Infinitesimal CR Automorphisms of Rigid Hypersurfaces

Infinitesimal CR Automorphisms of Rigid HypersurfacesAuthor(s): Nancy K. StantonSource: American Journal of Mathematics, Vol. 117, No. 1 (Feb., 1995), pp. 141-167Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2375039 .

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INFINITESIMAL CR AUTOMORPHISMS OF RIGID HYPERSURFACES

By NANCY K. STANTON

Let M be a real hypersurface through the origin in C"'n, or, more generally, an integrable CR manifold of hypersurface type. A smooth vector field X on M is called an infinitesimal CR automorphism of M if the local one-parameter group it generates is a local group of CR automorphisms of M. Fix p C M and let aut(M, p) denote the space of infinitesimal CR automorphisms of M which are defined in a neighborhood of p. If the point p is understood, we write aut (M) for aut (M, p). In this paper we consider the following local problem.

Suppose M is a rigid real analytic real hypersurface through the origin in Cn,1. Under what additional hypotheses is the space aut (M) finite dimensional?

Here, following the terminology of Baouendi, Rothschild and Treves [BRT], we call M rigid if there are coordinates (zi, ... Zn W) such that M is given by an equation of the form

(0.1) Imw = F(z,z-).

This is equivalent to the existence of a transversal infinitesimal CR automorphism on M. Rigid hypersurfaces are called regular by Tanaka [T] and T-regular by D'Angelo [D]. If M is locally CR isomorphic to the hyperplane

(0.2) Imw = 0,

the space aut (M) is infinite dimensional. If M is rigid, real analytic and not locally CR isomorphic to (0.2), it is of finite type (see [BG]). If, in addition, M c C2, aut(M) is finite dimensional. However, in higher dimensions, this is no longer true. For example, in C3, if M is given by Im w = IZ, 12, then M is of finite type, but aut (M) is infinite dimensional.

If Z is a holomorphic vector field defined in a neighborhood of 0 C Cn+l and X = Re Z, then the local one-parameter group of X is a group of biholomorphic transformations [KN, remarks preceding Proposition IX.2. 10]. Here, by

141

Manuscript received April 30, 1992; revised August 31, 1992. Research supported in part by NSF grants DMS 89-01547 and DMS 91-01113. American Journal of Mathematics 117 (1995), 141-167.



142 NANCY K. STANTON

holomorphic vector field, we mean a vector field of type (1, 0) with holomorphic coefficients. Hence, if M is a real hypersurface through the origin and X is tangent to M, then X C aut (M). Let hol (M) denote the space of all infinitesimal CR automorphisms X defined in some neighborhood of the origin in M which are of the form X = Re Z for some holomorphic vector field Z. Then hol (M) C aut (M). This raises the following questions.

Suppose M is a rigid real analytic real hypersurface through the origin in C'". Under what additional hypotheses is the space hol (M) finite dimensional?

Under what hypotheses is hol (M) = aut (M)?

The problems we consider are local, so we may assume that M is a rigid real analytic real hypersurface through the origin and p = 0. We introduce the geometric notion of holomorphic nondegeneracy. A real hypersurface M is holomorphically nondegenerate at the origin if there is no nontrivial holomorphic vector field tangent to M in a neighborhood of 0 (see Definition 4. 1). One of our main results, Theorem 4.16, is that for rigid hypersurfaces, holomorphic nondegeneracy at the origin is the necessary and sufficient condition for hol (M) to be finite dimensional. In all dimensions, essentially finite hypersurfaces are holomorphically nondegenerate (see [BJT] or Section 4 for the definition of essentially finite), as are hypersurfaces with somewhere nondegenerate Levi form. In C2, M is holomorphically nondegenerate if and only if it is not flat, i.e., if and only if it is not CR equivalent to a real hyperplane. In higher dimensions holomorphic nondegeneracy appears to be a new condition. For example, it is not the same as nonflat, finite type, essentially finite or somewhere Levi nondegenerate (see Section 7). If the function F of (0.1) is a polynomial, in principle-and often in fact-it is easy to check whether M is holomorphically nondegenerate at the origin. In general, it is very difficult to check directly whether hol (M) is finite dimensional.

A rigid hypersurface is called homogeneous if it is locally equivalent, via a CR diffeomorphism which preserves the origin, to

(0.3) Im w = p(z, z)

with p a homogeneous polynomial. This terminology comes from the fact that an equation of the form (0.3) with p a homogeneous polynomial is homogeneous with respect to a nonisotropic group of dilations (see Section 1). We obtain an answer in the case of holomorphically nondegenerate hypersurfaces to a question posed by Linda Rothschild:

How can you tell if a rigid hypersurface is homogeneous?

For the case of rigid hypersurfaces in C2, we answered Rothschild's question in [S2, S3] (see also [S1]). Also, in this case, we showed in [S3] that hol (M) =



INFINITESIMAL CR AUTOMORPHISMS 143

aut (M) and dim aut (M) < 8 for real analytic rigid hypersurfaces of finite type in C2.

After some preliminaries in Section 1, in Theorem 2.1 we give a sufficient criterion for a rigid hypersurface to be homogeneous. In Section 3 we show that hol (M) is finite dimensional if the Levi form of M is not everywhere degenerate. We discuss holomorphic nondegeneracy in Section 4 and prove Theorem 4.16 in Section 5. In Section 6 we show that aut (M) = hol (M) if M is essentially finite or somewhere Levi nondegenerate. We also answer Rothschild's question in these cases and, more generally, if M is holomorphically nondegenerate. We conclude with examples in Section 7.

I want to thank Linda Rothschild and Salah Baouendi for many helpful dis- cussions concerning this work. I also benefited from conversations with Tejinder Neelon, Andrew Sommese and Frangois Treves. I would like to thank the referees for helpful comments.

1. Preliminaries. Throughout this paper, unless we state otherwise, we will assume that M is a rigid real analytic real hypersurface of finite type m through the origin in C" 1 given by an equation of the form (0.1). We call such an equation a rigid equation. By finite type m we mean that the tangent space of M at the origin is spanned by commutators of length m of sections of T1 '0M ED T0'1M and it is not spanned by commutators of length at most m - 1. By finite type we mean finite type m for some m. We will work locally; all functions, maps, vector fields, etc. will be defined in sufficiently small neighborhoods of the origin. We can assume, after a change of coordinates if necessary, that M is given by an equation of the form

(I. 1) V = F(z,z-

where w = u + iv and F has no pure terms. This equation is in Bloom-Graham normal form [BG] (see also [BRI]). Because M is of finite type m, we can write

(1.2) F(z,z-) = p(z,z-) + O(m + 1)

where p is a nontrivial polynomial homogeneous of degree m having no pure terms. We call the hypersurface Mo given by

Im w = p(z, z)

the homogeneous part of M. Define nonisotropic dilations on Cn+1 by

(1.3) 6t(Z,gW) = (tz, tmw)

for t > 0. Then &t is a CR automorphism of Mo. We say a function h is homo-




geneous of weight j if h o Et = tPh. We say h = 0(j) if every term in its Taylor expansion about 0 has weight > j.

Definition 1.4. A vector field Y is homogeneous of weight j if

Y(f o 6t) = t-j(Yf) o 6t

where &t denotes the dilation (1.3).

We write Y = 0(j) if, when we expand the coefficients of Y in a Taylor series about 0, each term is a vector field of weight > j. If Y = 0(j) and f is a function with f = 0(k), it follows that Yf = 0(j + k). If X = 0(j) and Y = 0(k) then [X,Y] = O(j+k).

Definition 1.5. A vector field X has weight j if X = 0(j) and X 7 O(j + 1). Let w(X) denote the weight of X.

The vector field

YO=2Re (Zz$ + Mw )

is an infinitesimal CR automorphism of Mo. Because the one-parameter group generated by Yo is the group of nonisotropic dilations 6et, we call Yo an infinitesimal dilation. If X is a homogeneous vector field of weight j, then

[Yo, X] = jx.

Definition 1.6. A vector field Y C aut (M) is an approximate infinitesimal dilation if Y = Yo + 0(1).

Because hol (M) is a Lie algebra, if Y E hol (M) is an approximate infinitesimal dilation and X C hol (M), then [Y, X] c hol (M) and if X is nontrivial

(1.7) [Y,X] = w(X)X + 0 (w(X) + 1) .

2. A criterion for a rigid bypersurface to be homogeneous. In this section we prove Theorem 2.1, a sufficient condition for a rigid hypersurface to be homogeneous. Let M be a real analytic rigid hypersurface of finite type m given by a rigid equation of the form (1.1). If M a homogeneous, then there is an




approximate infinitesimal dilation Y C aut (M),

(nla a Y=2do3Re (zji +mw9w

where q: Mo -* M is a CR diffeomorphism with q(O) = 0. We do not know whether the existence of an approximate infinitesimal dilation Y is a sufficient condition for homogeneity. However, Theorem 2.1 shows that it is if, in addition, Y E hol (M) and hol (M) is finite dimensional.

THEOREM 2.1. Suppose M is a rigid real analytic real hypersurface of finite type m through the origin in Cn+1 given by a rigid equation of the form

Im w = p(z, -) + O(m + 1)

where p is a homogeneous polynomial of degree m having no pure terms. Suppose also that dim ( hol (M)) < oo. If there is an approximate infinitesimal dilation Y C hol (M), then M is homogeneous.

Proof. Because a change of variables of the form

(z', w') = (z, w +f(z)), f = O(m + 1),

takes an approximate infinitesimal dilation to an approximate infinitesimal dilation, after making such a change if necessary, we can assume that M is given by an equation of the form

v = F(z, z-)

where F has no pure terms. The first step in the proof is to show that there is X = + O( - m + 1) C hol(M) such that [Y,X] = -mX, so hol(M) has a subalgebra isomorphic to the Lie algebra generated by Yo and a. Suppose XI= 0u + Of-m + 1) C hol (M) satisfies

[Y,Xi] = -mXl +XI

where X' C hol (M) and w(X') = w > -m + 1. Let

X2=XI- X'. w+m




Then X2 = + O(-m + 1) c hol (M) because hol (M) is a Lie algebra. By (1.7),

[Y,X2] = -m (XI- X') + O(w + 1) w+m

= -mX2+O(w+ 1).

Because dim (hol (M)) < oo, wo = sup{w(V): V C hol (M), V # O} is finite. Thus, by induction beginning with X1 = 0 we can find X = '9 + O(- m + 1) c hol (M) with [Y, X] =-mX + X' and X' = O(wo + 1) C hol (M). Hence, X'- 0.

Now X = 2 Re W for some holomorphic vector field defined in a neighborhood of the origin. Because W does not vanish at 0, we can make a biholomorphic change of coordinates preserving the origin such that, in the new coordinates (z', w'), W = 0 . Then 0 is tangent to M and -0 - is not, so M is given by an equation of the form v' = F'(z', z'). By making a change of variables of the form (E, wv) = (z', w' +f(z')) with f holomorphic, if necessary, we can assume that F' has no pure terms. Write Y = 2 Re Z with Z holomorphic. Then [Z, W] =-mW, so

n&

Z = Efi(z) , + (mw' + h(z')) a ,

with fi(O) = 0 = h(0). Because Y is tangent to M,

&F'(z', Z') -,F'(z', zI) mF'(z', -') + Im h(zl) = ( +

The right side of this equation and the first term on the left have no pure terms, so h _ 0. Hence

n I (2.2) Z=Zfi(z'), + mw ,

Because F and F' have no pure terms,

(z', w') = q(z, w) = (Az + 0(2), bw + 0(m + 1)).

Hence, by (2.2),

n&& z = E (Z/ + gi(z')) + mw' O,'

where gi = 0(2).




By Poincare's Linearization Theorem [Po, Deuxieme Partie, Theoreme III] there is a biholomorphic change of coordinates

(z", w") = (z' + h(z'), w'), h = 0(2),

such that

Z = ?: ',i +mw" ,

Then M is given by an equation of the form v" = F"(z", z-") where F" has no pure terms. The equation 2 ReZ(v" - F"(z", -"))IM = 0 shows that F" is homogeneous of degree m, hence M is homogeneous.

3. Hypersurfaces with somewhere nondegenerate Levi form. To apply Theorem 2.1, one needs to know that hol (M) is finite dimensional. In this section, we show that it is finite dimensional if every neighborhood of the origin contains points where the Levi form is nondegenerate. We do not assume that M is rigid or analytic.

THEOREM 3.1. Let M be a smooth real hypersurface through the origin in Cn+' . Suppose that every neighborhood U of 0 contains a point p C M such that the Levi form of M is nondegenerate at p. Then dim ( hol (M)) < (n + 2)2 _ 1.

Proof. We use the method of [S3, Corollary 4.15]. Let LI,... L(n+2)2 C

hol (M). Choose a neighborhood U of the origin on which L1, ... , L(n+2)2 are defined. Let p C U n M be a point where the Levi form is nondegenerate. Then there is a neighborhood M' of p in U n M such that the Levi form is everywhere nondegenerate on M'. By making a biholomorphic change of coordinates if necessary, we can assume that M' is given by an equation of the form

n

(3.2) Im w= Zajk-zIzIk + 0(3). j,k=l

Let

Wj = {X C hol (M'): X = 0(j)}.

For X C Wj, write X = X + O+(j + 1) where Xo is homogeneous of degree j. Let

Mo be the hypersurface Im w' = aj -zIzIk. The linear map a,k

Tj : Wj /Wj+ I~ aut (M6O), T1(X/W1+1) = o




is injective. If the Levi form of M' has signature (p, q), aut (MO) = su(p + 1, q + 1), so dim( aut (MO)) = (n + 2)2 - 1, and if X0 0 0 C aut (Mo) then w(Xo) ? 2. Thus,

2

hol (M') W= e Wj/Wj+ . j=-2

Let T = EDTJ. Then T is injective, so dim ( hol (M')) < dim ( aut (MO)). It follows that LI,... , L(n+2)2 are linearly dependent on M'. Because they are analytic, they are linearly dependent on U n M. Hence, dim ( hol (M)) < (n + 2)2.

If M is analytic, the hypothesis on the Levi form in Theorem 3.1 is equivalent to the hypothesis that the determinant of the Levi form is not identically zero. In this case Tanaka [T, Proposition 14] proved the bound on the dimension of the Lie algebra of analytic globally defined infinitesimal CR automorphisms. In the rigid analytic case, Theorem 3.1 is due to Han [H].

4. Holomorphic nondegeneracy. In this section, we introduce the geometric notion of holomorphic nondegeneracy. We show in Theorem 4.3 that in C2 holomorphic nondegeneracy is equivalent to known nondegeneracy conditions. In higher dimensions, by Theorem 4.3, some known nondegeneracy conditions are sufficient for holomorphic nondegeneracy, but examples in Section 7 show that holomorphic nondegeneracy is a new nondegeneracy condition. In the rigid case, a criterion of Neelon (Proposition 4.9) gives a useful alternative characteri- zation of holomorphic nondegeneracy. The main result of the section is Theorem 4.16: hol (M) is finite dimensional if M is rigid, analytic and holomorphically nondegenerate. We prove the theorem in Section 5.

Definition 4.1. Let M be a real hypersurface in Cn+ 1. A nontrivial holomorphic vector field W is called a holomorphic tangent to M at the point p C M if W is defined in a neighborhood of p and WIM is tangent to M. The hypersurface M is holomorphically nondegenerate at p if M has no holomorphic tangent at p. Otherwise, M is holomorphically degenerate at p.

Note that we do not require a holomorphic tangent W to M to be nonvanish- ing.

Remark 4.2. The existence of a holomorphic tangent is a degeneracy condition on M in the following sense. If W is a holomorphic tangent to M at 0, then Re za W C hol (M) for all multiindices ae so dim ( hol (M)) = oo.

Let M be an analytic real hypersurface through the origin in Cn+I. Suppose M is given by the equation p((, ( ) = 0, where dpIo # 0, E Cn+l . Let

V= {4E Cn+l : p(q,rT)=0 if r1 C Cn+ 1 and p(O,rT)=0}.




Following the terminology of Baouendi, Jacobowitz and Treves [BJT], we call M essentially finite at 0 if V = {O}.

THEOREM 4.3. LetMbe an analytic real hypersurface through the origin in Cn' 1. If M is essentiallyfinite at 0 or if the Levi form of M is somewhere nondegenerate, then M is holomorphically nondegenerate at 0. If n = 1 thefollowing are equivalent.

(1) hol (M) isfinite dimensional;

(2) M is not flat;

(3) the Levi form of M is somewhere nondegenerate;

(4) M is holomorphically nondegenerate at the origin.

Proof. The following proof in the essentially finite case was suggested by Baouendi. By [BR2, Proposition 1.12], essential finiteness is an open property. Suppose there is a nontrivial holomorphic vector field W tangent to M in a neighborhood U of 0. Then there is p C U such that Wip -7 0 and M is essentially finite at p. We can make a biholomorphic change of coordinates taking p to the origin such that in the new coordinates (, W = ,9 . Because W and W are tangent to M, in the new coordinates the equation of M is p'((, ( ) = 0 with p' independent of (I and (I. Thus, {((l, O)} C V. This contradicts the essential finiteness of M at p. Hence, M is holomorphically nondegenerate at 0.

If M is somewhere Levi nondegenerate, it follows from Theorem 3.1 and Remark 4.2 that M is holomorphically nondegenerate.

Suppose n = 1. If M is flat, we can choose coordinates such that M is given by Im w = 0. Then Z = '9 is a holomorphic tangent to M. If M is not flat, we can find coordinates [CM, Section 3] such that M is given by an equation of the form v = F(z, -, u) with F(z, 0, u) = 0 = F(0, -, u) and F # 0. Suppose

z z~~~&

Z = f(z, w)y-) + g(z, w)&w

is holomorphic and tangent to M in a neighborhood of the origin. Applying Z to v - F(z, z, u) and evaluating on M gives

(4.4) g i = f (z u + iF(z, z, u)) (z, z, u)

+ g(z, u + iF(z, z, u)) OF -

2 au(Zu)

Because F(z, 0, u) = 0, substituting -z = 0 in (4.4) gives g(z, w) 0. Hence,

OF f(z, u + iF(z, z, u)) (z,z , u) -




so f - and thus Z- 0. This proves the equivalence of (2) and (4). By Remark 4.2, (1) implies (4). If M is not flat, the Levi form is not identically zero (see, e.g., [BR3, Proposition 1.4]), SO (2) implies (3). Finally, by Theorem 3.1, if the Levi form is somewhere nondegenerate, dim ( hol (M)) < 8. Thus, (3) implies (1).

The next two propositions give criteria for holomorphic nondegeneracy of rigid hypersurfaces. Neelon's criterion shows that if F is a polynomial, one can tell whether M is holomorphically nondegenerate at the origin by computing a finite number of determinants; if n and the degree of the polynomial are not too large, the computations are easy to carry out.

PROPOSITION 4.5. Let M be a rigid real analytic hypersurface through the origin in Cn+1 given by an equation of the form

Im w = F(z, z)

where F has no pure terms. Then M is holomorphically degenerate at 0 if and only if there is a nontrivial holomorphic vectorfield Z in a neighborhood of 0 C Cn = {Z}

such that

(4.6) ZF(z, z -) .

Proof. If there exists such a vector field Z then, because Z is independent of w, it is tangent to M. Suppose M has a holomorphic tangent W at 0. Write

W = Efi(Z, w) a + g(z, w) i=1 azi Ow'

Because W is tangent to M,

1 OF (4.7) ?ig (z, u + iF(z, )) = fj (z, u + iF(z, z)) a-(z, z).

Substituting z- = 0 in (4.7) gives g(z, u)- 0. Then substituting u = w - iF(z, -) in (4.7) shows that

(4.8) Zfj(Z w)%F(z ( 0 j=1 Izj

for all sufficiently small w c C and z c Cn. Let

ro =min {r: a

(z, 0) # O for some j}.




Applying 3 to (4.8) and setting w = 0 shows that

z = E ji(Z' 0)a

is a nontrivial holomorphic vector field in Cn satisfying (4.6).

PROPOSITION 4.9. (Neelon [N]). Suppose M is a rigid analytic real hypersurface through the origin in CnlI given by an equation of the form

(4.10) v = F(z, -) = , WV

where F has no pure terms. Then M has a holomorphic tangent at 0 if and only if for every n-tuple of multiindices A = {a1, . . ., an } the Jacobian matrix JA(Z) of the

map FA(Z) = (fa, (z), .. ., fan(z)) is singular for all z.

Proof. Throughout the proof, Z will denote a holomorphic vector field in a neighborhood of 0 C Cn. Suppose for some A the Jacobian matrix JA(Z) is not everywhere singular. Then there is a variety V such that in any sufficiently small neighborhood U of the origin, JA is nonsingular on U \ V. If ZF(z, z-) _ 0 on a neighborhood 0 C U then Zfa(z)- 0 for all ai. Hence, Z C kerdFA(z) for all z C U. Because JA is the matrix of dFA, kerdFA(z) = {0} for z C U \ V. Thus, ZjU\v = 0 so Z -0. By Proposition 4.5, M has no holomorphic tangent at 0.

Suppose for all A and all z the rank of JA(Z) is less than n. Let

r = max rank JA(Z). A,

After renumbering the zi if necessary, we can findfa,,.. .,far such that

h(z) = det (&fai 0 0. (aZi /ij<r

Let

= G(z) = (fa, (Z), ,far (Z), Zr+ l, l Zn).

Let C denote the Jacobian matrix of G and B = adj C the classical adjoint of C, i.e., the transpose of the matrix of cofactors of C, so

B(z)C(z) = det (C(z))I = h(z)I

where I is the n x n identity matrix. Because h X 0, we can find zo with h(zo) # 0.




Set (o = G(zo). Then G is invertible on a neighborhood V of (0, and on V

(4.11) (<) (z) = C(z)Y I 1 B(z). a (j ~~~h(z)

By (4.11),

(4.12) Z = h(z) E I )

is a holomorphic vector field in a neighborhood of 0. Also, by (4.12), for V,

(4.13) (G*Z) =h(G a

For any ao, the Jacobian matrix of

(fai (G-1 (()), ... ,f r (G-1 (()),fa (G-1 (())) = (4 , .r,fa (G- 1(()))

has rank r on V, so

(4.14) fae(G-1(()) = ga((I, , (r)

for some holomorphic function g, on V. From (4.14) and (4.13), we see that Zfo(z) = 0, z C G-1(V). Because G-'(V) is open and Zfa is holomorphic, Zfc (z) 0, all ae. Thus, ZF(z, -) 0 O and Z is a holomorphic tangent to M.

COROLLARY 4.15. Let M be a rigid analytic real hypersurface through the origin in Cn+l. Suppose M is holomorphically nondegenerate at the origin. Then M is holomorphically nondegenerate everywhere.

Proof. Suppose M is given by an equation of the form (4.10). Let p = (z?, wO) c M and let Mp be the hypersurface equivalent to M via the change of variables

z =zz, w' =w-w?-2i (F(z'+zo,zO)-F(z? z?)),




which takes p to the origin. Then Mp is given by the equation

V/ = F'(z',zE) = Ego(z, zO,z0)z/a ao

where

g~(z',z00) = 1 I0 0(+oc&F( oo))

Because F' has no pure terms, Mp is given by an equation of the form (4.10). Now, ga(z',/O,O) = fa(Z'), with fa as in (4.10). It follows from Proposition 4.9 and the definition of the g, that for sufficiently small zo, Mp is holomorphically nondegenerate at 0, i.e., M is holomorphically nondegenerate at p. Thus, the set of points where M is holomorphically nondegenerate is open. By the definition of holomorphic degeneracy, it is closed, so M is everywhere holomorphically nondegenerate.

The following theorem shows the importance of holomorphic nondegeneracy for rigid hypersurfaces.

THEOREM 4.16. LetM be a rigid analytic real hypersurface through the origin in Cn+ l. Then the space hol (M) isfinite dimensional if and only if M is holomorphically nondegenerate.

5. Proof of Theorem 4.16. To prove Theorem 4.16, we first need an approximate version of the theorem, Lemma 5.4, which requires a notion of approximate infinitesimal CR automorphism. Example 7.5 shows that in dimensions greater than 2, the approximation must include some higher order terms; the homogeneous part may not give a good approximation.

Let M be a rigid real analytic hypersurface of finite type m, given by

Im w = F(z, -) = E a.3 a -0

where F has no pure terms. We denote by MK the hypersurface

Imw = FK(z,Z-) = E: acozz m< jaj+j/j <K

Definition 5.1. A vector field X is a mod K approximate infinitesimal CR automorphism of M if X is the real part of a holomorphic vector field such that either X _ 0 or X is a sum of vector fields homogeneous of degree at most




w(X) + K - m and

(X(Im w - FK(Z, Z))) IW=U+iFK(Z,Z) -0 mod O(K + w(X) + 1).

The set of mod K approximate infinitesimal CR automorphisms of M is denoted holK (M). We say holK (M) is finite dimensional if its span is and we denote the dimension of its span by dim (holK (M)).

LEMMA 5.2. Suppose holK (M) is finite dimensional. Then hol (M) is finite dimensional.

Proof. Because holK (M) is finite dimensional, there is a k such that if Y C holK (M) then w(Y) < k. Let X = EW(x)?jXj # 0 C hol (M) where Xj is homogeneous of weight j. Then

W(X)+K-m

XK = E Xj C holK (M), j=w(X)

So W(X) = W(XK) < k. Let W be the space of real vector fields which are sums of vector fields homogeneous of degree at most k. If Y C W, the coefficients of Y are polynomials of degree at most k + m, so dim W < oo. Define a map T hol (M) -> W by

TIV) - JZEJ-W(X) Xj, X =W(x)?jXj X 0 ( - o, X--O.

Then T is linear. It is injective because w(X) < k and Xw(x) # 0 for X C hol (M), X # 0. Hence, dim (hol (M)) < dim W < oo.

Definition 5.3. A nontrivial holomorphic vector field Z in Cn = {z} is a mod K approximate holomorphic tangent to M if Z is a sum of vector fields homogeneous of degree at most w(Z) + K - m and

ZFK=_ OmodO(w(Z) + K+ 1).

If there is such a Z we say M has a mod K approximate holomorphic tangent.

LEMMA 5.4. The vector space span ( holK (M)) is finite dimensional if and only if M has no mod K approximate holomorphic tangent.

Proof. Suppose M has a mod K holomorphic tangent Z. Then for all multiindices ao, Xc, = 2 Re z'Z E holK (M) and w(X,) = w(Z) + I a 1. Hence dim holK (M) = 00.




Suppose there is no nontrivial mod K approximate holomorphic tangent to M. Let X E 0 e holK (M) with w(X) = k > 0. To show that holK (M) is finite dimensional, it suffices to show that k < (2m + 3)K. Write X = 2 Re Z where

Z = Ef1(z, w) + g(z, w) j=1

9 w

with fj and g holomorphic polynomials. Then, because X e holK (M),

(5.5) 2 (g(z, u + iFK(z, z))-g(z, u - iFK(Z, z)))

( &~~~M FK - -E yfi(z, u + iFK(Z, Z 9z,j (z, z

+ fj(Z, U F(Z,)) a (Z, ))mod O(k + K + 1).

Because FK has no pure terms, setting -z= 0 in (5.5) yields

n &FK

(5.6) g(z, u) _ 2iZ fj1(O, u) a (z, O) + g-(O, u) mod O(k + K + 1).

Setting z = 0 in (5.6) we obtain

(5.7) g(O, u) =g-(O, u) mod O(k + K + 1).

Substituting (5.6) and (5.7) in (5.5) gives

1 (5.8) 2 (g(0, u + iFK(z, z)) - g(O, u - iFK(z, z))

9FK + ( K0', U + iFK(Z, Z)) aZ (Z, 0)

+ fj(O, U - iFK(Z, Z)) a (?1 Z)

( OFK E (yf(z u + iFK(Z, Z a (Z, Z)

+ Mj( , U-iFK(z, Z)) F j(Z,Z )) mod O(k+ K+ 1).




Because k > 0, fj(O) = 0, j = 1,... , n, and g(O) = 0. Hence, if we set u = iFK(z, z-) in (5.8) we obtain

1 &~~~~ FK (5.9) 2jg(0, 2iFK(z, z)) + Zfj(O, 2iFK(z, z)) (z, 0)

- z (f.(z2iFK(z)))FK (z, )

+ f1(z,O) 2iF(z,z)) mod O(k+K+ 1).

Define holomorphic vector fields Zi inductively by

[9 aw] f[3w f] i_

so

(5.10) Z af(Z w)_ + , (z, w)

Let X = 2 Re Zj. By induction, we see that if Xj - 0, then w(Xj) > k - mj and

Xj is a sum of vector fields homogeneous of weight at most k - m(j + 1) + K. Because MK is rigid, ,9 is tangent to MK and

(5.11) XjlMK(Imw-FK(z,Z)) = i (X-IMK) (Imw-FK(z,)).

By induction, using (5.5) and (5.11), we see that

(5.12) XjIMK(Imw - FK(Z,Z-)) =Omod O(k - mj+K+ 1).

Write

(5.13) fi(z, w) = Zfij(z)w

where]i = max{j fij(z) - 0 for some i}. Let

(5.14) W fijand 11a=dw VWWI

so

Z= Wj +g(z, w) w

We need two auxiliary lemmas.




LEMMA 5.15. For each jo,

n &FK Ef j (z) AK (z _) i=1 i

is a linear combination mod O(k + K - mjo + 1) of

n . O~~FK (5.16) f fij()(FK(Z, -)y) (z, Z),

i=1~~~~~~~~~~~~z

(FK(Z, )-I &f (0, iFK(z, ZFZ)) (z, 0), & Zr

(F(, ) 0 ) (0, iFK(Z, Z)(FK(Z, Z)y j )T (O iFK(z )

_ &FK OFK (FK(Z, -z)j-jO &r (0, - iFK(Z, Z)) a (0,Z) :i >i O

Proof. We prove the Lemma by downward induction on jo. We have by (5.10), (5.12) and (5.13)

1 (&"g__- (5.17) au (, (z, u + iFK(z, ))- a' u - iFK(z,z)))

=jl! Efij (z) 9(z,z) +afiZ I (Z) (Z,z

mod O(k + K - mjl + 1).

Set u = 0 in (5.17). By (5.6) and (5.7) we can replace the left side of (5.17) by a linear combination mod O(k + K - mjl + 1) of

(5.18) J~ '~Ir (0, iFK(Z, -) aFK ____ (5. 18) { , (O, iFKZ Z)) a (z, 0) '&wi (0, iFK(Z, Z)),

__ _ a~~~~FK g ,9, O-iFK(z,) #) (OZ), ,9 (O,-iFK(z, Z))}

aw" zo r } O il

and then solve for the first sum on the right. This proves the lemma for jo ji1.




Suppose it is true forj2 + 1 ?j < jl. By (5.10) and (5.12) withj =12,

1 J2g__

2i (W12 Z + K( )) -

aW12 (Zu- iFK(z,

n &FK

-.E 2(2- 1)... (f-]2 + 1) Zfi ,(z) a (z, z)(u + iFK(z,Z))312 12< 3 =

& FK + Z (- 1). .(J-2. + 1 )Z Q I f(Z) a (z, z)(u- iFK(Z,' ))3'J2

12?3 i=~~~~~~~1 -i

n &FK + 12! Zfis2(z) a (z, z) mod O(k + K - mj2 + 1).

i=1 z

Set u = 0. By (5.6) and (5.7) we can replace the left side by a linear combination of terms of the form (5.18) with j2 substituted forjl. By the induction hypothesis for each . > ]2, because (j-. ) + (.1-j2) =i-j2, we can replace the first sum on the right by a linear combination of terms of the form (5.16) with jo replaced by 12. Now solving for the last sum on the right gives the lemma for jo = 2. Hence, by induction, the lemma holds for any jo.

LEMMA 5.19. Let

(5.20) f = (m + 1)K - m and f (j) = max f{degree (fij(z))}. I1<i<n

For each j, either f (j) < f or

n OFK (5.21) Zis(z) a (z,) 0 mod O(k - mj +K+ 1).

i=l aZi

Remark. Because we only know that w(WJ') > k - mi, we cannot conclude that if (5.21) holds Wj' is a modK approximate holomorphic tangent.

Proof. Fix jo with ?(jo) > ?. Because X is a sum of vector fields homogeneous of weight at most k + K - m, ?(jo) + mjo <k + K - m + 1. Hence, by (5.20),

(5.22) mK < ?(jo)-K + m < k-mjo + 1.

OF Let h(z, -z) be the degree d = K + k - mjo Taylor polynomial of Efijo(z) r (z -). By Lemma 5.15, h is a linear combination of the Taylor polynomials of degree d of terms of the form (5.16). The minimum power of z- in a term in (5.16) with j-jo > K is at leastj -jo > K. Set a(j) = k - mj+ 1. If j-jo < K, then by




(5.22), a(j) > a(jo) - mK > 0, so a(j) + j - jo > a(j) + j-jo Hence by (5.22), because w(X) = k, in a term in (5.16) with j -jo < K, the minimum power of z- is at least a(jo) > K. On the other hand, by the definition of h, the maximum power occurring in any term is at most the maximum power occurring in FK(Z,Z), SO at most K - 1. Thus h(z,z-) = 0 and (5.21) holds.

Proof of Lemma 5.4, continued. Suppose that for all j either Wj - 0 or w(Wj) > k. Then fi(z, w) = O(k + 2) for i = 1,.. .,n. Write

MI

g(O, w) = E cjw. j=MO

By (5.6), because w(Z) = k, C(k+m)/m 0 O. Equating terms of weight k + m in (5.9) gives

pC(k+m)/m(2iFK(Z z))(k+m)/m 0 mod O(k + m + 1),

which is impossible since C(k+m)/m /0. Hence, there is an i such that w(Wi) = k. By (5.14), w(Wi') = k - mi. By Lemma 5.19, ?(i) < C, since otherwise Wi' would be a mod K approximate holomorphic tangent to M.

Let

jo = min{j: cj 1 0 or Wj 1 0, ?(j) < t},

JO = min{j: Wj 0, w(Wj) = k},

.o = minj: cj /g0or Wj0O}

and

1 = max{j: cj /g0 or Wj - O}.

Then, .1 < jo < j. Becausefi = O(k+ 1) for all i, k+1 < (j)+mj, so if ?(j) < f

then k + 1 < f + mj. Also, because g = O(k + m), if cj / 0 then k + m < mj. Hence,

(5.23) k + f

jo. m

On the other hand, because the nonzero terms inf1 have weight at most k+K-m+ 1 and the nonzero terms in g have weight at most k + K,

(5.24) ? k+K m




Suppose

(5.25) K+2f +m < k.

By (5.23)-(5.25), we see that

(5.26) L = j-jo < K<jo-1. m

For Jo < j < jo, ?(j) > ?. Hence, by Lemma 5.19,

n &FK (5.27) Zfrj(z) a (z, Z) 0 mod O(k - mj + K + 1), 1 <Jo.

i=1 z

Taking the conjugate of (5.27), multiplying by wi and evaluating at -z= 0 gives

n &FK

(5.28) Z:frj(0)w' & K (z, 0)-0 mod O(k + K + 1), j <jo. i=1a Z

By (5.6), (5.7) and (5.28),

(5.29) g(z, w)- (2i _frj( O

(z 0) + cj) w1 mod O(k + K + 1).

Fix iv with 0 < iv < L. By (5.26) and (5.25), k - mv > 0. By (5.12), in equations

(5.5)-(5.9) we can replace X by X, and k by k - mv. Then (5.9), (5.10) and (5.29) give

J1 (n OFKN (5.30) &FK'Efr()~ZO) i(iK(,ZY,

- ( a Zr)aZ 2i J=Jo r=1 Zr

- bj,1,frj(z)(2iFK(z, z)y #) (Z, Z)

r=1 j -Oz

n &FK + ii! Zfr,v(Z) a) (z,z) mod O(k -mv + K + 1)

r=1

where

b - { (j-v)t Z-i B(2., te l 0, t >j.

By (5.26), zv < jo so, by (5.27), the last term in (5.30) is 0 mod 0(k - mv+K+ 1).




Now multiply (5.27) by (2iFK(z, -Z)y-' and substitute the result in (5.30). Multiply the result by (2iFK(z, z))" to obtain

(5.31) Ebj,v(2iFK(z, z)Y Z (frO(z) zFK 1=10

frj(0) K (Z 0)) - iCj) -E mod O(k + K + 1).

Let aj(z, z) denote the Taylor polynomial of degree k + K at the origin of the coefficient of bj,> in (5.31). Then aj is independent of iv and we can rewrite (5.31) as

1

E bj, ,,aj = O, zV = O, ... ., L. J=JO

This gives L+ 1 equations for the L+ 1 unknowns aj, j = jo,... , fI. The coefficient

matrix B has as its i,j entry

B

_=(1o +jI-

1)! (io +i-i)!

Then det B = (Ji -Jo)! (l -ljo - 1)! .. . 2! 1! ([M, pp. 322-323], [MM, p. 679]) so B is nonsingular. It follows that aj = 0 for all j. Hence, by (5.31), for all

J =Jo, * * ,J

(5.32) E tj(z) ,) (z, z)-frj(O) ar (z, 0)) --cj (5.32) ~r=1 r Zr&K

=OmodO(k-mj+K+. 1).

Equating terms in (5.32) which are divisible by some positive power of z, we see that for all j > jo

(5.33) aK (Z, ) Omod O(k-mj + K + 1).

Now, w(Wjl) = k, so w(WJ,) = k - mj'. By (5.33), because j' > jo, WI, is

a nontrivial modK approximate holomorphic tangent to M. Because M has no mod K holomorphic approximate tangents, we conclude that if X g 0 E holK (M), w(X) < K + 2f + m = (2m + 3)K - m. This proves the lemma.

Proof of Theorem 4.16. By Remark 4.2, if hol (M) is finite dimensional then M has no holomorphic tangent. Suppose M has no holomorphic tangent. We




can assume that M is given by an equation of the form (1.1) where F has no pure terms. By Proposition 4.9, there is an n-tuple A = (a, ... a, an) such that detJA(z) - 0. Hence, for some K,

(t fy -a' f(tna (5.34) det Iz j 'j) -z ...z detJA(z) 0 OmodO(K).

If MK is given by Im w = ZafK,,(z)z-, then by (5.34)

(5.35) det z z ... Z detJA(z) - O mod O(K).

Suppose Z = EZ1=j hi(z)70y is a modK approximate holomorphic tangent to M with k = w(Z). Then for all ae

(5.36) Z hj(z) (z)z O mod 0(k + K + 1). j=i j

Let

C(z, z) = (Cij(z, z)) - ( O i f (z)\

and let H(z) be the n x 1 matrix with jth row hj(z). By (5.36)

(5.37) C(z, z-)H(z) Omod 0(k + K + 1).

If D is the classical adjoint of C then by (5.37)

(5.38) D(z, -z)C(z, -z)H(z) = (det C(z, -z)) H(z)

- mod 0(k + K + 1).

By (5.35), det C(z, z-) - 0 mod O(K) so by (5.38), hi(z) 0_ mod 0(k + 2) for i =

1, . . . , n. However, because w(Z) = k, hi(z) 7/ 0(k + 2) for some i. Thus, M has no mod K approximate holomorphic tangent and, by Lemma 5.4, dim ( holK (M)) < oc. Thus, by Lemma 5.2, hol (M) is finite dimensional.

6. Rothschild's question and sufficient conditions for aut(M) = hol(M). We can now answer Rothschild's question and show that aut (M) = hol (M) if the hypersurface is essentially finite or the Levi form is not everywhere degenerate.




THEOREM 6.1. Let M be an analytic real hypersurface through the origin in Cn+1. Suppose that one of the following holds.

(1) M is essentiallyfinite;

(2) M is rigid and every neighborhood U of 0 contains a point p E M such that the Levi form of M is nondegenerate at p.

Then aut (M) is finite dimensional and aut (M) = hol (M).

Proof. In case (1), the result follows from Theorems 4.3 and 4.16 and [S3, Remark 2.5]. In case (2), it follows from [H, Lemma 6] and Theorem 3.1.

THEOREM 6.2. Let M be a rigid, analytic, holomorphically nondegenerate real hypersurface through the origin in Cn+l given by an equation of the form

Imw =p(z,z-) + O(m + 1)

where p 0 0 is a homogeneous polynomial of degree m having no pure terms. Then M is homogeneous if there is an approximate infinitesimal dilation Y E hol (M).

Conversely, if M is homogeneous, then there is an approximate infinitesimal dilation Y E aut (M).

Proof. The theorem follows immediately from Theorems 2.1 and 4.16 and the remarks at the beginning of Section 2.

Remark. By Theorem 4.3, Theorem 6.2 applies if M is rigid, analytic and either essentially finite or somewhere Levi nondegenerate. In these cases, by Theorem 6.1 the approximate infinitesimal dilation is in hol (M).

7. Examples. In this section, we give some examples to show how various conditions on hypersurfaces are related. The first example gives the simplest way there can be a nontrivial holomorphic vector field tangent to M.

Example 7.1. Let M be an analytic real hypersurface in Cn+. Suppose M is given by a defining equation which is independent of one of the variables,

Im w = F(z', -z', Re w)

where z' = (ZI, Zn- 1). Then W =9 is tangent to M. Hence, by Remark 4.2, dim( hol (M)) = oo.

The next example shows that the existence of a holomorphic vector field tangent to M is not equivalent to the existence of a defining function which is independent of one of the variables.




Example 7.2. Let M be the rigid hypersurface

Imw = JZIZ2 12

in C3. Then the holomorphic vector field Z = zi - Z2 is tangent to M. 9Zi 0Z2

However, there are no coordinates such that the equation of M is independent of one coordinate in a neighborhood of the origin.

In Example 7.2, although the equation of M is not independent of one vari- able, M contains the variety V = {zi = 0, w = 0}. The next example shows that the existence of a variety contained in M does not guarantee the existence of a holomorphic vector field tangent to M, even in the homogeneous case.

Example 7.3. Let M be the hypersurface

Imw = IZ, 14 + |Z1Z212

in C3. Then M is homogeneous and contains the variety V = {zi = 0, w = O}. The Levi form of M is not everywhere degenerate. Hence, by Theorem 3.1, hol (M) is finite dimensional. By Remark 4.2, there is no nontrivial holomorphic vector field tangent to M.

Example 7.3 is of finite type. However, it is not essentially finite. The next example shows that finite type is not a necessary condition for finite dimensionality of hol (M).


lmw = Re wIz12

in C" 1. Then the origin is a point of infinite type, and M contains the variety w = 0. The Levi form is nondegenerate in every neighborhood of the origin, so, by Theorem 3.1, dim(hol(M)) < (n + 2)2 _ 1. By the Remark 4.2, M is holomorphically nondegenerate.

The next example is a hypersurface M which is essentially finite, but its homogeneous part Mo is not. In addition, the Levi form of M is not everywhere degenerate, but the Levi form of Mo is.


Imw = IZ, | + JZ214

in C3. Then M is essentially finite and the Levi form is not everywhere degenerate.




Its homogeneous part Mo, given by Im w = Iz 12, is not essentially finite and has everywhere degenerate Levi form. By Example 7.1 dim(hol(Mo)) = 00. However, by Theorem 3.1, dim ( hol (M)) < oc.

The proof of Theorem 4.16 is quite complicated. However, in the case that the Levi form is not everywhere degenerate, finite dimensionality of hol(M) is easy. The next example shows that there are essentially finite rigid hypersurfaces with everywhere degenerate Levi form.


Im w = ((z + (Z2 + Z2

-((Zl + 1) + Z2) -(Z + +Z2) +

in C3. This example is equivalent to a tube. It is obtained by a change of coordinates from Freeman's example {(zi, Z2, Z3) : X3+ x + x3 = O} C C3 \ {O} of a real hypersurface which admits a complex foliation but no local straightening [F]. The Levi form of M is degenerate at every point. Thus, finite dimensionality of hol (M) does not follow from Theorem 3.1. Let V denote the set of z E C2 such that

(7.7) (ZI + (I + 1)3 + (Z2 + (2)3) 1/3

((Zl + 1)3 + Z)/3- l + 1)3 + 1)/3+I=

for all small ((, (2). By [BJT, Proposition 4.6], to show that M is essentially finite it suffices to show that the intersection of V with a small neighborhood of the origin is {0}. Suppose z E V. Set (2 = 0, (I = 4 in (7.7). The equation becomes

(7.8) (ZI +(+ 1)3 +Z = (((ZI + 1)3 +Z +

The coefficient of (2 on the left side of (7.8) is 3(zi + 1). On the right side it is 3 ((zi + 1)3 + Z3) 1/3. These must be equal, so Z2 = 0. Now set Z2 = 0, (I = 0 and

= ( in (7.7). This gives




If zl :/ 0, expanding (7.9) yields

(7.10) Zi + 1 = ((3 + 1)1/3 (Zi + (W3 + 1)1/3)

For (./0 the solution of (7.10) is

Zi = -(3 + 1)1/3-1

Thus there is no nonzero solution z of (7.7) which holds for all small (. Hence M is essentially finite.

The final example shows that in general aut (M) :' hol (M).

Example 7.11. Let M be the flat hypersurface

Imw =0

in C"'. Let

X=e1-/u2 (1a + & ) 1/U2& &w a w Au

when u :' 0 and X = 0 when u = 0. Then X is tangent to M. The one parameter group generated by X is of the form

(z, u) - (z, 0(t, u))

and hence is a one parameter group of CR automorphisms. Thus, X E aut (M) but X , hol (M).

In Example 7.11, both aut (M) and hol (M) are infinite dimensional.

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF NOTRE DAME, NOTRE DAME, IN 46556 Electronic mail: [email protected]

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