let r(x) be an irreducible germ of real analytic function defined at

0 G R*. Assume that V: r = 0 has dimension k ? 1. Then r is irreducible as a germ

ofholomorphic function at 0 G C*. Moreover, for any ball B = {x G R* | |x| < e}

there exists an open set U C Be, containing 0, such that for any real power series

R(x) which is convergent on Be and vanishes on a nonempty open subset ofUCiV*, r divides R on U.

Proof Let r =/f* /^m, where fj(x) are irreducible holomorphic functions

in x with^(O) = 0, and assume for the sake of contradiction that m > 1, that

is, that r is reducible as a holomorphic function at 0 in C". Since r is real, then

rearranging the order of^ gives fx =

/jf2, where // is a nonvanishing holomorphic function. One sees that /jff2 is real on R*, and it divides r. The irreducibility of

r then implies that m = 2 and dj = 1. Therefore,^!y

= 0 for y = 1,2, i.e., {f\

= 0}

and {f2 =

0} are the same complex variety because V is of dimension k?l. This

contradicts that f\ and f2 are relatively prime. Hence, r is irreducible as a germ

of holomorphic function. Choose an open set ? C {z G C* | |z| < e} containing the origin such that the smooth locus (Ve)* of Ve = ? fl {z G Ck \ r(z) =

0} is connected, and such that any holomorphic function on U which vanishes on

(Ve)* is divisible by r on ?. Let U = f/D R*. Assume that R(x) is a power series

convergent on Be and vanishes on a nonempty open subset of U fl V*. Then R

vanishes on Ve. Therefore, r divides R on ?.

Lemma 2.2. Let M be a real analytic hypersurface in Cn defined by r = 0.

Suppose that r is irreducible. Then there is an open set U C C" containing 0 such

that M fl U is Levi-flat if and only if one of the connected components ofM* (1 U

is Levi-flat.

Proof Fix e > 0 such that r(z9z) converges on Be C Cn. By Lemma 2.1 there

exists an open subset U of Be such that all real power series convergent on Be

and vanishing on a nonempty open subset of M* fl U are divisible by r.




To study the Levi-form of r, consider the following Levi-matrix of r

(2.1) L = Lr =

From [13], one knows that for p G M* with dr(p) j? 0, M is Levi-flat near p if

and only if the rank of L is 2. Fix a connected component K of M* D ?/. We want

to show that dr is not identically zero on K. Otherwise, all rZj,rZj

vanish on K.

Lemma 2.1 implies that rZj

= a/r. We now have

J2 zJrzj + Wj =

J2 (aJzJ + "a)'

The vanishing order of the left-hand side is the same as that of r, while the

right-hand side has a higher order of vanishing. This contradiction shows that rz does not vanish identically on K. We now take K to be a Levi-flat component of

M* n U. Let A be the determinant of any 3x3 submatrix of L. Then A|# =

0,

and Lemma 2.1 says that r divides A, i.e., A\mhu = 0- This proves that M* HU

is Levi-flat. D

We now turn to the classification of Levi-flat real quadrics in Cn.

Table 2.1 is a list of Levi-flat quadrics in Cn accompanied by the defining functions for both the quadrics and the corresponding Segre varieties ?o

Table 2.1. Levi-flat quadrics.

type normal form singular set ?o

Qo,2k

Qui

Q^2,A 6(0,1) 02,2

02,4

?{*? + +*?} z\ + 2z\Z\ +z\

z\ + 2\ziz\ + z\ (Zl +Z1XZ2+Z2)

Z\Zl ?

Z2Z2

Cn-k

empty Cn-\

R2 x C"-2

Cn-2

z2 + --- + z2k

z\

z\

Z\Z2

0

Theorem 2.3. The above table is a complete list of holomorphic equivalence classes of Levi-flat quadratic real hypersurfaces in Cn.

Proof In the table, Qy stands for a quadric defined by a quadratic form

q such that the Levi-form of q has rank /, and as a real quadratic form, q has

rank/ This shows that all quadrics in Table 2.1, except the family Qx2, are

not biholomorphically equivalent. The singular set of Qx2, defined by zi = 0,

is invariant under a complex linear transformation mapping from Qx2 to Qx'2, i.e., the transformation must be of the form zi ?> czi. Therefore, we may restrict

ourselves to the case n = 1 to distinguish the family Qx2. However, each Qx2




consists of two lines in C intersecting at the origin with the angle arceos A. Hence,

all Qx2 are not equivalent.

Next, we want to prove that a Levi-flat quadric is always equivalent to one

of the quadrics in Table 2.1. Let q(z,z) be a real quadratic form, which defines

a quadric Q in Cn of real dimension 2n ? 1. We first consider the case that q is

reducible with multiplicity two. In this case, q(z, z) is obviously equivalent, up to

a possible change of sign, to (z\ +zi)2 through a C-linear transformation. Assume

now that q = q\q29 where q\9 q2 are linearly independent R-linear functions. There

are two cases to be considered: (i) q$z,0)9q2(z,0) are C-linearly independent;

(ii) q\(z,0) =

/j,q2(z,0) with ?i G C \ R. For case (i), one can introduce C-linear

coordinates with zj

= qj(z,0)(j

= 1,2). Thus, Q becomes 02,2- For case (ii), one

may assume that |/i| = 1. One then obtains 5ft/x > 0 by changing the sign of q9 and

5/x > 0 by interchanging q\ and q2. Thus, Q is given by (zi + Zi)(/iZi +~?z$ = 0.

When \x is pure imaginary, Q becomes Qo,2; otherwise, Q is equivalent to Qx2 with A = $fy?.

We now assume that q is irreducible. Write

q(z, z) = 23? ]T ??/Z/Z/ + ^2 BfjZitJ

with Aij =

A?9B? =

B?. Consider

(2.2) Lq = Az + Bi BZZ J

Since q is irreducible, then qz ^ 0 on the smooth locus of Q. From [13], one

then knows that at the smooth points of Q, the rank of L is 2. Thus, the rank of

B?? is at most two.

We first consider the case B(z,z) = 0. Here, it is easy to see that Q is

equivalent to Qo,2k with k the rank of the complex quadratic form A(z). Next,

we assume that rank B^ = 1. Replacing r by

? r if necessary, one may further

assume that B(z,z) = zizi- Among the 3 x 3 submatrices of Lq9 we consider

0 A^+zi A-z\ (2.3) |AZl+zi 1 0

Azj 0 0

;>!.

Notice that the determinant ? |A^.|2

of (2.3) contains no harmonic terms. Since

L is of rank at most 2, then |AZ.|2 vanishes identically at the smooth points of

Q. Hence, |AZ.|2 =

Cjq(z,z), where Cj is a real constant. If Az. ^ 0, then this

would imply that in suitable coordinates, q(z,z) =

|zi|2. This contradicts that

dim Q = 2n ? 1. Therefore, A(z) is independent of z/ for j > 1, which leads to

another contradiction since q is irreducible. We now turn to the case rank B^ = 2.




One may assume that B(z, z) = z\Z\ +ez2Z2 with e = ?1. If e = +1 then the complex Hessian

(rZiZj) would have rank at least 1 on the leaf of the Levi foliation of Q

through any smooth point sufficiently close to 0. But this contradicts the fact

that r = 0 along such a leaf. Thus e = ? 1. The above argument shows that A is

independent of z/ for j > 2. Now the determinant of the first 3x3 submatrix of

(2.1) gives us

(2.4) - 1 \AZl +zi|2 + \AZ2

- z2|2

= Cq(z,z).

Comparing the holomorphic terms in A and in (2.4) gives us

(2.5) -

z\Azx -

z2AZ2 = CA(z).

The left-hand side of (2.5) is precisely ?2A(z), which implies that either A = 0, or C = ?2. For the former case, Q becomes 02,4- In the case C = ? 2 with A^O,

(2.4) reads

|Azl(z)|2-|A,2(z)|2 =

|zi|2-|z2|2.

Put A(z) = az2 + 2feiZ2 + cz2. One may assume that a,c are nonnegative. The

above identity then gives us ab = be and 4(a2 ?

|Z?|2) =

4(c2 ?

\b\2) = 1. Hence,

a = c > 0 and b is also real, and

2A(z) = ZU(t)Z< = ZU(t/2)(ZU(t/2))\

where Z = (zuzi) and

\b c) \sm\it cosh tj w

Notice that U(t) preserves |zi|2 ?

|z2|2- Therefore, q(z) is equivalent to !R{z2 +

z2} + |zi|2 ?

|z2|2. This shows that q is reducible, which is a contradiction. The

proof of the theorem is complete.

Proposition 2.4. Let M be an irreducible real analytic hypersurface defined

byr = 0, where r =

q(z, z) + 0(|z|^+1 ) and q is a homogeneous polynomial of degree k. Assume that {q

= 0} is of dimension 2n ?

1, and that q is reduced ifk > 2. If M

is Levi-flat, so is the tangent cone q = 0.

Proof. Let re(z,z) = r(ez, ez)/ek. Fix indices / = (ii, i2, ?3),^ =

(j\,J2J3) with

1 < iiji < n + 1. Denote by Uu the 3 x 3 submatrix consisting of entries in rows

*i> *2> h and in columns j 1J2J3 of (2.2). Since M is Levi-flat and r is irreducible, then

det Lrjj(z, z) = pu(z, z)r(z, z),




where p?j is a convergent power series. Hence,

ed det Lru(z, z) = Pu(ez, ez)r(ez, ez),

where d equals 3k - 4 for i\ +j\ = 2, 3k - 5 for i'i = 1 < yi or j\

= 1 < i\9 and

3& ?6 for i'u'i > 2, respectively. Obviously, e^-* divides pu(ez, ez), if d ? k > 0.

Letting e ?> 0, we see that for d > k9 detLqu(z,z) =

pu(z,z)q(z,z), where p/y is

the sum of homogeneous terms of /?/y of degree d ? k. Also, we have Lqn = 0 for

J < &. In both cases, we obtain that Lqu vanishes on q = 0. This shows that the

Levi-matrix of q has rank at most 2 on q = 0, i.e., {# =

0} \ {dq =

0} is Levi-flat.

On the other hand, by the assumptions we have a decomposition q = q\.. .q^9 where q? are irreducible. Let V) be the vanishing set of #7. From Lemma 2.1 it

follows that dim(V; fl Vj) < 2n - 1 for i ^j. From the proof of Lemma 2.2, one

sees that dim ({dqj = 0}D V,) < 2n-1. Hence, dim ({dq = 0}n{q = 0}) < 2n-1.

Therefore, q = 0 is Levi-flat.

Let us record here the following result about the case of r with generic

quadratic singularity, anticipating the proofs of Theorems 1.1 and 1.2 in Sections 3

and 4.

Corollary 2.5. Let M be a smooth Levi-flat real analytic hypersurface in Cn

defined by r = 0. Suppose that 0 is a nondegenerate critical point ofr. Then M is

biholomorphically equivalent to the real cone (1.1), or to the complex cone (1.2) with n-2.

Proof. By Proposition 2.4, the quadratic term q of r defines a Levi-flat quadric with isolated singularity at 0. Theorem 2.3 says that q is equivalent to 5?(z2 + z2 +

- + z2), or zizi ?

Z2Z2 with n = 2. Now Corollary 2.5 follows from Theorem 1.1

and Theorem 1.2.

3. Nondegenerate Segre variety at a singular point. In this section, we

shall first study the Segre variety go, where 0 is a singular point at which the germ

of a Levi-flat real analytic hypersurface M is defined. According to a theorem of

J. E. Fornaess (see [9], Theorem 6.23), such a Levi-flat hypersurface M always contains a complex variety of dimension n ? 1 which passes through the origin.

(The theorem is stated for M being the boundary of a domain with smooth and

real analytic boundary. However, the proof is valid without any change if M is

a germ of real analytic hypersurface with singularities.) We shall use this fact

to classify the quadratic tangent cones of Levi-flat hypersurfaces which are of

dimension less than 2n ? 1. The main result of this section is to determine when

a Levi-flat hypersurface is defined by the real part of a holomorphic function.

We shall denote by Ae the disc in C of radius e, and by A^ the product of k

discs A . We first need the following lemma.




Lemma 3.1. Let M be a smooth Levi-flat real analytic hypersurface in Cn de

fined by r = 0. Assume that r(z90) ^ 0 and M is of dimension 2n ? 1 at 0. One

branch Qf0 ofQo is contained in M. Furthermore, Q'0 is smooth, and it is the unique

germ of complex variety of pure dimension n ? 1 at 0 which is contained in M.

Proof. Choose holomorphic coordinates such that near 0, M is given by

5Rzi = 0. Then M contains the hyperplane zi = 0. Since r = 0 on M, then

r(z,z) = a(z,zWz\. Obviously, zi = 0 is a branch of go. For the uniqueness,

assume that a germ of complex variety V of pure dimension n ? 1 at 0 is inside

M. This means that zi is pure imaginary on V. Since 0 e V, then zi = 0 on V.

Therefore, V is the hyperplane z\ = 0.

Given a real analytic hypersurface M defined by r = 0, we shall denote by

Qw the Segre variety defined by r(z,w) = 0, and by g* the smooth locus of Qw.

We shall denote by Q'w the union of all branches of Qw which are contained in

M. Q'w could be an empty set when w e MS9 otherwise, it is a complex variety of

pure dimension n ? 1. Lemma 3.1 implies that if Qfw is nonempty, then Q'w DM*

is smooth.

The following result is essentially due to Fornaess [9]. We present a proof

here, since some arguments will be needed later on.

Lemma 3.2. Let M be a Levi-flat hypersurface defined by r(z9z) = 0 with

r(z, 0)^0.

(a) go is contained in M, ifQPj Pi

A"0 is contained in M for a sequence pj e M*

with pj ?> 0 and for a fixed q > 0.

(b) go is nonempty. The closure of each topological component ofM* containing 0 contains a component ofQ'0 provided dimM5 < 2n ? 4.

(c) If dim Ms < 2n?4(n > 2), Qf0 is irreducible and M* has a unique connected

component of which the closure contains 0. Furthermore, the dimension of the

singular locus ofQf0 is less than n ? 2, and M contains no germ of complex variety

of pure dimension n ? 1 at 0 other than Q'0.

Proof Since r(z, 0) ^ 0, one can introduce new coordinates such that r(zi, 0,0) =

zf. Choose small e, 6 such that

(3.1) ttw: Qw = QwHAex Ans~l -

A^"1, for w e An6

is a proper d-to-l branched covering, where nw is the restriction of the projection

Tr*(zi,z') -> z! to Qw. Let G\,...,Od be the elementary symmetric functions on

the symmetric power space C^, := Cd/&d, where 6? is the permutation group

operating on Cd. Then a = (a\,..., ad): Cdsym

? Cd is a homeomorphism (see

[14]). Counted with multiplicity, ir~l(zf) is a set of d points in Ae x A^"1, of

which the first coordinate of these d points form a point (f(z',w) e Cdym.

Notice

that the components of aocp(?9 w) are precisely the coefficients of the Weierstrass

polynomial of r(z,w) in zi, which depend on z', w holomorphically. In particular,




(p(z\pj) ->

<?>(z',0) in Cdsym

as pj -> 0. This implies that tt"1^')

- 7VQX(zf)

as

Pj -> 0. More precisely, ttq1(?)

= r\i>iUj>i7rPjx(z').

For the proof of (a), assume that QPj

f) A o is contained in M. Then in the

original coordinates, Qp is equal to

QPj n Aeo. In particular, *"}(?!) is contained

in M. Letting /?, - 0 shows that ̂ x(z') C M for z' G Anfx. Therefore, g0 fl

A x A?_1 is contained in M.

For the proof of (b), we take any sequence pj G M* with /?; ?? 0. Fix z7 and

take a point /?j E

7r~x(zf)nQfp.9 where

g^. is now the branch of

QPj fl A x

Anfx contained in M. One may assume that

pfj is convergent as j

? oo, of which the

limit is in 7r?"1(z/). Hence, KqX(z')P\M is nonempty for all z' G A"-1. Therefore, at

least one branch of go is contained in M. We now assume further that dim My <

2n ? 4 and K is a connected component of M* with 0 G A'. By choosing all pj in K9 the above argument shows that there exists a branch Q1 of go such that

g' n K is nonempty. Since the codimension of M5 fl Q' in g' is at least 2, g7 \ Af5 is connected; in particular, it is contained in K. Therefore, K contains Q1.

(c) follows from (b) and Lemma 3.1 immediately.

An immediate corollary of Lemma 3.2 (b) is the following.

Corollary 3.3. Let M\9M2 be two germs of Levi-flat real analytic hypersur

face at 0 G Cn withno common component of their regular points. Then the singular set of M\ U M2 is of dimension at least 2n ? 4.

The following gives all possible (quadratic) tangent cones and dimM5 when

M is the union of two smooth Levi-flat hypersurfaces. The notation is as in

Table 2.1 in Section 2.

Proposition 3.4. Let M be a reducible Levi-flat real analytic hypersurface

defined by r = 0, where r starts with nonvanishing quadratic terms. Then Ms is a

real analytic variety in Cn of codimension 2 and the quadratic tangent cone of M

is equivalent to one of Qq?, Qij, Qx2, Q2,2

Proof. Since M is reducible and r starts with nonvanishing quadratic form,

Lemma 2.1 implies that r is reducible. Hence, M is the union of two smooth

real hypersurfaces. Obviously the quadratic tangent cone of M9 denoted by C, is

equivalent to one of Qo,2, Q\,2, 02,2? and Qij, since r is reducible. All these

quadrics, except Qij, are the union of two transversal hyperplanes, and hence

M is also the union of two transversal smooth hypersurfaces, from which the

proposition follows. For C = Qij, one may assume

M:3fci -3?(zi+/*(z)) = 0,

where h(z) = 0(|z|2) ^ 0 is a holomorphic function. Ms is then the intersection

of x\ =0 with $th(iy\9z!) = 0. If h(09zl) = 0, Ms contains z\ = 0. Otherwise,




SR/i(0,z') = 0 defines a germ of Levi-flat hypersurface at 0 e Cn~l (see Proposi

tion 5.1). In fact, fixing any small y\9 Wi(iy\,z!) = 0 defines a germ of Levi-flat

hypersurface at 0 C1-1. Therefore, Wt(iy\,z!) = 0 is a germ of real analytic

variety of codimension one in R x Cn~l. The proof of the proposition is complete.

We now consider quadratic tangent cones of Levi-flat hypersurfaces which

are of small dimension, a case left open in Section 2 above.

Proposition 3.5. Let M be a Levi-flat real analytic hypersurface defined by

r(z, z) = q(z, z) + 0(|z|3)

= 0, where q is a real quadratic form. Assume that q = 0is

a real analytic set C of dimension less than 2n ? 1. Then after a possible complex linear transformation, and up to a possible change of sign, q is equivalent to

Zizi + Az2 + Az2, where 0 < A < 1/2.

Proof. By a theorem of Fornaess (see [9], p. 114), there is a germ of complex

hypersurface V satisfying 0 e V C M. Let C(V) be the tangent cone of V at 0 (see

[14]). Given p e C(V)\{0}9 there exists a sequence pj ?> 0 such that/?, G ^\{0}

andpj/\pj\ -

p/\p\ asj -> oo. Hence, we have 0 = KPj,Pj)/\Pj\2

-? q(P,P)/\p\2 asj ?? oo, i.e., C(V) C C. Since C(V) is a complex hypersurface, we obtain that

dim C = 2n ? 2, and q is a semi-definite real quadratic form of rank 2. Let us

assume that q > 0. Then the eigenvalues of the Levi-form of q are nonnegative. It is clear that the Levi-form of q is not identically zero. On the other hand, the

Levi-form cannot have more than one positive eigenvalue. Otherwise, the Levi

form is positive definite on a complex subvariety of V of positive dimension, which contradicts the fact that q vanishes on C(V). Therefore, we may assume

that q(z,z) = ZiZi + $iA(z), where A(z) is a complex quadratic form. For a fixed

zi, q is a nonnegative pluriharmonic function on Cn~l. Hence, q is independent of Z2> >Zn- Now, it is easy to see that C is given by z\ -

0, and q is equivalent to zizi + Az2 + Az2 with 0 < A < 1/2.

We consider a more general Levi-flat real analytic hypersurface defined by

(3.2) r(z, z) = q(z, z) + 0(|z|3)

= 0, rank^z, 0) > 0.

As a use of Segre varieties, we shall prove the following result on the connectivity of the smooth locus of M. The result will however not be used in this paper.

Proposition 3.6. Let M be a real analytic hypersurface in Cn defined by (3.2). Assume that dimMs < 2n ? 4. Then M* is connected.

Proof. More precisely, we would like to prove that any neighborhood U of the

origin contains an open subset Uf with 0 e U' such that i/'HM* is connected. To

this end note that Proposition 3.4 implies that r is irreducible. From Lemma 2.1

it follows that M is irreducible. Now a theorem of Bruhat and Cartan [4] says




that there exists [/' such that U' fl M* = i/i U U [/*, where Uj are connected

open sets with 0 G Uj. We need to show that k = 1.

One may assume that q(z\, 090) = z2. With the notations in the proof of

Lemma 3.2, the projection irw: Qw ?? A"-1, given by (3.1), is 2-to-l and proper. We also choose e96 so small that Ms is closed in Ae x

An6~x. Assume for contradiction that M* = M* n U' is not connected. Then Lemma

3.2 (b) says that M* has exactly two components M'9M"9 and go has two branches

go, go witri ?o c Ml and Q'? C Af". Take a sequence of points py ?? 0 in

Af' \ (go U Afy). In view of Lemma 3.2 (a), one may further assume that each Qp. is reducible. By Lemma 3.1, Qp. contains a unique branch

Qp. which is inside

M' and contains pj. Let Q'0, Qf?9 and Qp

be the graphs of holomorphic functions

f?>fo> and fj?7?\ respectively. Passing to a subsequence if necessary, we know

that^' converges ?o/q. Since 0 G Qf0 H Q'?9 we can find a complex line, say the

Z2-axis, which goes through the origin of A?_1

but is not contained in the complex

variety 7r(g? fl Q'?). By Rouch?'s theorem,^ ?f?f has zeros on the Z2-axis for

large j, and these zeros can be arbitrarily close to the origin as pj ?? 0. Hence,

Qp ^ ?o is a nonempty complex variety of dimension n ?

2, and it contains points

arbitrarily close to the origin as pj ? 0. In particular, dim My is 2n ? 4.

Next, we want to show that Qp.

D Q'? actually contains the origin for large

j. Since dim My < 2n ? 4, we may assume that Ms C V\ U ... U V*, where each

Vi is a germ of irreducible real analytic variety of dimension at most 2n ? 4 at

the origin. Let V? be the set of points in V? where V? is a real analytic manifold

of dimension 2n ? 4. Then there is a neighborhood U? of the origin such that

V[ fl Ui has only a finite number of connected components A,y [4]. Furthermore,

A,y is either empty when dim V,- < 2n ? 4, or 0 G A,y. Choose a large y such that

g =

Q'p n go intersects all of U\9..., f/?. It is clear that intersection of g with

one of A//s has interior points in g. Consequently, g contains one of A,y; hence,

0 G Aij C Qp . Notice the reality property of Segre varieties; namely, z G gw if

and only if w G Qz. Hence, we have pj G go, which is a contradiction. The proof of the proposition is complete.

We return to the main line of argument to show Theorem 1.1.

Lemma 3.7. Let M be defined as in (3.2). Suppose that one branch of Qo is

not contained in M. Then there exist e, 6 > 0 such that for w G AJ, Qfw is given

by z\ =f(z\ vv), where f is holomorphic in z! G Anfx for each fixed wGAJ?l M*.

Furthermore, f(z!,w) tends to f(z!) uniformly on An6~x

as w ?> 0 in M*, where

z\ =f(z') defines Q'0.

Proof Choose 6,8 as in the proof of Proposition 3.6 so that the projection

7rH given by (3.1) is a 2-to-l branched covering for w G A?. Since go is not

contained in M, then Lemma 3.2 (a) says that for w G M* D A^ close to the

origin, Qw consists of two branches, of which both are graphs of holomorphic




functions over A? l. Let Q'w be given by zi =/(z7,w), and let the other branch

Qw of Qw be given by z\ = g(z', w).

To see the uniform convergence of/(-,vv), we first notice that all the partial derivatives of f(-,w) are uniformly bounded on each compact subset of A?_1.

Hence, it suffices to show that/(-,vv) is pointwise convergent on a dense subset

of A^"1. Since Qf? is not contained in M, then E =

7r(Qbr\Qf?) is nowhere dense in

Anfl. Fix z! e An6~l \E. Then (g(z!, 0), z!) has a positive distance d to M. From the

Rouch? theorem (or the elementary symmetric functions argument in the proof of

Lemma 3.2), it follows that \f(z\w) ?

g(z',0)| > d/2 for w sufficiently close to

the origin; hence, \f(z!, w) ?f(z\ 0)| ?? 0 and | g(z!, w)

? g(z!, 0)|

? 0, as w ?? 0.

Lemma 3.8. Let p(z) be an irreducible germ of holomorphic function at 0.

Assume that r(z, z) is a real analytic function vanishing onV:p = 0. Then r =

apV?p

for some convergent power series a(z, z).

Proof. One may assume that p(z) is a Weierstrass polynomial z^+J^fs^1 Pj(z!)z!\. Since p is reduced, there is a complex variety B in Cn~l such that p(z\,z!)

= 0

has d distinct roots for each fixed z! ? B. Consider the complex variety

VccC"x Cn:p(z) = 0, p(w)

= 0.

It is clear that Ve is irreducible and of codimension 2 in C2n. We identify V with

the set Ve D {w =

z}. Then V* is a totally real submanifold in (Ve)* of maximal

dimension. Notice that r(z,w) vanishes on V*, hence also on Ve.

Applying the Weierstrass division theorem, one gets

d-\

r(z, w) = ad(z, w)p(z) + Yl aj(z\ w)z!x,

y=o

and

d-\

aj(zf9 w) = bM(z', w)p(w) +

^2 hk(z\ w')w\ k=o

for0<j<d-l. On Ve, one has

d-\

(3.3) EM^V)^ = 0.

?*=0

Fix (z',wf) i B x C""1 U C1"1 x B, where B = {z7 | z! e B}. Then there are

d distinct zeros zi,i,. . ,Z\,d of p(z\,zf), and d distinct zeros wij,... ,w\? of

p(w\,w'). In (3.3), replace zi by zi,/ with a fixed i, and w\ by w\?,.. .,w\?,

respectively. This gives us d linear equations in terms of ^jbj^z^w^zd

for




0 < k < d ? 1. The coefficient matrix of the d linear equations is a nonsingular Vandermonde matrix ( vvf

~ ) . Thus, we get

d-\

Y, bj,k(z\ w')z!u = 0, 0 < * < d - 1.

y=o

Now, fix k and vary /. The above are d linear equations in terms of b^k(z!, w'), ...,

bd-\,k(z!, vv'); one readily sees that ?/?(z', w') = 0 for 0 < j,k < d ? 1. Therefore,

we obtain a decomposition r = aop + bop for some convergent power series

a0(z, z), ?ofc z). Replacing a0(z, z) by the average (a0(z, z) + b0(z, z))/2 completes the proof of the lemma. D

Theorem 3.9. Let M be a Levi-flat real analytic hypersurface defined by (3.2) with dim My < 2n ? 3. Assume further that go is not a double hypersurface. Then

M is given by 3?A(z) = 0 with h a holomorphic function.

Proof Before we proceed to the details, we shall explain how the Segre varieties will be used in the proof. Roughly speaking, h is constructed as follows:

We pick one branch of Q'0. Then take a curve 7(0 in M* which is transverse to that branch of Qf0 at 7(0). One would hope that for any point z M close to the origin, one branch of Qz intersects 7 at a unique point 7(0; this is achieved

essentially due to the fact that go contains no component of multiplicity 2. Then, ?ti will be the value of h at z. Now the question becomes whether h extends to a

holomorphic function defined in a neighborhood of 0. To deal with this problem, we shall substitute z for 7(f) in the complexified function r(z9 z). Solving for t from r(z, 7(f))

= 0 would yield t = ih(z). We shall obtain the convergence of h(z) by choosing the curve 7 carefully to cope with the singularities of M.

The proof will be accomplished in two steps. The first step is to show that M contains go. The second step is to show the existence of the holomorphic

function h stated in the theorem.

Step 1. Assume for contradiction that go is not contained in M. In this case,

go consists of two smooth hypersurfaces, and only one branch Q'0 is in M. We have r(z,0) =

q(z,0) + 0(|z|3). Since q(z,0) ^ 0, then, by a parametric Morse lemma [2], one can find holomorphic coordinates such that r(z,0) = z2 +

?(z'). Since go are two distinct smooth hypersurfaces, then ?(z!) = b2(z!) ^ 0.

Assume that {zi + ib(z!) = 0} is inside M. Replacing zi by zi + ib(z!) yields

r(z,0) = zi(zi +p(zf)) with p = ?lib and Q'0:zi

= 0. Since M contains zi = 0,

by Lemma 3.8 M is now given by

M:3?{zi(zi+a(z,z))} = 0.




It is convenient to decompose

a(z, z) = p(z!) + b({9 z!) + c(z, z)

with

p(z') ? 0, b(z\ 0) =

c(z, 0) =

0, c(0, z', 0, z') = 0.

Let A: be the vanishing order of p(zf), and / the vanishing order of p(z') +

b(z!,z!). By the assumptions, we have 1 < I < k < oo. Denote by pkM the

homogeneous terms of p, b with degree k and /, respectively. Fix zf0 such that

(0, Z?) G g? \ Ms, and such that

(3.4) Pk(z!0) ? 0, /7/?z?)) + ?>/(4 zi)) ̂ 0.

A linear dilation (zi,z') ?> (e/+1zi,ez') transforms M into

(3.5) Me: re(z,z) = 5R{zi[p(z\ e) + ?(*\ z\ c)

+ e(zi + c(z,z,e))]} = 0

with c(z, 0, e) =

c(0, z\ 0, z7, e) = 0 and

(3.6) p({, 0) + b(z!, z', 0) = />/(z') + ?/(z', z').

Throughout the rest of the proof, we shall replace r by the equivalent r , and M

by the equivalent M , unless stated otherwise.

We shall find a real curve in M *

given by

(3.7) 7 = 7 : zi = i>(c)(i + ia(t, e)), z! =

Zq,

where a(0, e) = 0, a(t, e) = a(r, e), and p(e) is determined by

/?(Zo, e) + ?(zo,z(), e) = p(e)/J(e), p(e) = |/?(zo, e) + b(z!0,Zo> e)\

Substituting (3.7) into (3.5), one sees that 7 is contained in M, if and only if

(3.8) -p(e)a(t) + eU{zi(zi + c(zi,z?,zi,z?,e))} = 0,

in which z\ is replaced by ip(e)(t+ia(t)). From (3.4), it follows that p(e) is a real

analytic function bounded from below by a positive constant. Using the implicit function theorem, we get a unique real analytic solution a = a(t, e) with

(3.9) a(t,e) = 0(t2), a(t90)

= 0.




The Segre varieties g7(,) are given by r(z, 7(0) = 0, from which we want

to solve for t as follows. Regard r(z, 7(0) as a power series in r, z, and e. From

(3.5), (3.7) and (3.9), we get the following expansion:

r(z, 7(0) = -iJi(e)p(zo, e)t + 0(\z\ | + \tzf\ +1\

in which, in view of (3.4), the coefficient of t does not vanish. From the implicit function theorem, it follows that r(z, 7(0)

= 0 has a unique holomorphic solution

t = ih(z, e) such that h = 0 when zi = 0 and \z'\ is small. Therefore, the uniqueness

of the solution implies that the value of h(z, e) is uniquely determined under the

condition \h(z, e)\ < eo for z G A?o,

where eo, <5o are sufficiently small positive numbers. Here we remark that

A?o might be too small to contain the curve 7.

Next, we want to show that ih(z, e) is real-valued on M. Notice that 7(0) G M*, and 7 is transverse to Q'0. Hence, the union of g7(^ for ?

eo < t < eo contains a

neighborhood D of 7(0) in M*. In view of Lemma 3.7, we can choose a possibly smaller ?0 such that (f(zb,z),zb) G D for z G

A?o, where/ is given in Lemma 3.7.

In other words, for each z G M* fl A?o,

there is a real t with \t\ < eo such that Q'z intersects 7 at 7(f). Notice the reality property of Segre varieties; namely, z G Qw if and only if w G Qz. Also, passing through each point in M* there is only one complex hypersurface in M. Hence, we obtain z G

Q'^y Since t =

ih(z,e) is the unique solution to r(z, 7(0)

= 0, then ih(z,e) = t is real for z G M* fl

A?Q. Since dimMj < 2n ?

3, by Proposition 3.4 r is irreducible. Now Lemma 2.1 says that r divides 5ft/i(z, e). Thus, h(z, e) starts with terms of order at least 2 for each

fixed e. Expand r(z9 7(f)) as a power series in t and z. One first sees that in that

expansion, the linear terms in z must be zero since the coefficient of t is nonzero

and h starts with terms of order at least 2. The linear terms in t and the quadratic terms in z are given by

-i?(e)p(z'0, ?)t + (c + 0(e2))Zi + Zi X^ ^y(?> 4 Ozy.

Thus, the quadratic form of A(z, e) contains z2 when e 7^ 0; in particular, the order

of h(z, e) is 2. Fixing a small nonzero e, we obtain that 3ftA(z, e) =

w(z, zMz, z) with

w(0) t? 0. In particular, go, given by h = 0, is contained in M. This contradicts

our assumption. Therefore, go is contained in M.

Notice that the above argument is valid if go is reducible and the set of

points z E M* with Qz C M contains an open subset U with 0 G U. Assuming that go is reducible, we want to show such a set U always exists. Otherwise,

choose a sequence z7 ?> 0 in M* such that z7 ^ ^?<jQ? and gzy =

g^ U g^ with

g" ?? M. Without loss of generality, one may assume that Q'zj

C M approaches

to one branch Q'0 of go as j ?> 00. With the above set-up, one finds a real

curve 7 such that Q'j intersects 7(0 for some small real t as z7 ?> 0. One also

has a unique (complex-valued) solution t = *7z(z) to r(z, 7(0) = 0, where h(z) is




holomorphic on A?Q.

Now h(z) is pure-imaginary on Q^

D A?o. Hence, Uh = 0

contains infinitely many complex hypersurfaces g7, C M. Since z7 G g7, and

z7 ?> 0, the real analytic set {5ft/i =

0} n M has dimension 2n ? 1 at 0. Therefore, r divides 5?A, a contradiction. This gives us the conclusion of the theorem when

go is reducible.

Step 2. We now consider the case that go is irreducible. As seen in the

beginning of Step 1, one may change coordinates such that r(z,0) = z2 + ?2+p(z/)

with

p(z,) = 0(\z,\k\ P(Z2,0)

= 0.

In view of Lemma 3.8, we rewrite

M: r(z, z) = 3?{(z? + A + p(z!))(l + a(z, z))} = 0.

Rotating the zi and Z2 axes and dividing r by 11 +<z(0)|, one may achieve a(0) = 0.

A linear transformation

z* = ekzu zl

= ez2, z* = e2z/, j>2

transforms M into Me given by

(3.10) re(z,z) = 3?{z? + 4} +H(z,z,e) = 0, e ? 0,

where //(z, z, e) is a real analytic function in z, z, e with

(3.11) H(z,z,0) = 0.

Again, we shall denote re by r, and M by M.

We now parameterize the Segre varieties of M by a real curve

(3.12) 7:zi = Vl+it + a(t9e), z2 =

i2/k, Zj = 09 j>2

with a(t9 e) = a(t9 e) and a(0, e) = a(t9 0)

= 0. This amounts to solving for a in

the equation

(3.13) a = E(t9a)9 E(t9a) = -g

L_ (a2

+ if(7(0,7(0,c))

.

Obviously, ? is a convergent power series in i, a. E(t9 a) is also real-valued when

t,a are real. From (3.11), it follows that Ea = 0 for e,a = 0. By the implicit




function theorem, (3.13) has a real analytic solution a = a(t9e) with

(3.14) a(i,0) = 0.

In (3.10), substitute (3.12) for z. Then the Segre varieties g7(,) are given by

(3.15) T(t,z, e) = it - 2\/l -

ita(t9 e) -

a2(t9 e)

-zj-zk2-H(z,j(t,e),e) = 0.

Now, (3.11) and (3.14) yield Tt = / for e = 0. By the fixed-point theorem, (3.15) admits a unique holomorphic solution

(3.16) t = ih(z,e)

on a domain

Ds = {z\\zj\ <2,l <j Kn^zj + zti < s}

for a fixed small s. It is clear that when e is small relative to s, 7 = 7e c Ds, and

go = {re(z,0) =

0} H Ae x A^1 C Ds. Return to the Segre varieties g7(,) determined by (3.15). The solution (3.16)

means that h(z, e) is pure imaginary on each g7(,) for real t. It is clear that 7 is not contained in go. Thus, g7^ sweep out an open subset of M, on which

/i(z, e) is pure imaginary. For any small neighborhood U C Cn of 0, g7^ remains

in M and intersects U as t ? 0. As in step one, we conclude by Lemma 2.1

that 3?/?(z, e) = u(z, z)r(z, z) near 0. Obviously, 5ft/*(0, e) = 0, and the origin is a

critical point of h(z,e). Notice that h(z,0) = z2 + z%. Thus, the quadratic form of

h(z, e) is not identically zero for small e. Therefore, u is nonvanishing near 0, which shows that M is the zero set of !R/*(z, e) for small e^0. Since the original

hypersurface M is biholomorphically equivalent to Me9 this completes the proof of the theorem.

Now Theorem 1.1 follows from the following.

Theorem 3.10. Let M be a real analytic Levi-flat hypersurface defined by

(3.2). Assume that the complex quadratic form q(z,0) is of rank k > 2, and the

real quadratic form q(z, z) of rank at least 3. Then the Levi-form of q vanishes, and M is equivalent to a real analytic hypersurface defined by 3?{z2 + +

zf +

p(Zk+\, ? -, zn)}

= 0, where p is a holomorphic function starting with terms of order

at least 3.

Proof. Since the rank of q(z, 0) is greater than 1, then q(z9 0) is nondegenerate. This implies that go:?Kz,0) + 0(|z|3)

= 0 is not a double hypersurface. Ms is

contained in the set defined by rZj(z,z) = 0 for 1 <j<n, which has codimension

at least 3 in Cn. Hence, dim My < 2n ? 3. Applying Theorem 3.9, we can find a




holomorphic function h(z) = 0(|z|2) such that r(z,z) = u(z,z)$lh(z). This implies

that the Levi-form of q vanishes, and the rank of the quadratic form of h(z) is k. A

parametric Morse lemma [2] says that h is equivalent to z2+* -+zl+p(zk+\, , zn) for some holomorphic function p. This completes the proof of the theorem.

4. Rigidity of quadrics. The purpose of this section is to show the rigidity

property of quadrics 02,2, 02,4, and Qx2. At the end of the section we summarize

the relationship as we understand it up to now between irreducible Levi-flat

hypersurfaces with quadratic singularities and their possible tangent cones.

The following proposition gives a sufficient condition for go of a Levi-flat

hypersurface to be completely degenerate, i.e. go = Cn. This proposition also

shows that the invariant A in Proposition 3.5 vanishes.

Proposition 4.1. Let M be a Levi-flat real analytic hypersurface in Cn with

defining function r(z,z) =

q(z,z) + 0(|z|3). Assume that the quadratic form q is

positive definite on a l-dimensional complex linear subspace ofCn. Then r(z9 0) =

0, and Q'z is a smooth complex hypersurface passing through the origin for z G M*.

Moreover, one of the following occurs:

(a) The rank of the Levi-form ofratO is 1, and Ms contains a complex variety of codimension 2.

(b) The rank of the Levi-form ofratO is 2, and Ms is a complex submanifold of codimension 2.

In the latter case, M can be transformed into a hypersurface defined by

(4.1) &{zxz2(\ + a(z,z)) + ziz2b(z,z")} = 0,

where z" = (Z3,..., zn)> and a9 b are power series satisfying

(4.2) a(z, 0) = a(09 z) = b(z, 0) = 0.

Proof. Without loss of generality, we assume that q is positive definite on the

Zi-axis. Then a linear change of coordinates gives us

<7(zi,0,zi,0) = zizi + A(z? + z?) + 0(|zi|3), 0 < A < 1/2.

Obviously,

(4.3) r(z9z)>cx\zi\2

for |zi| > |z'|/c2, where ci,C2 are small positive numbers. Hence, for small <5,

7r:M ?> A^-1

is a proper mapping, where ir is the projection (zi,zO ?> z'. Thus,

(4.4) ttw: Q'w =

Qfwn(Cx Anfx) -

Anfx




is a branched covering. In particular, g^n{z7 =

0} is nonempty; hence, it follows

from (4.3) that 0 G Qw. This shows that Q'0 contains infinitely many complex

hypersurfaces, i.e., r(z,0) = 0.

We now know that

(4.5) r(z, z) = ft{zi?i (z)} + ̂ J2 W^ + 5fc *>,

where a\(z\, 0) = zi + 0(\z\ |2), and ?(z, z) contains no term of the form zazP with

\a\ < 1 or \?\ < 1. In particular, we have

(4.6) rZl(09w) =

al(w) + wl^0.

Since all branches of Qw go through the origin, then Q'w can have only one branch,

and it is the graph of a holomorphic function when w e M\ {w\ + a\(w) = 0}.

Another consequence of (4.6) is that

(4.7) M5c{zi+ai(z) = 0}.

Note that the latter is transverse to the zi-axis on which q is positive definite. In

particular, the singular set of M is exactly a complex variety of Cn of codimension

2, if there exist two complex lines so that q is definite on each of them.

For the proof of (b), we now assume that the Levi-form of r at 0 is of rank

k > 1. In view of (a), we may further assume that q(z, z) = zizi ? ... ? ZkZk- This

means that oy(z) =

?z/ + 0(|z|2) for 2 < j < k9 where a? is given by (4.5). We

have

Vzr(z,vv)|z=0 =

(wl9..., ?w*,0,... ,0) + 0(|w|2).

Obviously, one can find two points on M *

such that the corresponding two Segre varieties are transverse to each other at the origin. By mapping these two Segre varieties onto zi = 0 and Z2 = 0 respectively, we see that r vanishes on zi = 0,

and on Z2 = 0. Hence,

(4.8) r(z9z) = $l{ziZ2P(z,z) + ziz2b(z,z")}, p(0) = 1.

This shows that k = 2 and Ms is a complex variety of dimension n ? 2.

To eliminate the terms of p(z, z) ? 1 which are purely holomorphic in z, or

in z, we shall seek a holomorphic transformation

?>: z\ =

zi/i(z), z2 =

zi?2(z), Zj =

z/, j > 2.




We shall restrict ourselves to/\(0) =f2(0) = 1 and then solve the equation

fip(zifuz2f2,z",0) = 1, f2p(09zifuz2f2,z")

= 1.

By the implicit function theorem, there is a unique solution (f\,f2) with/i(0) =

/2(0) = 1. Now, (?>-1(M) is given by (4.1) and (4.2). D

To continue the proof of Theorem 1.2, we now assume that M is a Levi-flat

hypersurface defined by (4.1) and (4.2). We shall prove that M is actually the

complex cone 3?{ziZ2} = 0 in Cn.

Without loss of generality, we may assume that a9b in (4.1) and (4.2) are

small functions defined on \z\ < 2. Let 7 be the intersection of M with the

complex line zi = l,Zy

= 0 for y > 1. We shall seek a parameterization of 7 as

follows

(4.9) r-zi = l, Z2 = it + a(t)9 zj = 0, j > 2,

where a(t) is real-valued for real t. Substituting (4.9) into (4.1), we see that

a = a(t) must satisfy

(4.10) a + $t{(it + a)a(l9 it + a,0,1, -it + a,0)} = 0.

By the implicit function theorem, there is a unique solution a = a(t) to (4.10) with a(0) = 0.

The Segre variety g7(,) is implicitly defined by

(4.11) z2(l +?(7(0,z)) + zi (~it + a(t))(l+a(z?(t))) + (-/? + a(0)?(7(0,z")

= 0.

Using the implicit function theorem, we solve (4.11) for Z2 = h(t9zi,z"), where h

is a convergent power series in r,zi, and z". It is clear that

/*(i,0,0) = 0.

Expand

00

h(t9zuz") = J2hk(t>z"rt

k=0

In (4.1), substitute h(t9z\,z!') for Z2- This gives us r(z\,h9z",Z\,h,z"), which, as a power series in zi,z",zi,z" and t9 is identically zero. In view of (4.2), terms

of the form zi(zi,z/r)a^ in r(z\,h9z",z\,Kz") give us

ziMr,0,0) + /*(i,zi,z") = 0.




In particular, h(t9z\,z") does not depend on z", and

(4.12) ho(t,zf,) = ^hx(t,0) = 0, hj(t,z") = 0, j>2.

To compute h\(t,0), we set z" = 0 in (4.11) and collect terms which are linear in

Zi and Z2, which yields

l+fl(7(0,0)

Thus, we have

hu(0,0) =

i-a'(0) t^O.

Combining this with (4.12), we obtain

g7(0:^2 = im(t)zu

where m(t) is a real power series with m7(0) ^ 0. On the other hand, Z2 = im(t)z\

is also contained in the complex cone 3?{ziZ2} = 0. Therefore, M coincides with

a portion of the complex cone. Since M and the complex cone are irreducible,

then M is actually the complex cone 3?{ziZ2} = 0. The proof of Theorem 1.2 is

complete. We now want to show the rigidity of the degenerate quadric 02,2- Consider

a real analytic hypersurface in Cn given by

(4.13) r(z,z) =

(zi +zi)(z2 +z2) + H(z,z) = 0,

where H(z,z) = 0(|z|3) is a real analytic function.

Theorem 4.2. Let M be a Levi-flat real analytic hypersurface in Cn defined by

(4.13). Then M is biholomorphically equivalent to 02,2

Proof. If go is irreducible, then it follows from Lemma 3.2 (b) that go is contained in M. Now, Lemma 3.8 implies that the Levi-form of r is zero,

which is a contradiction. Therefore, go is reducible. It is clear that go consists

of two smooth hypersurfaces intersecting transversally at the origin. By (b) of

Lemma 3.2, one of the hypersurfaces is contained in M. Change the coordinates

so that go consists of zi = 0 and Z2 = 0 with the latter being contained in M.

This means that M is given by

(Z\+Z\)(Z2+Z2) + H(Z,Z) = 0

with

(4.14) H(z,z)\Z2=o = 0.




We may assume that H is a small function defined on \z\ < 3.

Consider the parameterization

7- z\ = 1, Z2 = it + a(t)9 Zj = 0, j > 3

with a = a(t) satisfying

4a(0 + H (7(f), 7(0) = 0, a(t) = a(t).

Since H is a small function vanishing for Z2 = 0, there exists a unique solution

a with a(0) = 0 and a'(0) small. We now consider parameterized branches Q/ ^

determined by

(4.15) (1 + zi)(z2 - it + a(t)) + if(z, 7(0) = 0.

Rewrite the above equation in the form

., ,? A gfo7(0) it

- a(t)

= z2 + ?:-,

1 +Z\

which, by the fixed-point theorem, has a unique solution

t = ih(z), for |zi -

1| <3/2, |z| < 1.

From (4.14), one sees that A|Z2=o = 0. Hence,

A(z) = cz2(l + 0(|z|)), c^0.

In particular, Q^(t) C {|zi -

1| < 3/2} x {|z'| < 1} is given by

Z2 = -it/c + 0(\(t9zuz")\2),

where t is small and real, |zi ?

1| < 3/2, and \z"\ < 1. Obviously, the above

expression shows that as part of M, & ^ sweep out a smooth hypersurface M'

containing the origin of C\ Hence, r is reducible. Write r = r\r29 where r\, r2 are

real-valued analytic functions. Since M is Levi-flat, then each ry defines a smooth

Levi-flat real analytic hypersurface. Therefore, Cartan's theorem says that there

are holomorphic functions (pj such that ry =

wy^R^y, </?y(0) = 0. By the uniqueness

of factorization for quadratic forms, one may assume that ry(z,z) =

zy+zy + 0(|z|2). Hence, <?>j(z)

= CjZj + 0(|z|2) for some nonvanishing real constants c7; the inverse

of z ? (</?i(z), y2(z),z!') then transforms M into 02,2- The proof of Theorem 4.2

is complete.




As a consequence of Theorem 3.9, we know that there is no Levi-flat real

analytic hypersurface M with dimMs < 2n ? 3 such that M is a higher order

perturbation of Qx2 for 0 < A < 1, provided go is not a double hypersurface. We now turn to the case that go is a double hypersurface.

Theorem 4.3. Let M be a Levi-flat real analytic hypersurface in Cn with a

defining function

r(z,z) = z\ + 2AziZi +z\ + 0(|z|3), 0 < A < 1.

Assume that the Segre variety go of M is a double hypersurface. Then the dimension

ofMs is at least 2n ? 3.

Proof. We will seek a contradiction under the hypothesis that M is smooth, or dim M5 < 2n ? 4. By the assumption, we have

r(z,0) = (zi+O(|z|2))2.

Hence, one may find new coordinates such that r(z,0) = z2, while the quadratic form of r remains unchanged. M is then given by

M: r(z9 z) = ?{zi (zi + Azi + a(z, z))}

= 0, a(z, 0) =

0,

where a(z,z) = 0(|z|2). The intersection of q(z,z)

= 0 with the zi-axis consists of two real lines, of which one is parameterized by

zi = pt9 z! = 0

with

(4.16) p2 = -X + i\/l -A2, Vl

- A2 > 0, Up>0.

We need to find a real analytic curve in M of the form

7: zi = iit(l + ia(t))9 z' = 0, a(t) = ?(0 = Oflf |).

This leads to the equation

(4.17) lr(7(0,7(0) = i(ji2

- 7?V(0 + ?3?{7i(0^(7(0,7(0)}

= 0. tL tL

Obviously, r(7(0,7(0)A2 = F is a real analytic function in t9 a with

Fa(090) = i(fi2-JI2)?0.




By the implicit function theorem, (4.17) has a unique real analytic solution a.

Furthermore,

rZl (7(0,7(0) = (2ji + 2X?)t + 0(t2).

Hence, 7 \ {0} is contained in M*. Notice that

(4.18) r(z,7(0)|r=0 =

Zi.

The Weierstrass preparation theorem then gives us

(4.19) r(z,7(0) = z\ + 2X?t(\

- ia(t))zx +?2t2(l

- ia(t))2

+ zia(z,7(0) + M'(l -

ia(t))??y(t)9z) = w(z, 0(zi + 2b(z\ t)z\ + c(z\ 0).

We need to compute the discriminant A = b2 ? c of the Weierstrass polynomial.

From (4.18), it is clear that

(4.20) b({9 t) = tbx (z', 0, c(z\ t) = tcx (z', 0, u(z, 0) = 1

and

(4.21) A(z', 0 = t2b\({9 0

- ici(z', 0 = t(tdx(z\ 0

- d0(z'))

with d0(zf) = ci(zf,0). Setting zi = 0 in (4.19) yields

7x?(l -

ia(i))2 +7??(l -

/a(i))?(7(0,0,z') =

w(0,z',?)ci(z',?)?.

Expanding both sides as power series in t, the linear terms give us

(4.22) a(0,0, z') =

M"(0, z', O)do(z')

In particular,

(4.23) ci(0) = 0.

Expanding (4.19) as power series in t,zi,z! and collecting the coefficients of z\t

only, one has

2Xp = 2?i(0)w(0) + ci(0)wZl(0).

Now, it follows from (4.20) and (4.23) that

h(0) = Xfl.




Collecting the coefficients of the term t2 from (4.19), we get

-?2 = ci (0)11,(0) + n(0)cif,(0) = cu(0)>

where the last identity is obtained from (4.23). Thus, we obtain

(4.24) di(0) = b2(0) -

cu(0) = (A2 -

l)?2 ? 0.

Next, we want to show that A has a square root a/?(z', 0, which is a conver

gent power series near (z7,0 = 0. From (4.21) and (4.24), it follows that A(z7,0

has a square root if and only if do(zf) = 0 = a(0909zf) near (z',0

= 0. Assume

for contradiction that do is not identically zero. Note that

r(z?(t))\Zl=0 = ?2t2(l

- ia(t))2 + Jit(l

- /a(0)?(7(0,0,z')

has the expansion

t [tZ5(0,

0,z!) + QJ2 + 0(\z'\))t + 0(t2)]

= tf(z!91).

Since a(09 0, z') is not identically zero, one can use Rouch?'s theorem to verify that

f(zf,t)9 and hence r(0,z',7(0), vanishes at a point z' near 0 whenever t is small.

Therefore, g7(,) intersects with go for all small t. Let Nt c Cw_1 be the complex

variety defined by/(z',0 = 0, and N the union of Nt. Note that/(0)

= ~?2^0.

Hence, for a fixed z'0 G Nto with |*o|, \zq\ small, the equation /(z7, t) = 0 can be

solved for t = to + h(z!) with h a holomorphic function defined near z?- Since

?(0,0,zO ^ 0, h is not constant. Thus, N contains a Levi-flat hypersurface of

Cn~l defined by Qh(z') = 0. Notice that

g7^ can only intersect with go at

singular points of M. Denote by T the set of real numbers t such that g7(,) is

contained in M. Then T has no interior points, since dim My < 2n ? 3 = dimAf.

Now for t ? T9 g7(i) is reducible with one branch in M and another branch not

fully in M. For t G T9 we choose a sequence of tj ? T with tj ?> t. Since g7(?) is a

2-to-l branched covering over a fixed domain D c Cw_1, one can represent g7(,.) as the graph of a holomorphic function/(z7). By passing to a subsequence, one

may assume that the sequence f(z!) converges to a holomorphic function /o(z') The graph zi =fo(zf) remains inside M and contains 7(0, so it is one branch of

g7(,). This shows that either g7(,) =

Q? ^ as a set, or g7(^ is reducible with two

branches.

Without loss of generality, we may assume that ?/ofe,0) ^ 0. Using (4.24),

it is easy to verify that for small t ^ 0, all zeros of d\(z2,0)t ?

do(z2,0) near

Z2 = 0 are simple. Hence, for each fixed small t ^ 0, there exists z'0 such that

A(z?, 0 = 0, but Vz>A(z?, 0 ^ 0- Clearly, ( ?

b(zb, 0, z'0) is a smooth branch point of the branched covering g7(,) over D. In particular, all g7(,) for small t are

irreducible, which contradicts our earlier conclusion.




We now know that V?(z', t) is a convergent power series. We take the root

with y/?(z\t) = ?

i?Vl ?

A2(r + 0(t2)). Next, we want to show that the branch

of g7(,) which contains 7(0 is given by

(4.25) zi + b(z\ t) + VA(z\ 0 = 0.

Equivalently, we need to show that 7(0 is not on the other branch, i.e.,

7i(0 + K0,0-V?(0,0^0

for t ^ 0. A simple computation shows that

71 (0 + b(09 0 -

VA(0,0 = ~?(?2 + A + iVl -

A2)?2 + 0(t3).

From (4.16), we get /x2 + A + iy/l - A2 = 2iVl

- A2 ^ 0, i.e., (4.25) holds.

Finally, for a small neighborhood D of the origin of R x Cn~x9 we define a

real analytic C/? mapping F: D ? C" by

F(r,z') =

(-Mz',0-VA(z',0,z').

From (4.25), it is clear that Mi = F(D) is contained in M and F(t9 0)

= 7(0,

F(0,z') = z'. Therefore, F is a real analytic embedding and Mi is a smooth Levi

flat real analytic hypersurface. This means that r\ divides r, where r\ is a defining function of Mi with dr$0) ^ 0. Write r = r\r2. Then r2 = 0 defines another

smooth Levi-flat real analytic hypersurface M2. Obviously, n(z,0)r2(z,0) = z2; in particular, each My contains go. Take a point zo G go \ M,y. Since My, M are

smooth Levi-flat hypersurfaces near zo with My c M, then each My coincides

with M near zo; in particular, Mi = M2. By Cartan's theorem, ry =

UjWi9 where h

is a holomorphic function with h(0) = 0. Thus, q(z,z) =

Uj(Q)(?h$29 where h\ is

the linear part of h. Obviously, this contradicts the assumption 0 < A < 1. The

proof of Theorem 4.3 is complete.

Recall that Proposition 3.4 gives all possible quadratic tangent cones to a

reducible Levi-flat hypersurface. Let us now summarize what we have found

about possible quadratic tangent cones to an irreducible Levi-flat hypersurface. Let M be such a (nonsmooth) Levi-flat hypersurface, and C the quadratic tangent cone of M. Note that by Propositions 2.4, 3.5, and 4.1, one may assume that C

is zizi = 0 or one of the quadrics in section 2. Now, one of the following holds:

(a) C = Qo,ik (k > 2) and M.y is a complex variety in Cn of codimension > k.

(b) C and the range of d = dimM^ are given by Table 4.1.

Note that x2 +y\ = 0 is a Levi-flat hypersurface for which the tangent cone is

Qi,i and the singular set is of dimension 2n ? 2. One can construct other examples of Levi-flat hypersurfaces M with dimM? described in (a) and (b) by pulling




Table 4.1. Quadratic tangent cones.

c

d even

< 2n - 6 2n - 4 2n - 2

d odd < 2n - 5 2n-3

00,2

Qi,i

Q\,2 02,4 z\z\ = 0

/

**

X

X

X

/

**

X

/

/

*

*

X

*

X

**

X

X

X

**

**

**

X

*

</ = existence ** occurs.

x = non-existence *, ** = unknown, Qq a double hypersurface if

back the real cone (1.1) or the complex cone (1.2) through suitable holomorphic

mappings. For instance, the Levi-flat hypersurfaces with C = Qo,2k (k > I) and

dimM5 = d can be constructed by pulling back (1.1) through (zi,...,z?) ?

(zi,... ,z?,z?+i,. >?2_?/2>0). The remaining statements in (a) and (b) are the

content of Theorems 3.9-10,4.2-3, and Proposition 4.1. The details are left to

the reader.

5. Levi-flat hypersurfaces and meromorphic functions. In this section, we shall discuss the singularity of Levi-flat hypersurfaces arising from meromor

phic functions and then give a sufficient condition for Levi-flat hypersurfaces to

be equivalent to real algebraic hypersurfaces. Recall that a germ of meromorphic function k is given by k =

f/g9 where

/, g are two germs of holomorphic function at 0 G Cn which are relatively prime. We shall define Ck: KJfc(z) = 0 by the real analytic set $l{f(z)g(z)} = 0. We shall

also denote by Cf/g the complex variety defined by fdg

? gdf

= 0. Then on

each component of Cf/g9 f/g is a constant meromorphic function, i.e., g

= 0,

or / ?

eg = 0 for some constant c. We shall denote by C*, the union of the

indeterminacy variety /* = {f

= g = 0} with all the components of

Cf/g9 on which

f/g is pure imaginary. We remark that C/ is precisely the critical variety of the

holomorphic function/. As germs at 0, one has Cf

= Cf C {f =/(0)}.

Proposition 5.1. Letk =f/g be a germ of meromorphic function withf(0) = 0

or g(0) = 0. Then Ck is a real analytic Levi-flat hypersurface of which the singular

set is the complex variety C'k.

Proof We first need to show that Ck is of dimension 2n ? 1. Let V be the

union of the hypersurfaces / = 0 and g = 0. Put V = V* U VS9 where Vs is the

singular locus of V. Take any point p eV*. One may further assume that p = 0,

/(0) = 0, and g(0) ^ 0. Choose local coordinates so that

(5.1) fc(z) = zT.




Near p = 0, Ck is given by zm + z = 0, which is a real hyperplane when m = 1, or the union of m real hyperplanes with {zi

= 0} being contained in (Ck)s when

m > 1. Thus, dimCk = 2n ? 1. The above arguments also show that Ck \ Ck is

smooth and Levi-flat. Since the real dimension of Ck is less than 2n ? 1, Ck is

Levi-flat.

One sees that p G (Ck)s\h if and only if either k or \/k is holomorphic near

p such that p is a critical point for which the critical value is pure imaginary.

Therefore, (Ck)s \ h equals C'k \ 4. Since both (Ck)s and C'k contain /?, this

completes the proof of the proposition. D

Corollary 5.2. Let M be as in Theorem 3.9. Assume further that dimM^ < 1

ifn > 3. M is biholomorphically equivalent to $lh(z) = 0, where h is a polynomial

with isolated singularity at 0.

Proof. From Theorem 3.9, it follows that M is defined by $lh = 0 with h

a holomorphic function. We also know that as a germ at 0 the singular locus

Ms is precisely the critical variety of h. Since dimM5 < 1, then the singular locus of h = 0 is isolated. By a theorem of Arnol'd [1] and Tougeron [12], h is

holomorphically equivalent to a finite jet of h.

We remark that not every irreducible real analytic Levi-flat hypersurface can

be defined by the real part of a meromorphic function. To see this, we need the

following.

Lemma 5.3. Let M be defined by ̂ (f/g) = 0 withf(0) = g(0) = 0. Then for all values ofceR, the subvariety f/g

= ic C M passes through each point of the indeterminacy locus of f/g, and for an open set of z G M*, the irreducible

component ofQz passing through z also passes through 0.

Proof. Indeed, for the first statement, we can assume without loss of generality that the dimension n = 2, and that the indeterminacy locus of f/g is just the origin.

We can blow-up C2 suitably so that we obtain a proper modification

Tr : ?2 - C2

with 7T_1(0) := E a (connected) union of copies of P1 such that the function F :=

7T*(f/g) extends holomorphically across E. On one of the irreducible components,

say Fi, of F the map F is a finite ramified covering onto P1. Let 7 C ?i be

the real analytic curve F~x({$lz = 0 U 00} C P1), and let <f> be the map F

restricted to 7. Note that (j) is surjective onto {!Rz = 0 U 00} c P1, and that for

all but a finite number of points zo in 7, </>(zo) is a regular value of 0 and hence

(locally) of F. In a neighborhood of such a point, the real subvariety 5?F = 0

is smooth, and the holomorphic curve F = F(zo) intersects 7 transversally at zo

We now conclude, after pushing these curves down to M C C2 using the map 7r,

that every level f/g = ic passes through 0, and that for an open set of z G M*




the irreducible component of Qz passing through z also passes through 0. This

shows the openness in the statement of the lemma when n = 2. The general case follows by a similar argument, but involves the resolution of singularities in higher dimensions, which is much deeper than the elementary result in two

dimensions. The case n = 2 will suffice for our purposes below. The lemma is

proved.

Proposition 5.4. Let C be a germ of real analytic curve at 0 G C Then C x

Cn~x is the zero locus of the real part of a meromorphic function if and only if C is conformally equivalent to m straight lines ^St(p!z\)

= 0, where p = il/m and

j =

0, ...,m- 1.

Proof. We may assume that 0 is an isolated singular point of C. The singular locus of M = C x Cn~x is 0 x Cn~~x. The branch of Segre variety Qz containing z =

(ci,... 9cn) G M* is the hyperplane zi = c\9 which does not pass through the origin. Lemma 5.3 implies that M cannot be defined by the real part of a

meromorphic function. Assume now that M is defined by 5r/ = 0, where / is

a germ of holomorphic function at 0. Proposition 5.1 implies that the critical

variety of/ is zi = 0, i.e., f(z) =

u(z)z% with u(0) ^ 0 and m > 1. Now,

C: 5?(w(zi,0)zi*) = 0 is conformally equivalent to the germ of the m lines stated

in the proposition.

Note that the claim in the introduction, that the Levi flat hypersurface M :=

{jc2+j3 = 0} cannot be defined by the vanishing of the real part of a holomorphic

or meromorphic function, follows directly from any of the results above.

We conclude the paper by noting that there are obviously several questions left unanswered by what we have done. We point out a few of them here.

(1) Do there exist real analytic Levi-flat hypersurfaces in Cn for which the

singular point is isolated and the tangent cone is Qi,i? More generally, can one

find a real analytic hypersurface with isolated singularity which is the union of

a family of smooth complex hypersurfaces? (Note that V: x2 + zy3 + z2 = 0 in

R3 (resp. C3) is a union of smooth real (resp. complex) hypersurfaces in R3

(resp. C3) parameterized by t = z, for which Vs = {0}.)

(2) Is the Levi-flat hypersurface $l(f/g) = 0 finitely determined if it has an

isolated singularity at 0, where f/g is a germ of meromorphic function at 0? Are

there topological invariants of a Levi-flat hypersurface near an isolated singular

point, analogous to the Milnor number of a complex hypersurface with isolated

singularity?

(3) What are the singularities and rigidity properties of Levi-flat real analytic varieties of higher codimension or of lower CR dimension?

Department of Mathematics, University of Michigan, Ann Arbor, MI

Electronic mail: [email protected]

Electronic mail: [email protected]




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singular levi-flat real analytic hypersurfaces

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