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Ohio State University Mathematical Research Institute Publications 9
Editors: Gregory R. Baker, Walter D. Neumann, Karl Rubin

Ohio State University
Mathematical Research Institute Publications
1 Topology '90, B. Apanasov, W. D. Neumann, A. W. Reid,
L. Siebenmann (Eds.)
2 The Arithmetic of Function Fields, D. Goss, D. R. Hayes,
M. I. Rosen (Eds.)
3 Geometric Group Theory, R. Charney, M. Davis, M. Shapiro (Eds.)
4 Groups, Difference Sets, and the Monster, K. T. Arasu, J. F. Dillon,
K. Harada, S. Sehgal, R. Solomon (Eds.)
5 Convergence in Ergodic Theory and Probability, V. Bergelson,
R March, J. Rosenblatt (Eds.)
6 Representation Theory of Finite Groups, R Solomon (Ed.)
7 The Monster and Lie Algebras, J. Ferrar, K. Harada (Eds.)
8 Groups and Computation III, W. M. Kantor, A. Seress (Eds.)

Complex Analysis and Geometry
Proceedings of a Conference
at the Ohio State University
June 3-6, 1999
Editor
Jeffery D. McNeal
W
DE
Walter de Gruyter • Berlin • New York 2001

Editor
Jeffery D. McNeal
Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA
Series Editors
Gregory R. Baker
Department of Mathematics, The Ohio State University, Columbus, OH 43210-1174, USA
Karl Rubin
Department of Mathematics, Stanford University, Stanford, CA 94305-2125, USA
Walter D. Neumann
Department of Mathematics, Columbia University, New York, NY 10027, USA
Mathematics Subject Classification 2000: 32-06; 32A07, 32A10, 32A40, 32G08, 32G15,
32H02, 32H35, 32T25, 32T27, 32W05
Keywords: Pseudoconvex domains, Proper mappings, d-Neumann problem, The d-problem,
hypoellipticity, Levi-flat hypersurfaces
© Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability.
Library of Congress — Cataloging-in-Publication Data
Complex analysis and geometry : proceedings of a conference
at the Ohio State University, June 3—6, 1999
/ editor, Jeffery
D. McNeal.
p. cm. — (Ohio State University Mathematical
Research Institute publications ; 9)
ISBN 311016809X (alk. paper)
1. Mathematical analysis — Congresses. 2. Functions
of complex variables - Congresses. 3. Geometry - Con-
gresses. I. McNeal, Jeffery D. II. Series.
QA299.6 .C6577 2001
515-dc21
2001047181
Die Deutsche Bibliothek —
Cataloging-in-Publication Data
Complex analysis and geometry : proceedings of a conference
at the Ohio State University, June 3-6, 1999 / ed. Jeffery D.
McNeal. — Berlin ; New York : de Gruyter, 2001
(Ohio State University Mathematical Research Institute
publications ; 9)
ISBN 3-11-016809-X
© Copyright 2001 by Walter de Gruyter GmbH &
Co. KG, 10785 Berlin.
All rights reserved, including those of translation into foreign languages. No part of this book may be
reproduced in any form or by any means, electronic or mechanical, including photocopy, recording, or
any information storage and retrieval system, without permission in writing from the publisher.
Printed in Germany.
Cover design: Thomas Bonnie, Hamburg.
Typeset using
the authors' TEX files: I. Zimmermann, Freiburg.
Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen.

Preface
A conference on Complex Analysis and Geometry was held at the Ohio State University
from June 3 to June 6, 1999. This volume contains ten articles written by some of the
principal speakers at this conference.
The conference was an exciting event and showcased some of the new ideas which
continue to fuel the strong interaction between analysis and geometry in several com-
plex variables. The articles are mostly oriented toward researchers in the field, but
the authors were invited to include more expository and illustrative material in their
papers than they might include in a normal journal article. It is hoped, therefore, that
the volume will also serve as an introduction to students to some of the active areas
in complex analysis.
Besides the contributors to this volume, I would like to thank
the many mathe-
maticians who attended the conference and made it such a stimulating event. I also
gratefully acknowledge the support of the National Science Foundation and the Math-
ematical Research Institute at Ohio State, which helped make the conference possible.
Finally, I thank Manfred Karbe and Annette Kolbl of Walter de Gruyter & Co. for their
professional and patient help in bringing this book to completion.
J.
D. McNeal

Table of contents
Preface v
M. S. Baouendi, Linda Preiss Rothschild and Dmitri Zaitsev
Points in general position in real-analytic submanifolds in
and applications 1
David E. Barrett
Holomorphic motion of circles through affine bundles 21
Bo Berndtsson
Weighted estimates for the 3-equation 43
Michael Christ
Hypoellipticity in the infinitely degenerate regime 59
Michael Christ
Spiraling and nonhypoellipticity 85
John P. D'Angelo
Positivity conditions for real-analytic functions 103
Peter Ebenfelt and Xiaojun Huang
On a generalized reflection principle in C2 125
Siqi Fu and Emil J. Straube
Compactness in the 3-Neumann problem 141
Joseph J. Kohn
Hypoellipticity at
non-subelliptic points 161
Yum-Tong Siu
Very ampleness part of Fujita's conjecture and multiplier ideal
sheaves of Kohn and Nadel 171

Points in general position in real-analytic submanifolds
in C^ and applications
M. S. Baouendi, Linda Preiss Rothschild and Dmitri Zaitsev
1. Introduction
Two pairs (M, p) and (M', p') of germs of real (locally closed) submanifolds M, M' c
C^ at distinguished points p e M and p' e M' are said to be biholomorphically
equivalent (or just
equivalent for short) if there is a biholomorphic map H between
open neighborhoods of p and p' in CN sending p to p' and mapping a neighborhood of
p
in M onto a neighborhood of p' in M'. We write (M, p) ~ (M', p') for equivalent
pairs and//: (CN, p) —>
(CN, p') for a map between open neighborhoods of p and
p' in CN sending p to p'.
It is easy to construct germs of smooth real curves in C that are not equivalent. In
contrast, any two germs of real-analytic curves in C at arbitrary distinguished points
are always equivalent, since any real-analytic diffeomorphism between them extends
to a biholomorphism between some open neighborhoods in C.
The simplest example of non-equivalent real-analytic submanifolds of the same
dimension is given by (C, 0) and (M2, 0) both linearly embedded in C2 in the standard
way. More generally, it is easy to see that two germs at 0 of real linear subspaces of
CN are equivalent if and only if they can be transformed into each other by a complex
linear automorphism of C^.
In this paper we give a local description of a real-analytic submanifold M c CN at a
"general" point (see Theorem 2.5 below). This description is based on various notions
of nondegeneracy and is of interest in its own right. An important application is that
at a "general" point p e M,
the germ (M, p) is equivalent to another germ (M', p') if
and only if (M, p) and (M', p') are "formally" equivalent (see Theorem 6.1 below.)
This result is the main theorem in [BRZ 2000]. (It should be noted that there exist
pairs (M, p) and (M', p') which are "formally" equivalent, but not biholomorphically
equivalent; see §4.1.) We also address here the case of real-algebraic submanifolds and
their algebraic equivalences, which was not studied in [BRZ 2000]. (See Theorem 9.1
below.)
We mention briefly that the study of biholomorphic equivalence of real submani-
folds in CN goes back to Poincaré [P 1907] and E. Cartan [Ca 1932a], [Ca 1932b],
and [Ca 1937]. In their celebrated work Chern and Moser [CM 1974] solved the
equivalence problem for germs of Levi nondegenerate real-analytic hypersurfaces in
Complex Analysis and Geometry
Ohio State Univ. Math. Res. Inst. Publ. 9 © Walter de Gruyter 2001

2 M. S. Baouendi, Linda Preiss Rothschild and Dmitri Zaitsev
CN and showed in particular that in this context the notions of formal and biholomor-
phic equivalence coincide. We will mention more recent work related to Theorem 6.1
later in this article.
2. Structure decomposition results for points in general position
If M is a connected real-analytic submanifold of CN, we say that a property holds for
p e M in general position if it holds for all p outside a proper real-analytic subvariety
of M. A number associated to M is said to be a biholomorphic invariant if it is
preserved by biholomorphic equivalences of germs of M at any point in M.
2.1. Generic and CR submanifolds. A smooth real submanifold M c CN is called
generic (or generating, in some translations) if TpM + JTpM — TpCN for all p e M,
where J: TCN —>• TCN denotes the standard complex structure of CN, and TpM
denotes the (real) tangent space of M at p. More generally, if the space TqM + JTqM
has constant dimension for q near p,M is said to be CR at p (or p is a CR point of M).
If M is CR at every point, it is said to be a CR submanifold. (For CR manifolds, the
reader is referred e.g. to the books [J 1990], [Bo 1991], [Ch 1991], [BER 1999a].)
If M is real-analytic, the set of all non CR points of M is a nowhere dense proper
real-analytic subvariety of M.
Examples 2.1. In C all nontrivial smooth submanifolds are generic. More generally,
the graph of any (smooth) map between open sets in C" x RN~n (0 < n < N) and
iRN~n is generic. For N >2, complex submanifolds of CN of positive codimension
and their real submanifolds are never generic. The submanifold
M :={w= \z\2} CC2,
where (z, w) are taken as coordinates in C2 , is generic and CR everywhere except at
the origin in C2.
The role of generic points is illustrated by the following property.
Proposition 2.1. If M c CN is a connected real-analytic submanifold, there exists
an integer 0 < r\ < N such that for p e M in general position
(M, p) ~ (Mi x {0}, 0), Mi x {0} C C"-''1 x C1, (2.1)
where M\ c CN~ri is a generic real-analytic submanifold through 0. The number r
with this property is unique and is a biholomorphic invariant.
Remark 2.2. The points p for which the conclusion of Proposition 2.1 holds are in
fact the CR points of M.

Points in general position in real-analytic submanifolds in CN and applications 3
The number r\ is called the excess codimension of M (cf. [BRZ 2000], §2). It
is equal to the maximal codimension of a complex submanifold of CN containing an
open subset of M. For p e M in general position, M is CR at p and there exists
a complex submanifold of CN of codimension r\ that contains a neighborhood of p
in M. This complex submanifold is unique in the sense of germs and is
called the
intrinsic complexification of M at p.
2.2. Finite and minimum degeneracy. Finite nondegeneracy is a higher order gener-
alization of Levi nondegeneracy. For a smooth CR submanifold M c we denote
by TCM the real subbundle given by TpM = {X e TpM : JX e TpM}. We consider
the (0, 1) vector fields on M, i.e., the sections of the subbundle
T0AM := {X + iJX : X e TCM} c TCM <g> C.
Then M is Levi nondegenerate at p if for any (0, 1) vector field L with L(p) ^ 0,
there exists a (0, 1) vector field L\ such that
[Li,!](/>) £TcpM®£.
This condition is equivalent to the nondegeneracy of the Levi form defined as the
(unique) hermitian form
JLP: Tf1M x :1M (TpM/ TcpM) ® C (2.2)
satisfying
Xp(Ll(p),L(p)) = ^ji[Ll,L](p) (2.3)
21
for all (0, 1) vector fields L, Lu where n: TM ®C (TM/TCM) (8) C is the
canonical projection.
The more general concept of finite nondegeneracy can be defined in a similar way
as follows.
Definition 2.1. A smooth CR submanifold M c
is called finitely nondegenerate
at p if there exists I > 0 such that for any (0, 1) vector field L on M with L(p) ^ 0,
there are (0, 1)
vector fields on M, L\,..., L*, 0 < k < I, such that
[Li,..., [Lk, L]... ](p) ? T^M <8 C. (2.4)
If I is minimal with this property, M is called I-nondegenerate at p.
Remark 2.3. It follows that M is 1-nondegenerate if and only if it is Levi-nondege-
nerate. Also M is 0-nondegenerate if and only if it is totally real, i.e., TpM = {0}.
To check Definition 2.1 it suffices to assume that L, L\,..., in (2.4) are all taken
from a fixed local basis of (0, 1) vector fields on M near p.

4 M. S. Baouendi, Linda Preiss Rothschild and Dmitri Zaitsev
The condition of finite nondegeneracy was given in [BHR 1996] for hypersurfaces
and can be found in [BER 1999a] for CR submanifolds of higher codimension. The
formulation given in Definition 2.1 is equivalent to that in the reference above (see § 3
or Proposition 1.24 in [E 2000]). In
[BRZ 2000] this notion was extended to arbitrary
real-analytic submanifolds. As in the case of the Levi form, higher order tensors can
be used to give an alternative definition of finite nondegeneracy (see §3).
Example 2.4. Let M c C2 be a real-analytic hypersurface through 0 given by Im w —
</>(z, z, Re to) where 0 is a real-valued real-analytic function defined near the origin
inCxR satisfying <p(z, 0, 0) = 0. Then M is finitely nondegenerate at 0 if and only
if at least one of the partial derivatives <pz ^(0) (1 < k < oo) does not vanish. If k is
the smallest such integer, then M is ^-nondegenerate at 0.
The role of finite nondegeneracy is illustrated by the following property.
Proposition 2.2. If M C CN is a real-analytic submanifold, then there exists an
integer 0 < ri < N such that for p e M in general position
(M, p) ~ (M2 x C2,0), M2 x {0} C CN~r2 x C2, (2.5)
where M2 C CN~ri is a real-analytic CR submanifold finitely nondegenerate at 0.
The number r2 with this property is unique and is a biholomorphic invariant.
The number r2 is called the degeneracy of M. It is equal to the maximal dimension
of the leaves of a holomorphic foliation in a neighborhood of a point p e M such
that a neighborhood of p in M is saturated (i.e., is a union of leaves) (see [F 1977]).
The points p e M for which the conclusion of Proposition 2.2 holds are said to be of
minimum degeneracy in M (see [BRZ 2000], §2).
Example 2.5. The hypersurface M c C3 given by Im w = \ziz2\2 in the coordinates
(zi, z2, w) is of minimum degeneracy 1 outside the plane H := {(0, 0)} x E. For
p i H, (M, p) is equivalent to (Mi x C, 0) with M\ := {(z, u))eC2:Imu) = |z|2}.
2.3. Finite type and CR orbits. A smooth CR submanifold M c CN is said to be
of finite type at p (in the sense of Kohn [K 1972] and Bloom-Graham [BG 1977]) if
all (0, 1) vector fields on M and their conjugates, together with all their higher order
commutators span the space TpM (8) C. For general CR submanifolds, the condition
of finite nondegeneracy and that of finite type are independent, i.e., one condition
does not necessarily imply the other. (See Examples 2.6 and 2.7.) However, for
hypersurfaces finite nondegeneracy at a point
implies finite type at that point.
Example 2.6. The hypersurface M c C2 given in Example 2.4 is of finite type at 0
if and only if at least one of the partial derivatives (0), 1 < k, I < oo, does not

Points in general position in real-analytic submanifolds in C^ and applications 5
vanish. Hence, for instance, the hypersurface given by
M = {(z, w) e C x C : Im w = |z|4}
is of finite type at 0, but is not finitely nondegenerate at 0.
Example 2.7. For any Hermitian form (, >: C x C C2 the quadric
M: ={(z,w)eCxC2:lmw = (z,z)}
is not of finite type at any point. If, in addition, the vector-valued Hermitian form
(z, z) does not vanish identically, then M is finitely nondegenerate at every point.
Proposition 2.3. If M c CN is a connected real-analytic submanifold, then there
exists an integer 0 < r3 < N such that for p e M in general position
(M, p) ~
(M3, 0), M3 C CN~n x IT3, (2.6)
with(0,u) e Mt, for u e sufficiently small, and for such fixed u, MT,C(<Cn ~r3 x {«))
is a CR submanifold of finite type at (0, u). The number r3 with this property is unique
and
is a biholomorphic invariant.
The number r3 is called the orbit codimension. The CR orbit of a point p e M
is the germ at p of a (real-analytic) submanifold of M through p of smallest possible
dimension to which all the (0, 1) vector fields on M are tangent. The number is
equal to the minimal codimension of any CR orbit. It is also equal to the maximal
number r such that there exists a holomorphic submersion of a neighborhood of a
point p e M in CN onto C sending M into the real part W. (See [BER 1996]
for the discussion in the algebraic case.) The points p for which the conclusion of
Proposition 2.2 holds are said to be of minimum orbit codimension (see [BRZ 2000],
§2). The germ of the submanifold M3 n (C3 x {«}) at (0, u) is the CR orbit of M3 at
this point.
In contrast to the situation of Proposition 2.2, in general the submanifold M3 cannot
be "flattened", i.e., written as M3 x with M^ c CN~n a real-analytic submani-
fold. One obstruction to such a "flattening" is the possible nonequivalence of the CR
orbits M3 n (C^-'3 x {M}) for different u e Rr3. In order to give an example of such a
submanifold that cannot be "flattened" at any point, we introduce some preliminaries.
Consider for any real number u the quadric Mu c C6 of codimension 2 defined
in the coordinates (z, w) € C4 x C2 by Im w — hu(z, z), where hu is the C2-valued
Hermitian form
hu(z, x) •= (ziXl + Z3X3 + 24X4' Z2X2 + 23X3 + "^4X4)- (2-7)
The following lemma will be used to construct our example.
Lemma 2.4. For any u, u' e M with u, u' > 1 the Hermitian forms hu and hu' are
equivalent if and only ifu = u'. That is, if there exist a 2x2 invertible real matrix B

6
M. S. Baouendi, Linda Preiss Rothschild and Dmitri Zaitsev
and
a 3x3 complex invertible matrix A such that
Bhu>(z, X) = hu(Az, AX) for all z, X e C4, (2.8)
then u — u'.
Proof Let [e\,..., be the standard basis of C4. We claim first that if {uj,..., 114}
is a basis of C4 for which hu(vj, Vk) = 0, ; ^ k, then there is a permutation a
of {1, 2, 3, 4} and complex numbers kj 0 such that vj = Xjea(j), j = I,... ,4.
To
prove the claim, let </>„ be the positive definite (scalar) Hermitian form given by
<Pu(z, x) = hl(z, x) + hl(z, x)> where h\,hare the components of hu. Then the
4x4 diagonal matrix D with diagonal entries (1,0, 1 /2, 1 /(I + «)) satisfies
<pu(Dz, x) = hlu(z, x) for all z, X e C4. (2.9)
Hence <pu(Dvj, v^) = 0, j / k, and since <f>u is positive definite, it follows that
Dvj — IjVj, for some lj e C, j = 1,..., 4. In particular, since D is diagonal and its
eigenvalues are distinct, any eigenvector of D is a nonzero multiple of one of the ej.
Since the Vj are all eigenvectors of D, the claim is proved.
Assume that (2.8) holds. Since hu>(ej, e^) = 0, 7 / k, we may apply the claim
to the vectors vj — Aej and conclude that Aej = Xjea(j), j = 1, ..., 4. From
this and (2.8) applied to z = x = ej> we have Bhu>(ej, e,) = \Xj\2hu(eaU), ea(j)),
j = 1, ..., 4. Hence the real 2x2 invertible matrix B maps each vector in the
set Su>
:= {(1, 0), (1, 1), (1, u'), (0, 1)} into a positive multiple of a vector in the set
Su := {(1,0), (1, 1), (1, m), (0, 1)}. Since each of the vectors (1, 1) and (1, u') is a
linear combination with positive coefficients of the vectors (1,0) and (0, 1), the
same
is true of their images under the linear map B. It follows that each of the vectors (1,0)
and (0, 1) is mapped into a positive multiple of one of the same two vectors. Similar
reasoning shows that up to positive scalar multiples B sends the vectors in the set Su>
into those of Su either by preserving the order of these vectors or reversing it. In either
case, a simple calculation shows that necessarily u = u'. •
As a consequence of Lemma 2.4, it follows that for any u, u' e R, u, u' > 1 and
p
e Mu, p' e Mui the germs (Mu, p) and (Mu>, p') are equivalent if and only if
u
— u'. Indeed since (Mu, p) ~ (Mu, 0) for any p e Mu and any u G M, it suffices
to assume p = p' = 0. If (Mu, 0) and (Mu<, 0) are (biholomorphically) equivalent,
their Levi forms hu and hu> are linearly equivalent, i.e., they must satisfy (2.8). Hence
u
= u' by the lemma. Moreover, for any uel and any p € Mu, the reader can easily
check that Mu is of finite type at p. We may
now give an example of a manifold which
cannot be "flattened" at any point, as announced above.
Example 2.8. Let hu(z, z) be defined by (2.7). Consider the generic submanifold
M cC7 of codimension 3 given in the coordinates (z, w, u) e C4 x C2 x C by
lm«=0, Im w = hu(z, z).

Points in general position in real-analytic submanifolds in C^ and applications 7
Then the CR orbits of M are M n (C6 x {u}),u el By the observation preceding
this example, the CR orbits of M are pairwise nonequivalent for u > 1. Hence, for
any q = (p, u) e M, with p e Mu, u > 1, (M,
q) is not equivalent to any product
(M'
x R, 0) with M' C C6 a CR submanifold of finite type. Indeed, the CR orbit of
(0, u') in M' x R is M' x {u1} and hence such orbits are equivalent to each other for
different values of u'.
2.4. A structure result for points in general position. Putting Propositions 2.1, 2.2
and 2.3 together we obtain:
Theorem 2.5. Let M c C^ be a connected real-analytic submanifold. Then there
exist integers 0 < r\, ri, rj < N, such that for p e M in general position
(.M, p) ~ (M x C2 x {0}, 0), MxC2x {0} c (C x x C2 x C1, (2.10)
where r := N — r\ — r2 — r-t,, M C C x Rr3 is a finitely nondegenerate generic
submanifold of C+r3 through 0 such that for u e Rr3 near 0, the point (0, u) is in M
and
M fl (Cr x {«}) is a CR submanifold of finite type at (0, u).
2.5. The hypersurface case. In the case M c CN is a hypersurface, Theorem 2.5 can
be reformulated in a simpler form. First, M is generic in C^, so that r\ = 0. Also, as
mentioned in §2.3, for N > 2, any hypersurface which is finitely nondegenerate at 0
is necessarily of finite type at 0, so that r3 = 0. Hence we have
Corollary 2.6. LetM c CN be a connected real-analytic hypersurface. Then exactly
one of the following alternatives holds.
(a) There exists an integer 0 < r2 < N —
2 such that for all p e M in general
position, (M, p) ~ (M x C2, 0), where M c CN~n is a finitely nondegenerate
hypersurface (and hence of finite type) through 0.
(b) For all p e M in general position, (M, p) ~ (R x C^-1, 0).
3. Finite nondegeneracy and higher order tensors for CR
manifolds
Let M c CN be a smooth CR submanifold, and p e M. The following construction
generalizes the Levi form as given by (2.2). For s > 1, consider the linear subspace
Fp c Tp'1 M of all possible values L(p) of a (0, 1) vector field L such that (2.4) fails

8 M. S. Baouendi, Linda Preiss Rothschild and Dmitri Zaitsev
to hold for all k < s and all (0, 1) vector fields L\,... ,Lk. We obtain a decreasing
sequence of subspaces
T^M = Flp D • • • D Fsp d • • • •
Then there exists a unique multilinear map
Xsp: (T°'lM)s {TpM/TcpM)® C
satisfying
Xsp(Li(p),..., Ls(p), L(p)) = it[L\,..., [Ls, I]... ](/>), (3.1)
for all (0, 1) vector fields L, L\,..., Ls with Hp) e Fp, where n
is as in (2.3).
Indeed, it follows from the construction of the subspaces Fp that the right-hand side
of (3.1) is multilinear in its arguments over the ring of smooth complex functions. The
tensor Xsp is complex-linear in the s first arguments and antilinear in the last one. For
s — 1 we obtain a multiple of the Levi form; i.e., Xp(X\, X2) = 2iXp(X\, X2).
The tensors analogous to Xp were introduced by Ebenfelt [E 1998], For s = 2,
the definition of X2p is due to Webster [W 1995] (in the case M is a hypersurface),
where X2 is called the cubic form. The subspaces Fp and the tensors Xsp are also
related to the submodules Ns, 1 < s < 00, of the C°°-module of all (0„ 1) vector
fields defined by Freeman [F 1977] inductively as follows. Let N\ be the module of
all (0, 1) vector fields on M. If Ns-\ is defined, let Ns C Ns-\ be the C°°-submodule
consisting of all (0, 1) vector fields L such that (2.4) fails to hold for all k < s, all
(0, 1) vector fields L\,..., Lk and all p e M. (Actually, Freeman's submodules are
the conjugates of the Ns's defined here.) If all subspaces F* have constant dimension
for q in a neighborhood of p, Ns consists precisely of their sections. However, in
general, a subspace Fp may be
nontrivial even if Ns = 0. For instance, this happens
for s = 2 if the Levi form Xp is nondegenerate for p in general position but degenerate
at
some point.
Example 3.1. The hypersurface M C C2 given by Im w = \z\4 is finitely degenerate
on {0} x K and finitely nondegenerate outside this line. For p = Owe have Fq =
TQ
'1M for all s > 1. On the other hand, the submodules Ns are all zero for s >2 and
thus "don't notice" the degeneracy.
We have the following characterization of fc-nondegeneracy:
Proposition 3.1. If M c is a smooth CR submanifold and p e M, then the
following are equivalent
for k > 1:
(i) M is k-nondegenerate at p,
(ii) k is the smallest integer for which Fp+l = 0,

Points in general position in real-analytic submanifolds in CN and applications 9
(iii) Fp ^ {0} and for any X € F*, X £ 0, there exist X\,...,Xk e Tp'1M with
£kp(Xu...,Xk,X)^0.
We now give an equivalent definition of finite nondegeneracy in terms of a defining
function of M. If M c CN is a smooth CR submanifold of codimension d and p e M,
a (smooth) defining function for M near p is a real smooth map p = (p1,..., pd)
of rank d defined in a neighborhood of p in CN such that M is locally defined near
p
by p(Z, Z) = 0. We write pJz = (pJZt,..., PJZN) f°r the complex gradient of pK
1
< j <d. We have the following.
Proposition 3.2. Let M c CN be a smooth CR submanifold and p a smooth defining
function of M near p e M. Then for an integer k > 0, the following are equivalent.
(i) M is k-nondegenerate at p.
(ii) The integer k is minimal such that the collection of vectors (L\ ... LspJz) (p, p)
span CN for 1 <j<d,0<s<k, and all choices of (0, 1) vector fields
L\,... ,LS on M.
As mentioned in Remark 2.3 above, ^-nondegeneracy at p can be checked using
any local basis of (0, 1) vector fields on M. Also, one can directly check that condition
(ii) of Proposition 3.2 does not depend on the choice of the defining function p and
that in this condition, if suffices to take the vector fields L \,..., Ls from a fixed local
basis of (0, 1) vector fields on M near p. To check the equivalence of (i) and (ii)
in the case of a generic submanifold, the reader can take "normal" coordinates as in
[BER 1999a] and do the calculation using the basis of vector fields given by (11.2.18)
in [BER 1999a] and the defining function p given there. (A related calculation is
done in the reference cited here.)
4. Different notions of equivalences for real submanifolds
4.1. Formal equivalence. In practice, biholomorphic equivalence of two germs of
real submanifolds in can be hard to check. For instance, in the work of Chern-
Moser-[CM 1974] the so-called formal equivalence is established first and then the
convergence of the formal map is proved, yielding biholomorphic equivalence.
Definition 4.1. A formal equivalence between two germs, (M, p) and (M', p'), of
real-analytic submanifolds in
CN is an invertible formal power series map
H(Z) = p' + J2 a„(Z - p)a, aae <C\ Z = (Zlt..., ZN), (4.1)
l«l>i

10 M. S. Baouendi, Linda Preiss Rothschild and Dmitri Zaitsev
sending M into M' in the "formal
sense", i.e., satisfying
p'(H(Z(x)),H(Z(x))}=0
(4.2)
for some real-analytic parametrization x Z(x) of M near p = Z(0) and some
real-analytic defining function p'{Z, Z) of M' near p'.
Equality in (4.2) is understood in the sense of formal power series in x. The
formal map H given by (4.1) is invertible if the vectors {dH/dZj(p), 1 < j < N] are
linearly independent in It is not hard to see that if (4.2) holds for some choice of
parametrization Z(x) of M and defining function p' of M', then it holds for any other
choice. We shall say that the germs (M, p) and (M', p') informally equivalent if
there exists a formal equivalence between them. Formal equivalence of (M, p) and
{M', p') translates into the existence of solutions for an infinite system of polynomial
equations which the coefficients of H must satisfy. Formal equivalence is often easier
to check than biholomorphic equivalence. One of the main results in [BRZ 2000]
(see Theorem 6.1 below) states that for points in general position these two notions of
equivalence coincide. Moreover, it is shown that any given formal equivalence can be
"corrected" to a convergent one without changing terms of a prescribed finite order.
The assumption that the point p be in general position cannot be dropped. Indeed,
in C2 there exist a pair of germs of formally, but not biholomorphically, equivalent
2-dimensional real-analytic (non CR) submanifolds. Moser and Webster proved in
[MW 1983], Proposition 6.1, that no neighborhood of 0 in the 2-dimensional (non
CR) submanifold McC2 given
by
w = \z\2 + yz2 + yz3z
can be biholomorphically transformed into the hyperplane C x R for y > 1/2. On
the other hand, they show that M is formally equivalent to a 2-dimensional real-
analytic submanifold contained in
C x R provided y is not exceptional, i.e., if
(1/jr) arccos(l/2y) is not a rational number. The authors of the present paper are
not aware of any example of pairs of germs of real-analytic CR submanifolds that are
formally but not biholomorphically equivalent.
4.2. CR equivalence and ¿-equivalence. A CR function on a smooth CR submani-
fold M c CN is a smooth (C°°) complex-valued function defined on M satisfying
the Cauchy-Riemann equations restricted to M. More precisely, / is CR on M if
Lf = 0 for every (0, 1) vector field L on M. If M and M' are germs of smooth CR
submanifolds of CN at p and p' respectively, we say
that (M, p) is CR equivalent to
(M',
p') if there is a CR diffeomorphism (a diffeomorphism whose components are
CR functions) between open neighborhoods of p in M and of p' in M' respectively
and taking p to p'. Such a diffeomorphism is called a CR equivalence.
It is known that if / is a CR function defined in a neighborhood of p in M, then there
is a formal (holomorphic) power series ca
(Z — p)a, Z = (Z\,..., ZN), ca e C,
whose restriction to M coincides with the Taylor series of / at p. Moreover, if M is

Points in general position in real-analytic submanifolds in CN and applications 11
generic, then such a formal power series is unique. (See e.g. [BER 1999a], Propo-
sition 1.7.14 for the generic case.) It follows that if h is a CR equivalence between
two real-analytic germs (M, p) and (M', p') of CR submanifolds of CN, then the
corresponding vector-valued formal power series H of the form (4.1) (obtained from
the components of h by the Taylor series property of CR functions mentioned above)
satisfies (4.2) and can be assumed to be invertible; hence H is a formal equivalence
between (M, p) and (Mp'). On the other hand, the restriction to M of a biholo-
morphic equivalence between (M, p) and (M', p') is obviously a CR equivalence.
Thus the notion of CR equivalence lies between that of formal and biholomorphic
equivalence:
biholomorphic equivalence =>• CR equivalence formal equivalence.
A weaker notion than that of formal equivalence is that of ¿-equivalence for an
integer k > 1.
Definition 4.2. For an integer k > 1 we say that two germs, (M, p) and (M\ p'),
of real-analytic submanifolds of CN are k-equivalent if there exists a biholomorphic
map H between neighborhoods of p and p' in CN, with H(p) = p', such that
P'\H(Z(x)), H(Z(X))) = 0(|x|*)
for some real-analytic parametrization x Z(X) of M near p = Z(0) and some real-
analytic defining function p'(Z, Z) of M' near p'. Such an H is called a k-equivalence
between (M, p) and (M', p').
Again here the definition of ¿-equivalence is independent of the choice of the
parametrization Z(x) of M and of the choice of the defining function p' of M'. We
also note that if H is a ¿-equivalence (or a formal equivalence), by taking its Taylor
polynomial of order k — 1, we can find another ¿-equivalence whose components are
polynomials. Hence in Definition 4.2 we could have assumed that H is a biholomor-
phism with polynomial components. Similarly, we could have also assumed that H
is just a formal invertible mapping, rather than a biholomorphism. Then any formal
equivalence may be considered as a ¿-equivalence for every k.
Example 4.1. ^ > 0 is an integer, then the identity map is a 2¿-equivalence between
the germs at 0 of the real hyperplane M :=CxlcC2 and the hypersurface M' given
by Im w = \z\2k• However, it is easily checked that there is no formal equivalence
between (M, 0) and (M\ 0).
The example shows that even very different looking germs of submanifolds can
be ¿-equivalent for some fixed ¿ without being formally equivalent. The situation
becomes rather different if we require (M, p) and (M', p') to be ¿-equivalent for
every k. This means the existence of a sequence of biholomorphic maps H^ each
sending (A/, p) into (Mp') up to order ¿, as in Definition 4.2. In particular, as noted
above, formal equivalence implies the existence of such a sequence. On the other

12 M. S. Baouendi, Linda Preiss Rothschild and Dmitri Zaitsev
hand, given a sequence of ¿-equivalences, in general, one cannot put them together
to obtain a formal equivalence. Nevertheless, for points in general position, the main
result in [BRZ 2000] states that the existence of such a sequence implies that (M, p)
and (M', p') are formally and even biholomorphically equivalent. The authors of the
present paper are not aware of any example of pairs of germs (M, p) and (Mp')
of real analytic submanifolds in CN which are ¿-equivalent for every k > 1, but not
formally equivalent.
5. Structure decompositions and ¿-equivalences
In this section we
consider the extent to which the decompositions given by Proposi-
tions 2.1, 2.2 and 2.3 (and summarized in Theorem 2.5) are invariant under different
notions of equivalences.
We first consider invariance under biholomorphic equivalences. We already re-
marked that the numbers r\, r%, r3 introduced in §2 are biholomorphic invariants.
Write r N — r\ — r2 — r^ for brevity. Now assume that we have two germs at 0 of
real-analytic submanifolds in CN of the form M x C2 x {0} and M' x C2 x {0} with
M, M' C C x as in Theorem 2.5. That is, both M and M' are finitely nondegener-
ate generic submanifolds through 0 containing all points of the form (0, u) for u e
near 0, and such that M fl (C x {m}) and M' fl (C x {u}) are of finite type for u small.
We fix local holomorphic coordinates Z = (Z°, Z3, Z2, Z1) e C x C3 x C2 x C1
near 0 and similarly we write H
= (H°, H3, H2, Hl) for the components of a
valued map H.
Proposition 5.1. Let H
: (CN, 0) (C^, 0) be a biholomorphic equivalence be-
tween the
germs at 0 of M x C2 x {0} and M' x C2 x {0}. Then we have:
(i) Hl (Z°, Z3, Z2, 0) s 0, i.e., H sends C x C3 x C2 x {0} into itself,
(ii) Z3, Z2, 0) = 0, |g(Z°, Z3, Z2, 0) s 0, i.e., H preserves the affine
subspaces given by
Z1 = 0, (Z°, Z3) = const;
(iii) (Z°, Z3, Z2, 0) s 0, i.e., H also preserves the affine subspaces given by
Z1 = 0, Z3 = const.
In particular, the restriction of(H°, //3) to C x C3 x {0} x {0} is a biholomorphic
equivalence between (M, 0) and (A/', 0).
Remark 5.1. Proposition 5.1 can be reformulated "geometrically" as follows. A bi-
holomorphic equivalence between real-analytic submanifolds preserves their intrinsic
complexifications, maximal tangent holomorphic foliations and CR orbits.

Points in general position in real-analytic submanifolds in C^ and applications 13
A statement similar to Proposition 5.1 also holds for formal equivalences. How-
ever, both statements for formal and biholomorphic equivalences are in fact special
cases of more general invariance properties under ¿-equivalences. In the following
proposition, as was illustrated by Example 4.1, it is crucial to require the existence of
a ^-equivalence for every k.
Proposition 5.2 ([BRZ 2000], Proposition 4.1). Suppose that (M, p) and (M', p')
are k-equivalent for all k > 1. Then the numbers r\,r2, r3 in Theorem 2.5 for M
coincide with the ones for M'.
Proposition 5.3 ([BRZ 2000], Lemma 4.4, Lemma 5.3). Under the assumptions of
Proposition 5.1 suppose that M' is l-nondegenerate and let H be a k-equivalence
between M x C2 x {0} and M' x C2 x {0}. Then we have:
(i) H:(Z°, Z3, Z2,
0) = 0(\Z\k);
(ii) |g(Z°, Z3, Z2,
0) = 0(\Z\k-'~l), Z3, Z2, 0) = 0(\Z\k~'-1) pro-
vided k > I;
In particular, the restriction of(H°, H3 )
to C x C3 x {0} x {0} is a k-equivalence
between (M, 0) and (M0).
6. Comparison of different notions of equivalences
The following, which is one of the main results of [BRZ 2000], states that the four
notions of equivalence discussed above actually coincide at all points in general posi-
tion.
Theorem 6.1 ([BRZ 2000], Corollary 14.1). LetM c CN be a
connected real-analy-
tic
submanifold. Then for any p € M in general position and any germ (Mp' ) of a
real-analytic submanifold in CN , the following conditions are equivalent:
(i) (M, p) and (M', p') are k-equivalent for all k > 1;
(ii) (Af, p) and (M', p') are formally equivalent;
(iii) (M, p) and (Af', p') are CR equivalent;
(iv) (M, p) and (M', p') are biholomorphically equivalent.
As mentioned in §4, the implications (iv) => (iii) =>• (ii) => (i) hold
trivially. It was shown in [BER 1999b] that if M and
M' are real-analytic generic
submanifolds which are finitely nondegenerate and of finite type at p and p' respec-
tively, then any formal equivalence H between (M, p) and (M', p') is necessarily

14 M. S. Baouendi, Linda Preiss Rothschild and Dmitri Zaitsev
convergent. In particular, one obtains the equivalence of conditions (ii), (iii) and (iv)
for M a connected real-analytic generic submanifold which is finitely nondegenerate
and of finite type at some point (and hence at all points in general position).
In the case that M is a real-analytic hypersurface, one can use the result in
[BER 1999b] mentioned above together with Proposition 5.3 to prove the equiva-
lence of (ii), (iii), and (iv) of Theorem 6.1 as follows. We begin with the structure
theory, Corollary 2.6, for hypersurfaces at points in general position. Since the fact
that (ii) (iii) <=>• (iv) in case (b) of Corollary 2.6 can be easily proved, we
may assume that condition (a) of that corollary holds. Hence we maj£ assume that
(M, p) = (M x C2, 0) and that (Af, p') = (M' x C2, 0), where M and M' are
finitely nondegenerate hypersurfaces and hence of finite type at 0. Let H be a formal
equivalence between (M, p) and (M', p'). By Proposition 5.3 (here r\ = rj = 0,
r = N - r2, and H — (H°, H2)), we conclude that the restriction Hoi H° to C x {0}
is a formal equivalence between (M, 0) and (A/', 0).
Since M and M' are finitely non-
degenerate and of finite type at 0, it follows from the result in [BER 1999b] mentioned
above that H must be already convergent. It is then easy to extend H to a holomorphic
equivalence between
(M x C2, 0) and (M' x C2, 0). In fact, one may choose such
a biholomorphic equivalence in such a way that its Taylor series coincides with that
of H up to any preassigned order.
For submanifolds of higher codimension it is not possible to reduce to the results of
[BER 1999b], even to prove that the notions of formal and biholomorphic equivalence
coincide, since the submanifold M given in Theorem 2.5 need not be equivalent to a
product of a CR manifold of finite type and Mm for some m. Indeed, if M is the finitely
nondegeneratej>eneric submanifold of codimension 3 in C7 given in Example 2.8, then
M = M, but M is not equivalent to such a product.
We now present some of the ideas involved in the proof of (ii) =>• (iv) in
Theorem 6.1, as given in [BRZ 2000], for a general real-analytic submanifold M. By
making use of Theorem 2.5 and Proposition 5.1, we first reduce to the case where
p
= p' = 0, M — M and M' = M', with M and M' as in Theorem 2.5. (That is, we
assume r\ = r2 = 0 in Theorem 2.5.)
The next step is to obtain a parametrization of all formal equivalences H between
(M, 0) and (M', 0) by their (formal) jets along the linear subspace C := {0} x C3 c
CN~n x C3, which is transversal to the CR orbits of M. In the coordinates (Z°, u) e
~n x C3 the required parametrization has the form
H(z°,
u) = r((aa//(o, z°, u), (6.1)
where k is a number depending only on the dimension N, T is a C^-valued holomor-
phic map defined in some neighborhood of jkH(0) x {0} in the space Jk(CN, CN) x
C3, with
jkH{0) = (daH{0, 0))|a|<(t and Jk{CN, CN) denoting the space of Jt-jets
at 0 of holomorphic maps from CN to C^. Here it is important to note that F does
not depend on the formal mapping H. Equality in (6.1) is in the sense of formal

Points in general position in real-analytic submanifolds in CN and applications 15
power series in Z° and u. The identity (6.1) is a simplified version of the statement
of Theorem 10.1 in [BRZ 2000],
The third step is to use (6.1) to obtain a system of holomorphic equations which
must be satisfied by the formal series components of (3"//(0, u))\a\<k- To this sys-
tem we apply Artin's approximation theorem [A 1968] to conclude that there exists
a convergent solution for this system of equations. This yields the existence of a
holomorphic map H : (CN, 0) —> (C^,0), which is the desired biholomorphic
equivalence between (M, 0) and (M', 0). In fact, we can even choose H in such a
way that its Taylor series coincides with that of H up to a preassigned order.
7. General conditions for the convergence of formal equivalences
We shall give here a result about convergence of formal equivalences between two
germs (Af, p) and (Mp') for p and p' in general position, more general than that
given in [BER 1999b] mentioned above. We restrict ourselves to the case where M
and M' are generic submanifolds of CN, i.e., r\ — 0. By making use of Theorem 2.5
we may assume that (M, p) = (M x C2, 0) such that M c CN~r3 is as in Theorem 2.5
(but with r\ = 0). Similarly we assume (M', p') = (M' x C2, 0), with again M'
as in Theorem 2.5, with the same integers r2 and We shall assume (M, p) and
M', p') have this form for the remainder of this section.
Theorem 7.1 ([BRZ 2000], Corollary 10.3). Let (M, p) and (M', p') be as above.
Then there exists an integer k > 0 such that a formal equivalence H between (M, p)
and (M', p') is convergent if and only if for
Z = (Z°, Z3, Z2) e CN~r2~r3 X C3 x C2, H = (H°, H3, H2),
both of the following conditions are satisfied.
(i) All partial derivatives —3(,zf)'a (0, Z , 0) are convergent for |a| < k ;
(ii) H2(Z) is convergent.
In fact, k can be chosen to be 2(d + 1)/, where d is the codimension of M in CN~r2
and I is chosen such that M is l-nondegenerate.
In the case where r2 — 0, the proof of Theorem 7.1 follows immediately from the
parametrization of all formal mappings given
by (6.1), since in that case condition (i)
of the theorem implies that the right hand side of (6.1) is convergent and hence so is
the left hand side, H(Z). For the general case where ri need not be 0, one also needs
to use Proposition 5.3.

16 M. S. Baouendi, Linda Preiss Rothschild and Dmitri Zaitsev
8. Real-algebraic submanifolds
A submanifold M c is real-algebraic if it is contained in a real-algebraic subvari-
ety of the same dimension. The basic example is given by the sphere \zi I2 — ' •
Most examples of real submanifolds included in this article are real-algebraic. The
study of local biholomorphic maps sending pieces of spheres into each other goes
back to Poincaré [P 1907] and Tanaka [T 1962], Webster proved in [W 1977] that
local biholomorphic maps sending open pieces of Levi-nondegenerate real-algebraic
hypersurfaces into each other are complex-algebraic, i.e., their graphs are contained
in complex-algebraic subvarieties of the same dimension. The algebraic properties
of holomorphic maps sending one real-algebraic submanifold into another reveal the
optimal nondegeneracy conditions for points in general position and have been inten-
sively studied (see [S 1991], [H 1994], [BR 1995], [S 1995], [Z 1995], [BER 1996],
[SS 1996], [Mi 1998], [CMS 1999], [Z 1999]).
If M is a connected real-algebraic submanifold of CN, we say that a property
holds for points in general algebraic position if it holds for all p e M outside a
proper real-algebraic subvariety of M. Also, a stronger notion of equivalence, that
of algebraic equivalence, can be naturally considered. Two germs of real-analytic
submanifolds of C^, (M, p) and (M\ p'), are said to be algebraically equivalent if
there exists a biholomorphic equivalence between them which is complex-algebraic.
Then the analogues of Propositions 2.1, 2.2, 2.3 and hence of Theorem 2.5 also hold
in the category of real-algebraic submanifolds and algebraic equivalences for points
in general algebraic position. The proof is based on the algebraic version of the im-
plicit function theorem and other elementary properties (see e.g. [BER 1999a], §5.4).
In particular, the algebraic analogue of Proposition 2.1 follows from [BER 1999a],
Proposition 5.4.3 (d). The algebraic analogue of Proposition 2.2 can be obtained by
repeating the proof of Proposition 3.1 in [BRZ 2000]. In contrast to this, the argu-
ment of the proof of Proposition 3.3 in [BRZ 2000] cannot be directly adapted to the
algebraic case since the algebraic version of the Frobenius theorem does not hold. The
algebraic analogue of Proposition 2.3 was shown in [BER 1996] (Lemma 3.4.1) by
using the Segre sets rather than the Frobenius theorem. In particular, the CR orbits of
real-algebraic CR submanifolds are algebraic ([BER 1996], Corollary 2.2.5), whereas
the orbits of single vector fields in TCM (the real parts of (0, 1) vector fields) need not
be algebraic.
Example 8.1. Consider the real-algebraic hypersurface M c C2 given in real coor-
dinates by y2 = x2y\ where (z\, zi) = (x\ + iy\,X2 + iyi)- The sections of the
complex tangent subbundle TCM are spanned at each point by the vector fields
3 9 8 9 8 9
X = + - yi—, Y = JX\ =x2— + — +y\-r- •
0X2 ox\ 9yi dy2 9)>i ox
The integral curve C for X through po = (jcj\ x®, y®, y2) € M is given by
x2=x%exi-x°, yi=y^-x\ y2 = yl

Points in general position in real-analytic submanifolds in CN and applications 17
Hence C is not algebraic if x® 0 or y^ ^ 0. (In contrast, the orbits of Y are all
algebraic.) However, the CR orbit of any point po = (.Xp x®, y®) e M withx^ ^ 0
is (M, po), while when x® = 0 (and hence y® = 0) the CR orbit is (C x {0}, po).
Thus the algebraicity of these CR orbits cannot be proved by showing the algebraicity
of the integral curves of the basis of sections of TCM given by X, Y.
9. Algebraic equivalence for real-algebraic submanifolds
The following strengthens Theorem 6.1 in the case of real-algebraic submanifolds.
Theorem 9.1. Let M C <CN be a connected real-algebraic submanifold. Then for any
p e M in general algebraic position and any germ (M', p') of a real-algebraic sub-
manifold in CN, the equivalent conditions (i)—(iv) of Theorem 6.1 are also equivalent
to
(v) (M, p) and (M', p') are algebraically equivalent.
The proof of Theorem 9.1 can be obtained by adapting the proof of Theorem 6.1
given in [BRZ 2000] to the algebraic case. It is sufficient to prove that (iv) implies
(v). As in the proof of Theorem 6.1 the first step is to reduce the general situation by
a complex-algebraic change of coordinates to the case where p = p' = 0, M = M
and M' = M' with M and M' as in Theorem 2.5, or rather its real-algebraic analogue
mentioned above. Here again, r\ = r2 = 0. The second step is to obtain an algebraic
parametrization of all biholomorphic equivalences H between (M, 0) and (M', 0) by
their jets along the linear subspace C = {0} x C3. This parametrization takes the form
(6.1) with T a map which is not only holomorphic but also complex-algebraic, defined
in some neighborhood of jkH{0) x {0} in the jet space Jk(CN, CN) x C3. Note
that in this case the right hand side of (6.1) is a convergent power series in (Zo, «).
If u i-> (daH(0, u))\a\<k is complex algebraic, then the algebraicity of H follows
immediately from the analogue of (6.1). Hence in the special case when M is of finite
type, i.e., rj — 0, H is parametrized by the single jet jkH( 0) = (3a//(0))^|<((; so that
the biholomorphic equivalence H is algebraic itself. Ifr3 > 0, the jet [daH(0, "))|a|<jt
and hence H need not be algebraic. Then we have to modify (daH(0, u)). to
obtain an algebraic equivalence. This is done by using an algebraic version of Artin's
approximation theorem [A 1969]. Here again we can choose the Taylor series of the
algebraic equivalence to coincide with that of H up to a preassigned order.
Example 9.1. Let Mo C C^-1 be an arbitrary finitely nondegenerate real-algebraic
submanifold through 0 and set M = M' := M0 x R c CN. Then both M and M'
are finitely nondegenerate of infinite type and any biholomorphic map H(z, w) :=
(z, 4>(w)) with 0: (C, 0) —(C, 0) sending M into itself is a biholomorphic equiv-
alence between (M, 0) and (M', 0). It is clear that if (¡> is not algebraic, then H is

18 M. S. Baouendi,
Linda Preiss Rothschild and Dmitri Zaitsev
not algebraic. On the other hand, the identity mapping is an algebraic equivalence
between (M, 0), and (M\ 0), and there are many others.
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277-291.
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janvier 1937; Selecta, 113-136; Œuvres complètes, Part. II, Vol. 2, Gauthier-
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Nauk 46 (1) (1991), 81-164 = Russ. Math. Surveys 46 (1) (1991), 95-197.
[CMS 1999] Coupet, B., Meylan, F., Sukhov, A., Holomorphic maps of algebraic CR mani-
folds, Internat. Math. Res. Notices 1 (1999), 1-29.
[E 1998] Ebenfelt, P., New invariant
tensors in CR structures and a normal form for real
hypersurfaces at a generic Levi degeneracy, J. Differential Geom. 50 (1998),
207-247.
[E 2000] Ebenfelt, P., Finite jet determination of holomorphic mappings at the bound-
ary, preprint, 2000, Asian J. Math., to appear; http://xxx.lanl.gov/abs/
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[T 1962] Tanaka, N., On the pseudo-conformal geometry of hypersurfaces of the space of
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[W 1977] Webster, S., On the mapping problem for algebraic real hypersurfaces, In-
vent. Math. 43 (1977), 53-68.
[W 1995] Webster, S., The holomorphic contact geometry of a real hypersurface, in: Mod-
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Ann. 302 (1995), 105-129.

20 M. S. Baouendi, Linda Preiss Rothschild and Dmitri Zaitsev
[Z 1999] Zaitsev, D., Algebraicity of local holomorphisms between real-algebraic sub-
manifolds of complex spaces, Acta Math. 183 (1999), 273-305.
Department of Mathematics
University of California at San Diego
La Jolla, CA 92093-0112, U.S.A.
sbaouendiSucsd.edu
IrothschildQucsd.edu
Mathematisches Institut
Universität Tubingen
Auf der Morgenstelle 10
72076 Tübingen, Germany
dmitri.zaitsevQuni-tuebingen.de

Holomorphic motion of circles through affine bundles
David E. Barrett *
1. Introduction
The pivotal topic of this paper is the study of Levi-flat real hypersurfaces S with
circular fibers in a rank 1 affine bundle A over a Riemann surface X. (To say that S
is Levi-flat is to say that S admits a foliation by Riemann surfaces; equivalently, in
the language of [SuTh], S may be said to prescribe a holomorphic motion of circles
through A.)
After setting notation and terminology in §2 we proceed in §3 to examine the Levi-
form of a general real hypersurface with circular fibers, emphasizing the connection
with curvature considerations.
In §4 we focus on the Levi-flat case. In Theorems 5 and 6 we construct moduli
spaces for Levi-flat S attached to a fixed underlying line bundle L in the compact and
non-compact cases, respectively. In particular, when X is compact we show that the
existence of a Levi-flat S implies that 0 < deg L < 2 genus(X) — 2. (The bound is
sharp.)
Theorem 7 in §7 states that when S is Levi-flat, the Levi-foliation on S extends
to a holomorphic foliation of the CP1 bundle obtained from A by compactifying
the fibers. In the general case, the extended foliation in constructed by looking for
holomorphic sections of A whose distance from the center is harmonic with respect
to the appropriate metric. In §7 we show that this construction produces a foliation
even in
some cases where 5 "disappears into the recomplexification of A."
§6 looks at general holomorphic foliations (transverse to fibers) of compactified
rank 1 affine bundles; in particular, it is shown that such foliations are classified up to
equivalence by a "Schwarzian derivative" and a "curvature function." An Addendum
to Theorem 7 shows how to recognize when such a foliation arises from a Levi-flat
hypersurface.
The remaining sections contain postponed proofs.
*Supported in part by the National Science Foundation.
Complex Analysis and Geometry
Ohio State Univ. Math. Res. Inst. Publ. 9 ©Walter de Gruyter 2001

22 David E. Barrett
2. Notation and terminology
2.1. Affine bundles. Let L be a holomorphic line bundle over a Riemann surface X;
L can be defined by local trivializations with transition functions of the form
(Z, W) (tpa,fi(z), Xa,fi(z) ' w) • (2-1)
An affine bundle A over X associated to L can be defined by local trivializations
with transition functions of the form
(Z, W) h-» (<f>a,p(z), Xa,fi(z) • W + (Ta,p(z))
satisfying the appropriate cocycle condition. Over each point £ e X we have a well-
defined subtraction operation Af x L^ defined in local bundle coordinates
by
((z(f), Wl), (z({), W2) (z(C), W\ - W2).
We will use the term L-shear to refer to a biholomorphic map between affine
bundles A and A! over X associated to L taking each fiber Aj to the corresponding
fiber A j and preserving the subtraction operation.
Let y be a smooth section of an affine bundle A associated to L. Then dy defines
a section of L <g> r*(CU)(X).
The following result follows easily from the definitions.
Proposition 1. Let A \ and Ai be affine bundles over X associated to a fixed line
bundle L, and let Yj be a smooth section of Aj.
Then the following conditions are
equivalent:
(1) there is an L-shearfrom A\ to A2 carrying y\ to yi
(2) dyi = dy2.
Conversely, let co be a smooth section of L
<g> Then using a system of
local solutions of du — co we may construct an affine bundle A associated to L and a
section y of
A satisfying dy = co. Alternatively we may accomplish the same end by
taking the total space of A to be the total space of L equipped with the unique complex
structure Jm satisfying:
• J0J coincides with the standard structure Jo
on vectors tangent to fibers;
• a (local) section y of L is JM-holomorphic if and only if it solves dy = —co
(with respect to the standard structure Jo).
Using local coordinates (z, w) coming from a local trivialization of L we find that
the (1,0) tangent vector fields for the
structure JM are spanned by and

Holomorphic motion of circles through affine bundles 23
3z — J=) The integrability of Jw can be checked directly; alternately we
may note that a solution of 3 y = -co on a open set U induces a biholomorphic map
(L\u> Jo) -»• (L\u' J")
(.z, w) (z, w + y(z)).
It follows that (L, Jw) is an affine bundle over X associated to L\ since the induced
Cauchy-Riemann operator 3m satisfies d0) = do + co we find that the zero section of
L provides a distinguished smooth section of (L, J0J) satisfying = co.
2.2. Bundle metrics; hypersurfaces with circular fibers. Let A be an affine bundle
over X associated to the line bundle L and let y be a smooth section of A. Suppose now
that the line bundle L is equipped with a Hermitian metric h. Then we may consider
the real hypersurface S = Sy h c A whose fiber over f e X is the circle centered at
y(t;) with unit radius with respect to h. Using bundle coordinates (z, w) and writing
h = eu<z^ \dw\ we find that Syh is given by the equation |u; — y(z)\ = e~u<zK
Proposition 2. The map
[SK,„] (3y,h)
is a bijection between
{real hypersurfaces with circular fibers in affine bundles associated to L]
{L-shears]
and
[smooth sections of L ® T*<0,I>(X)} x {Hermitian metrics on L}.
In particular, Sy h is equivalent up to L-shears to the unit circle bundle E^ h C
{L>hyY
3. Levi-form computations
Remarks on terminology. If L is a (not necessarily holomorphic) line bundle given
by transition functions (2.1), we let
\L\ denote the line bundle given by transition
functions
(z, w) h-> (0a,0(z), lx«,/j(z)| • w) .
We will frequently find it convenient to identify \L\2 with L <g> L.
A metric on L may be viewed as a section of |L|_1.

24 David E. Barrett
Proposition 3. Let L be a line bundle over a Riemann surface X, let co be a smooth
section ofL <g> 7,*(0'1)(X), and let h be a Hermitian metric on L. Then the unit circle
bundle in (L, JM) is
strictly pseudoconcave
pseudoconcave
Levi-flat
pseudoconvex
strictly pseudoconvex
if and only if
> ih2co AM + h~l \d(h2a))
§© > ih2co ACO + h~l ¡9 (h2(o)j
±© = h2co A co, 9 (h2co) =0
< ih2co am -h~l |a (h2co) |
< ih2coAco-h~l \d (h2co)
here© = — 23 9 log/i is the curvature (1, )-form for the metric h, and the inequalities
are taken with respect to the standard orientation on X.
To further explain the above equations, note that
• /¡2fflAwisa section of \L\~2 ®{L® r*(0'1}) ® (L <g> r*(1'0)) = r*(U);
• h2co is a section of the anti-holomorphic bundle \L\~2 ®L® T*(0'l> = L 1 ®
7*( o,i).
• 3 (h2co) is a section of I"1 ® r<u>;
• \d(h2co)\is a section of \L\~l ® T^1-1);
• h~l \d(h2co)\ is a section of r*(U).
By Proposition 2, the results of Proposition 3 also describe the pseudoconvexity
properties of hypersurfaces SYih with dy = co. Passing to local coordinates as in §2.2
this translates to the statement that the hypersurface \ w — y(z) \ = e~u(z) is
strictly pseudoconcave
pseudoconcave
Levi-flat
pseudoconvex
strictly pseudoconvex
if and only if
-«« > e2" \2 + e" IYzz + 2uzVï
-Uzz > e2" \Vi\2 + IVzz + 2"zVzl
-Uzï = e2u\Yz\2, Yzz+2UZYZ=Q
-uz-z<e2u\Yz\2-eu\Yzz + 2uzYz
-Uzz <e2u\2 -eu\Yzz + 2uzYz
For a proof in this framework see [Ber, Prop. 2.3] and the references cited there.
For co = 0 Proposition 3 reduces to the following (quite classical) result.

Holomorphic motion of circles through affine bundles 25
Corollary 4. Let L be a line bundle over a Riemann surface X equipped with a
Hermitian metric h. Then the unit circle bundle Eo,/i in L is
strictly pseudoconcave
pseudoconcave
Levi-flat
pseudoconvex
strictly pseudoconvex
if and only if the curvature form 0 satisfies
i& > 0
i@ > 0
0-0 .
10 < 0
«0 < 0
For vector bundles of higher dimension, curvature conditions for Hermitian and
Finsler metrics are related to the theory of interpolation of norms [Roc].
4. The Levi-flat case
Recall that a real hypersurface in a complex manifold is said to be Levi-flat if its Levi-
form vanishes identically. A real hypersurface is Levi-flat if and only if it admits a
(uniquely-determined) codimension-one foliation with complex leaves [Kra, p. 308].
In the situation of Proposition 3, if E^,/, c (L, JM) is Levi-flat then h2co is a
holomorphic section of L"1 <g> T*<l-0)(X).
If h2aj = 0 then by Proposition 1 we may take A — L; also the curvature 0 of the
metric h vanishes identically so that h is flat. In the case where X is compact we thus
have deg L = 0 [GrHa, §1.1]. (Recall that every degree 0 line bundle admits a flat
metric, unique up to scalar multiples [GrHa, §1.2].)
To analyze the case where h2co does not vanish identically, note that 2h~1 \h2to\ =
2h\co\ is a non-negative section of | |; it may be viewed as a conformal metric
on X with a so-called conical singularity of total angle 2n(j + 1) at any point where
h2to has a zero of order j (see for example [HuTr, §2]). The corresponding area form
is 2ih2a> A co, and away from the degeneracies the scalar curvature is given by
2/93log (2h~l\h2co) _ 2iddlogh
2ih2a) A m 2ih2co A co
<S>/2
h2co A CO
= -1.

26 David E. Barrett
To take proper account of the degeneracies we may compute the Gauss-Bonnet form
in the sense of distributions:
-2/33log (2h~l\h2co(j
= —2ih2co Aco — 2jt ^ (order of vanishing of h2co at f) • ,
[(eX-M()=0)
where S^ denotes a unit point mass at f. (See for example [Bar, Lemma 11].)
In the case where X is compact, invocation of the Gauss-Bonnet theorem yields
47t(1 — genus(X))
= - J 2/33 log (2/T1 \h2cof)
= —l 2ih2co A co — 2tz ^ (order of vanishing of h2co at £)
Jx l(€X-M()= 0)
= - J 2ih2co A ft) — 2n deg (l~1 <8> r*(ll0)(X))
= - / 2ih2co A co - 2n (2(genus(X) - 1) - deg L)
Jx
so that
1 f 2
tie e L = — I ih co A co > 0.
TT Jx
Conversely, suppose that X is compact and that deg L > 0. Then for any non-
trivial holomorphic section / of L~x ® r*(1-0)(X) the results of[HuTr, Thm. B]
allow
us to construct (uniquely) a conformal metric h on X of curvature — 1 with conical
singularities of total angle 2n(j + 1) at points where / has
a zero of order j. Setting
h — 2h~l\, w = we find that C (L, Jis Levi-flat.
Let LFC(X, L) denote the space
{Levi-flat hypersurfaces with circular fibers in affine bundles associated to L}
{¿-shears}
Then we may sum up the preceding discussion as follows.
Theorem 5. Let X be a compact Riemann surface and let L be a holomorphic line
bundle on X.
If deg L < 0 then LFC(X, L) = 0.
If deg L = 0 then the map [S^/,] ih is a bijection between LFC(X, L) and the
one-dimensional space of flat metrics on L.
If deg L > 0 then the map i->- h2dy is a bijection between LFC(X, L) and
the space of non-trivial holomorphic sections ofL~l ig)

Holomorphic motion of circles through affine bundles 27
Note that if deg L > 2genus(X) - 2 then deg (L~] <g> T*(l>°\X)) < 0 so that
there are no non-trivial holomorphic sections of L~x & r*(1,0'(X); thus in this case
we again have LFC(X, L) = 0.
Note also that for deg L > 0 we never have A — L, else we would have
0 = -J d(h2dy)y = j h2\dy\2,
x x
forcing h2dy = 0.
To
treat the case of non-compact X we will get a simpler-to-state result by working
modulo not just shears but arbitrary fiber-preserving biholomorphic maps —
following
terminology in dynamics [AnLe] we will refer to such maps as overshears.
Theorem 6. If A is a rank 1 affine bundle over a noncompact Riemann surface X
then the map [Sy,/i] 2h \dy | is a bijection from
{Levi-flat Sy h C A: y not holomorphic}
{overshears}
to the space
{conformal metrics on X of curvature —1 with all total angles E 2TTN}.
Proof Recall from Proposition 2 that Syuh\ and Sn^2 are equivalent modulo shears
if and only if hi =
/i2 and dy\ = 3/2- Similarly, it is straightforward to check that
SYuh\ and Snth2 are equivalent modulo overshears if and only if there is g : X
C\{0} holomorphic with h \ = \g\h2, dyi = g~]dy2', the latter condition implies that
2hi \dy\ | = 2h2\dy2\, showing that our map is well-defined.
To check injectivity, note if , and Sn<h2 are Levi-flat then equality oilh\ \dy\
and 2h2\~dy2\ forces the holomorphic sections h\ 3/j and h\ dy2 to have the same
zeros (counting multiplicities) so that h\
dyl = gh\ dy2 for some holomorphic g :
X -> C \ {0}. Thus
21^97!| (2hi \dyi)2
showing that SYlihi and are equivalent modulo overshears, as required.
To prove surjectivity, let h be a conformal metric on X of curvature — 1 with all
total angles e 27rN. By the WeierstraB Product Theorem [For, Thm. 26.5] and the
triviality of L~x <g> [For, Thm. 30.3] we may pick a holomorphic
section
/ of L-1 (81 T*(l'°\X) so that / vanishes to order j at £ if and only if h has total
angle 2n(j + 1) at f. Let h = 2h~l |/|, co = JJ. Then c (L, Jw) is Levi-flat,
and our mapping takes [Eo>J to h, as required. •

28 David E. Barrett
Turning to the flat case, a standard argument shows that when X is non-compact,
SYth with y holomorphic are classified up to overshears by the associated monodromy
homomorphism n\ (X) ->• S1 [CaLN, Chap.
V],
We close this section with consideration of the special case where L — T^-^iX)
(with X not necessarily compact) and the (l,l)-form co is positive. Then h2to is both
holomorphic and positive, hence equal to a constant C/2; it follows that h is -/C
times the metric on 7*(1'0,(X) induced by the conformal metric on X with area form
co. Moreover, 2h\co I = Ch~x has curvature —1, so the conformal metric on X with
area form co has curvature — C.
5. Extension of Levi foliations
If A is an affine bundle over X we will denote by A the CP1 = C = CU {oo} bundle
over X obtained by adding a point at infinity to each fiber Af.
Theorem 7. Let S — Syh be a Levi-flat hypersurface with circular fibers in an affine
bundle A over a Riemann surface X. Then the Levi-foliation of S extends uniquely to
a holomorphic foliation of A.
The extended foliation !Fs is transverse to the fibers .
If the corresponding line bundle metric h is flat then !Fs is described by the condition
(CM) the graph of a local holomorphic section v of A lies in a leaf if and only if
|| v
— y || is constant.
If the corresponding line bundle metric h is not flat then !Fs is described by the
condition
(LHM) the graph of a local holomorphic section v of A lies in a leaf if and only if
log || v — y || is harmonic on X(v — y)~] (0).
Theorem 7 will be proved in §9.
6. Foliations of compactified affine bundles
6.1. Residues. Let A be an affine bundle over X associated to a line bundle L and
let !F be a holomorphic foliation of A tranvserse to fibers Aç. For
Ç e X let vç be
the unique (germ of a) meromorphic section of A with graph contained in a leaf of T
satisfying (Ç) = oo.
If Vf has a simple pole at Ç then the residue Res^ vç defines an element of ®
Lç. (To be specific, we may choose a small loop Cç about Ç and a local holomorphic

Holomorphic motion of circles through affine
bundles 29
0) —l
section Y of A; we then define a functional Yj on <g> L^ by the formula
(O
for u> a local holomorphic section of <g) L~~x. Yf is clearly linear, and it is
easy to check that the right
hand side depends only on a>(f) and in particular does not
depend on the choice of Q or y. Then we can define Res^ v^ to be the element of
T^'0)(X) (g> representing Yj.)
We may define a section K$r of T*(]'0)(X) ig> L~x by setting
= —2/(Resj vf)_1
when Vf has a simple pole at t, and kjr (f) = 0 when vj has a multiple pole at f.
Proposition 8. For 7 as above, /cjr « a holomorphic section ofT*^l'°\X) <g> L_1.
Proof. Since the fibers of A are compact, the transversality hypothesis guarantees that
!F is locally equivalent to a product foliation onlxC [CaLN, Chap. V], Thus,
choosing bundle coordinates (z, w) for the restriction of A to a small open set U c X
we find that there are holomorphic functions a(z), b(z), c(z), d(z) with ad — be = 1
so that leaves of !F are given by equations of the form
a(z)w + b(z)
c(z)w + d(z)
or
= C (6.1)
d(z)C —
b(z)
w = , (6.2)
-c(z)C+a(z)
C constant. To get w = oo when z = C we must set C = = a(£)/c(£) so that
= a(K)d(z) ~ b{z)c{Q
fl(z)c(?)-a(f)c(z)"
A short computation reveals that
KF =2i-W(a,c) (6.3)
where W(a,c) denotes the Wronskian a dc — cda\ it follows immediately that K? is
holomorphic. •
6.2. Schwarzians. Let L be a holomorphic line bundle over a Riemann surface X and
let A be a non-vanishing section of (X) ® L. With respect to a local coordinate
z and a corresponding local non-vanishing holomorphic section r] of L we may write
A = A(z) dz <8> rj. Replacing z and rj by z = <p(z) and rj = ft] we find that
Hz)=lO)<P'(z)f

30 David E. Barrett
and
- i
(log • {4>'{z)f + (log4>'(z))zz - (log4>'(Z))1 •
Thus ¿RA = (log A.(z))zz - ± (log k(z))f defines a section of the affine line bundle
Aj(X) with transition functions
(z, W) I </>(;), ^
1 (4>'(z))2
If co is the area form of a conformal metric then ¡Ru> is holomorphic if and only if
that metric has constant curvature. (See Lemma 15 in §9.)
def
If / is meromorphic and non-constant then the Schwarzian derivative Sf =
Sl(df) defines a meromorphic section of A$(X). The standard transformation law
[Leh, II. 1.1] may
be written
S(T
of)-Sf = (ST o /) (df )2 ;
here T is a non-constant meromorphic function on a domain containing the range of
/, the subtraction of two sections of (X) on the left-hand side
results in a section
of the associated line bundle (t*(1.°)(T))2 of quadratic differentials, and ST is the
"classical" (scalar-valued) Schwarzian derivative of T. Note that S(T o f) = Sf if
and only if ST = 0 if and only if T is a linear fractional transformation.
A result of Laine and Sorvali [LaSo, Cor.
4.8] states that if X is simply-connected
then a meromorphic section r of A<$ (X) is the Schwarzian derivative of a non-constant
meromorphic function on X if and only if the following condition holds:
(LS) at each pole f of r there is a holomorphic coordinate z vanishing at f and an
integer k > 1 so that the representation of r with respect to z takes the form
1
- k2
—-—+ c-iz~l + C0 + • • • + ck-2Zk~2 + ••• (6.4)
with
det
/2(l — k) 0 0 ... 0
c_ i 2(4-2 k) 0 ... 0 c0
c0 c_i 2(9-3 k) ... 0 ci
c\ co c_ i ... 0 C2
Cfc_3 4 ck-5 • • • 2((fc - l)2 - (k - 1 )k ck_3
\ ck-2 ck-3 Ck-A ••• C-1 ck—l)
= 0.

Holomorphic motion of circles through affine bundles 31
If the condition in (LS) holds at £ for a fixed coordinate z then it will hold (with
the same value of k) for any other holomorphic coordinate vanishing at f; k is in fact
the multiplicity at f of any solution / of -8 f = r.
Returning now to the notation of the proof of Proposition 8 let us examine the
function Cz = a(z)/c(z). It is easy to check that replacing w by M(z)w + B(z) in
(6.1) induces no change in a(z)/c(z), whereas changing the representation (6.1) by
replacing C by T(C) for some fixed linear fractional transformation T has the effect of
replacing a(z)/c(z) by T (a(z)/c(z)). Thus a/c is determined up to post-composition
with a linear fractional transformation by the foliation 3r.
In view of (6.1) and (6.3), a/c is constant if and only if K? = 0 if and only if the
infinity-section w = oo is a leaf of T. If this does
not occur then the Schwarzian
derivative <8{a/c) gives rise to a global meromorphic section of Aj(X) satisfying
the condition (LS).
Lemma 9. Let vj, V2> v3 be distinct local sections of A with graphs contained in
leaves of?. Then sF =
Proof. We may assume that v\,V2, V3 are defined by (6.2) with C — 0, 00, 1, respec-
tively. Then = s. •
J VI— V3 c
Lemma 10. If K? is not = 0 then the only over shear from A
to A taking ¥ to ¥ is
the identity map.
Proof. Choose f 1, £2
so that vf, and vf2 are distinct meromorphic sections of A defined
on a open set U containing £1 and
Then for generic f e U the overshear in question must fix the two distinct finite
points Vf, (X) and t>f2(£)> forcing the overshear to be the identity map. •
Proposition 11. If X is a Riemann surface then the map 1—> <Sjr is a bijection
from
U {holomorphic foliations T of A transverse to fibers with Kjr ^ 0}
A
affine over X
{overshears}
to
{meromorphic sections ofAj(X) satisfying (LS)}.
Proof The preceding discussion shows that our map is well-defined.
To prove injectivity, note that if !F and F are two candidate foliations with =
Sjr then
for a pair of representations of the form (6.1) on the same coordinate patch
we have ¿(a/c) = 4(a/c), so that
a/c = T o (a/c) for some linear fractional
transformation T; changing the representation of !F by replacing C by TC we may
arrange that a/c = a/c. Then an elementary calculation shows that the overshear
(z, w) (z, aJ~Zcb~w + bJ~ldb?) maps F to ?. By Lemma 10 there are no nontrivial

32 David E. Barrett
locally defined overshears mapping f to T, so the overshears mapping T to <F are
locally unique and therefore patch together to define a global overshear.
To prove surjectivity, recall from §6.2 that given any meromorphic section r of
A$(X) satisfying (LS) and any £ e X we may pick / holomorphic and non-constant
on a neighborhood V of f with Sf = £f~l = x on V. Then the foliation iy on
V x C defined by = C satisfies Syr = x. The argument of the preceding
paragraph shows that Tf is determined by x up to overshears, so choosing a family of
local solutions covering X the overshears defined on overlaps can be used to construct
the desired affine bundle A and foliation T. •
Proposition 12. For a fixed line bundle L over a Riemann surface X and a fixed
meromorphic section x of A#(X), the map h-> KJT is a bijection from
1J {holomorphic foliations !F of A tranvserse to fibers with Sjr = x]
A associated to L
{L-shears}
to
[f e //(7*(1'0)(X) ® L"1) : / vanishes to order k - 1 at $ & (6.4) holds at f}.
Proof. The map in question is well-defined since L-shears do not affect the construc-
tion of Res j Vf.
To see that our map takes its values in the prescribed space, note that the quotient
rule d(a/c) = in conjunction with (6.3) shows that Kjr must vanish to order
k — 1 when a/c has multiplicity k, and (as mentioned earlier) this in turn will happen
precisely when (6.4) holds at the point in question.
Using the notation of §2.1 it is easy to see that any overshear between affine bundles
A i and A2 associated to L is equivalent modulo L-shears to a map Fg : (L, JQJ)
(L, Jgco) that dilates each fiber Lf by the factor g(X)\ here g is a holomorphic map
from X into C \ {0}. The bijectivity claimed in Proposition 12 now follows easily
from Proposition 11 and the transformation law — g~lKjr. •
6.3. Recognizing extended Levi-foliations. In §9 we will prove the following.
Addendum to Theorem 7. If !F is the extended Levi-foliation of a Levi-flat hyper-
surface SY,h then kjt = —2ih2co and Sy = 3l(2ih2co A co) = Sl(co), where co — dy.
Suppose we are given a rank 1 affine bundle A over a Riemann surface
X and a
holomorphic foliation y of A transverse to fibers. How can we determine whether or
not !F is the extended Levi-foliation !Fsy h for some Levi-flat Syj1?
If X is non-compact then Theorem 6, Proposition 11, and the Addendum to Theo-
rem 7 combine to yield the (somewhat tautological) conclusion that F is an extended
Levi-foliation if and only if

Holomorphic motion of circles through affine bundles 33
• K? = 0 and "F has a leaf with unitary holonomy projecting bijectively onto X,
or
' -5jr = ¿R( A) where A is the area form of a conformal metric on X of curvature
—
1 with all total angles e 2nN.
If X is compact then Theorem 5 shows that will not be an extended Levi-foliation
unless the degree of the corresponding line bundle L
is > 0.
If deg L = 0 then the only possible extended Levi-foliation is that on A = L
induced via condition (CM) by the unique (up to positive constants) flat metric on L.
If deg L > 0 then Theorem 5, Proposition 12 and the Addendum to Theorem 7
combine to show that T is an extended Levi-foliation if and only if Syr = ¿R(A),
where A is the area form of the metric h constructed from the divisor of Ka in the
proof of Theorem 5.
7. Foliations from "phantom hypersurfaces"
Let A be an affine bundle associated to the cotangent bundle
r*(i.o)(Cpi) of the
Riemann sphere CP1. Since deg J^^CP1) = -2, Theorem 5 shows that A does
not contain a Levi-flat hypersurface with circular fibers. On the other hand, if co is the
area
form for the usual spherical metric on CP1 then taking A = (L, Jw), y = 0, and
h to be the metric on 7*(l-0)(CP1) induced by the spherical metric on CP1, it turns
out that the condition (LHM) from Theorem 7 still defines a holomorphic foliation !F
on A transverse to fibers. Comparing with the last paragraph of §4 we see that T
is
formally jh - but of course the radius of the fibers is not allowed to be imaginary!
More generally we have the following.
Theorem 13. Let A be an affine bundle over X associated to a line bundle L equipped
with metric h, and let y be a smooth section of A. Suppose that a> = dy is nowhere-
vanishing. Then the following conditions are equivalent:
(1) Condition (LHM) of Theorem 1 describes a holomorphic foliation ¥ of A with
leaves transverse to fibers.
(2) Kw = -iBS1°sco is a holomorphic section of L~x <g> r*(1-0)(X) and h2\o)\ is a
flat metric on L ® T<h0)(X).
The notation km is motivated by the fact that if o is a positive (l,l)-form then Km
is the curvature of the conformal metric with co as area form. The notation k? from
§6.1 was motivated by the Addendum to Theorem 14 found at the end of this section.
Theorem 13 will be proved in §9 essentially as a special case of the following
result which allows for zeroes of co.

34 David E. Barrett
Theorem 14. Let A be an affine bundle over X associated to a line bundle L equipped
with metric h, let y be a smooth section of A, and let co = dy. Assume that co is not
= 0. Then the following conditions are equivalent:
(1)
Condition (LHM) of Theorem 1 describes a holomorphic foliation T of A with
leaves transverse to fibers.
(2) There is an open set U C A with tt(U) = X such that condition (LHM) of
Theorem 1 describes a holomorphic foliation F of U with leaves transverse to
fibers.
(3)
For each f e X there is a neighborhood V of p together with
(a) a holomorphic section v of A
on V with v — y non-vanishing
and
(b) a non-vanishing holomorphic section T]
of L on V
such that log || v — y || and r]/(v — y) are harmonic. (Note that t)/(y — y) will
in general be C-valued.)
(4) o) and h admit local representations of the form
r]dg
a)
— ~
(f-8)2
h
= -g\ fi;
here f and g are holomorphic functions with f — ~g
non-vanishing, rj is a
non-vanishing holomorphic section of L and /z is aflat metric on L.
(5) (a) h2\co\ is a flat metric on L® off of the zero set of w,
(b) 3UD is a meromorphic section of A#(X) satisfying condition (LS) from
§6.2;
(c) if has a pole at f and z is a local coordinate vanishing at $ then
oo = zk-X<p (7.1)
for some smooth <p defined near f with ^ 0.
Remark 1. In condition (5) above, the values of k in (7.1) and condition (LS) will
coincide wherever co vanishes.
When co = 0, the existence of section v (not = y) satisfying the condition in
(LHM) implies that h is flat. In this case (CM) defines a foliation but (LHM) does not.

Holomorphic motion of circles through affine bundles 35
Addendum to Theorem 14. For as in Theorem 14 we have
Sjr = Ä(aäiogo))
on{Ç eX: 0)(f ) £ 0}.
Theorem 14 is proved in the next section; the Addendum will be proved in §9.
Remark 2. A Levi-flat S C A given by the equation h2 \ w — y \2
= 1 is the pullback
via the map Id x^Id : A
—> A xx A of the hypersurface h2(w — y)(w — y) — 1.
In view of the Addendum to Theorem 7, a foliation !F constructed from the condition
(LHM) may be viewed as stemming from the hypersurface iic?-(w—y)(w—y) = 23 y,
though this hypersurface may not intersect (id x^Id) (A).
8. Proof of Theorem 14
(1) ^ (2): Take U = X. •
(2) (3): According to (2), for £ e X we may find a neighborhood V of f and
a
biholomorphic map ^ from V x A to an open subset of A with n o — n such
that for fixed *!*(•, £) is a holomorphic section with log H^G, £) — y(-)|| harmonic.
(Here A is the unit disk in C.)
Let i>( ) = *!>(-, 0) and let rj be the non-vanishing holomorphic section of L over
V given by f)||=o-
Working with bundle coordinates (z,w) on n~l(V) we may write *I>(z,£) =
(z, £)), h = eu(z) \dw\ to obtain
0 =
ddlog{e2uiz)(z,l)-y(z)\2)
=
2 ddu + dd log (if -y) + dd log (r/f - y)
for each fixed It follows that
0 = 2 99 (¿"^o) + 99 log(Vf" + 99 (i
=
o + dd(W(v-y)) + o,
as required. •
(3) (4): We may locally represent rj/(v — y) as / — g with /, g holomorphic,
f —
g non-vanishing. Thus
V
v-Y = 7—=
/ ~g

36 David E. Barrett
and
Moreover,
is harmonic, so
h = -g(i
where \x = ev ]\ 1 is a flat metric on L. •
(4) =>• (1): Suppose that in one of the hypothetical local representations the function
g is constant. Then co = 0 on the open set in question, and an analytic continuation
argument shows co = 0 globally, contrary to hypothesis. So
g must be non-constant
in each of the hypothetical local representations.
Let v be a holomorphic section of A with log ||v — y || harmonic on a connected
open set V c X on
which co and h admit the prescribed representations. Since
3 = co = dy
f~g
we have that
1
f-g
is a holomorphic section of L.
Thus
\v-y\=h-\v-y
= \ f-g\v P +
f~g I
= M I(f~g)p + T)\ .
On the set
V = tt e V :/£>(?) ^ 0}
we have
\v-Y\=tAp\if-g) + ri/p\.
Since g is non-constant, the set
T/// def .
V" = R e V : -(C) + f(0 ^ *(?)}

Holomorphic motion of circles through affine bundles 37
is dense in V'. On V" we have
0
= 93 log ||v - /||
= -33 log ( — + / — g | + -33 log ( — + / — g ) (since dp = 0 and /x is flat)
2
\p J 2 \p J
_d(l + f)ATg dgAd^ + f)
2
(i + f-lj2 2(jTf~g)2
so
¿(2 + /)^ dgAd(* + f)
(i + f-s) (? + /-«)
If ^ + / is non-constant then we may apply 33 log to both sides of (8.1) to obtain
d(* + f)Adg _dgAd(j + f) ^^
(yn-s)2
on a dense subset V'" of V".
Combining (8.1) and (8.2) we find that
liiizHL o
on V'", hence on V". Since g is non-constant, j + / must be constant on V", hence
also on V'.
Thus p — -^rj on V'\ if p does not vanish identically then p — ^y also
on V (so in fact V' = V). Thus
v = Y + 7—= + 7' (8-3>
f~g C-f
taking C = oo we obtain the remaining case p =
0, v — y + jz^-
This describes the required foliation on A | v, and the local uniqueness shows that
these foliations patch together to give the required foliation on A. •
(4) => (5): h2\a>\ — n2i\dg\ is flat off of the zero set of dg. Straightforward
computation (see (9.1) below) shows that ¿RTD = Sl(dg) = $g so that [LaSo, Cor. 4.8]
shows that ¡Rxo satisfies (LS). Condition (5c) holds by inspection. •

38 David E. Barrett
(5) (4): Let £ e X. By [LaSo, Cor. 4.8]
there is a meromorphic function g
defined near £ with
SU5 = 3t(dg) = Sg-, (8.4)
if Slco has a pole at £ then the multiplicity of g at f is the integer k from (6.2). Post-
composing g with a fractional linear transformation we may assume that = 0.
We focus first on the case where ¿Red has no pole at Then we
may take g to be
our local coordinate z. Fixing a local non-vanishing holomorphic section
£ of L we
set &> = k(z)dz<8>£.
We wish to arrange that (logA)z(f) ^ 0. If this is not true we may replace g by
J^J. We find then that k(z)
is replaced by (1 + z)_2A(z) and that (log X)z is replaced
by —2(1 + z) + (1 + z)2(logA.)z so that (logA.)z(f) is now non-zero.
With our choice of coordinate now fixed we have
(Rm = Sg = Sz = 0
and so
/_} \ (logX)zz _ 1
V(logI)Jz (logX)2 2'
Thus
i_ z-7
(log^)z 2
with / holomorphic, f(p) / g(p). This yields
(logX), =
Z- f
and hence
logX= -2log(z+ (8-5)
h holomorphic. Then
(1) =
(.f~g)2
setting rj — eh% we have the desired local representation for co, and condition (5a) sets
up the corresponding representation for h.
We turn now to the case where ¡RH> has a pole at f, recalling that in this case g has
a zero of multiplicity k at £, where k is the integer from (6.2).
Here g cannot serve as a coordinate at f but using g as a (non-univalent) coordinate
z in a punctured neighborhood of f and replacing z by J/z in (5c) to obtain
X(z) = k-l<p(zl/k),<p( 0)^0 (8.6)

Holomorphic motion of circles through affine bundles 39
we have
(log*)? = 0(ZH). (8.7)
If (logX)j 0 as we approach £ then replacing g by as before we may
arrange that (log tends to a non-zero limit as we approach thus we may assume
that
(log does
not approach 0 at f. (8.8)
Following our previous computation we set
f(z) =
so that
Hz)
oo for (log = 0.
-iMBk for(logÀ)^0
0 for (log A.)j = 0.
Our earlier work shows that the continuous function 1 // is holomorphic where it
is non-zero, so Rado's Theorem [Nar, 11.8] shows that 1// is in fact holomorphic in
a deleted neighborhood of p. In view of (8.7) and (8.8), after shrinking our neighbor-
hood we may assume that / is holomorphic in a neighborhood of f with / ~ cg^k
for some c ^ 0 and some integer 0 < j < k — 1. Consequently we may also assume
that f ^g except perhaps at
Continuing on to (8.5) we find that we must replace h by h + ^ log g to ensure
that h is single-valued near We thus have
g2Hkeh%~dg
with h holomorphic in a punctured neighborhood of f. In view of (8.6), h must have
a removable singularity at Letting tj = eh% as before we find that
gTllkr\Tg
a)=iniW'
The smoothness of <p in condition (5c) implies that of
/ hdg
gj/k V ft) gitk'
differentiating k times it follows
that j = 0; thus / ^ g at
As before, the local representation for co and condition (5a) induces the corre-
sponding representation for h. •

40 David E. Barrett
9. More proofs
Proof of Addendum to Theorem 14. Comparing the formula (8.3) describing T to the
definitions in §§6.1 and 6.2 we find that
V( = Y + t—= +
D ^
? df
KT
f-g m-f
2 idf
K(o — 2/ •
V
•&F = &f-
On the other hand, from the representation in condition (4) of Theorem 14 we have
aai ^df Adg
99 logo; = -2^—^
1
¿R(dd log u>) = Sf.
(The computation of ¿R(dd log to) is facilitated by taking / to be the coordinate func-
tion z - permissible away from critical points
of / - leading to
as claimed.)
Thus everything matches. •
Lemma 15. If to is a non-vanishing section ofL (g>7,*i0'1) (X) then Sitò is holomorphic
if and only if kuj is holomorphic.
Proof. This follows from the easily-checked identity
(rtâjJ^iûjfoak. (9.2)
•
Proof of Theorem 13. This is now immediate from conditions (1) and (5) from Theo-
rem 14 combined with Lemma 15. •
Proof of Theorem 7 and Addendum to Theorem 1. If h is flat then in local coordinates
we have h — eRt f1^ \dw\, f holomorphic. The hypersurface is given by eRe | w —
y(z) | = 1, the leaves of the Levi-foliation are defined by equations of the form
ef(z) (Vj _ y(z)) = el6°, and leaves of the extended foliation are given by equations
of the form (w — y (z)) = C. If v is holomorphic then the holomorphic function

Holomorphic motion of circles through affine bundles 41
ef(z) —
y(z)) will be constant if and only if ||v — y || = (v(z) — y(z))| is
constant.
If h is not flat then h2cô is holomorphic so that condition (5a) of Theorem 14 holds.
Also, away from zeros of co we have 0 = 33 log h2co so that
33 log = -233 log h = ©
0
—
= 2h co Aw
and Ko, = —2ih2cô is holomorphic. By Lemma 15, Stcô is holomorphic away from
zeroes of co.
The fact that h2co is holomorphic also easily implies that condition (5c) holds and
that StcD has poles at
zeroes of co.
Consulting Theorem 13 we see that the condition (LHM) provides the desired
extended foliation away from zeroes of co. By the Addendum to Theorem 14 we have
Kjr = kw — —2ih2co and S? — St (33 log«) — SI (2h2co A To) = Si (&J).
If <w(zo) =
0 then picking distinct sections vi, vi, V3 near zo whose graphs are
leaves of the original Levi-foliation we have from Lemma 9 that St (cô) = S jr —
so that St (co) satisfies condition (LS) from §6.2. Thus condition (5) of
Theorem 14 holds, so condition (1) must hold as well. •
References
[ AnLe] E. Andersen and L. Lempert, On the group of holomorphic automorphisms of Cn,
Invent. Math. 110 (1992), 371-388.
[Bar] D. Barrett, On a class of conformal metrics arising in work of Seiberg and Witten,
Math. Z. 233 (2000), 149-164.
[Ber] B. Berndtsson, Levi-flat surfaces with circular sections, Several Complex Variables,
Math. Notes 38, Princeton University Press, 1993, 136-159.
[CaLN] C. Camacho and A. Lins Neto, Geometric theory of foliations, Birkhäuser, 1985.
[For] O. Forster, Lectures on Riemann surfaces, Grad. Texts in Math. 81, Springer-Verlag,
1981.
[GrHa] P. Griffiths and J. Harris, Principles of algebraic geometry, John Wiley & Sons, 1978.
[HuTr] D. Hulin and M. Troyanov, Prescribing curvature on open surfaces, Math. Ann. 293
(1992), 277-315.
[Kra] S. Krantz, Function theory of several complex variables (2nd ed.), Wadsworth &
Brooks/Cole, 1992.
[LaSo] I. Laine and T. Sorvali, Local solutions of w" + A(z)w = 0 and branched polymor-
phic functions, Results Math. 10 (1986), 107-129.
[Leh] O. Lehto, Univalent functions and Teichmüller spaces, Grad. Texts in Math. 109,
Springer-Verlag, 1987.

42 David E. Barrett
[Nar] R. Narasimhan, Complex analysis in one variable, Birkhauser, 1985.
[Roc] R. Rochberg, Interpolation of Banach spaces and negatively curved vector bundles,
Pacific J. Math. 110 (1984), 355-376.
[SuTh] D. Sullivan and W. Thurston, Extending holomorphic motions, Acta Math. 157
(1986), 243-257.
Department of Mathematics
University of Michigan
Ann Arbor, MI 48109-1109, U.S.A.
barrettQumich.edu

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“Stow the guff, Dick,” roared Peebles; “it wasn’t nobody’s fault, as
far as I can see. Instead of talkin’ what we oughter do is to go after
this ’ere Tarzan feller and take the bloomin’ gold away from ’im.”
Flora Hawkes laughed. “We haven’t a chance in the world,” she
said. “I know this Tarzan bloke. If he was all alone we wouldn’t be a
match for him, but he’s got a bunch of his Waziri with him, and there
are no finer warriors in Africa than they. And they’d fight for him to
the last man. You just tell Owaza that you’re thinking of going after
Tarzan of the Apes and his Waziri to take the gold away from them,
and see how long it’d be before we wouldn’t have a single nigger
with us. The very name of Tarzan scares these west coast blacks out
of a year’s growth. They would sooner face the devil. No, sir, we’ve
lost, and all we can do is to get out of the country, and thank our
lucky stars if we manage to get out alive. The ape-man will watch
us. I should not be surprised if he were watching us this minute.”
Her companions looked around apprehensively at this, casting
nervous glances toward the jungle. “And he’d never let us get back
to Opar for another load, even if we could prevail upon our blacks to
return there.”
“Two t’ousand pounds, two t’ousand pounds!” wailed Bluber.
“Und all dis suit, vot it cost me tventy guineas vot I can’t vear it
again in England unless I go to a fancy dress ball, vich I never do.”
Kraski had not spoken, but had sat with eyes upon the ground,
listening to the others. Now he raised his head. “We have lost our
gold,” he said, “and before we get back to England we stand to
spend the balance of our two thousand pounds—in other words our
expedition is a total loss. The rest of you may be satisfied to go back
broke, but I am not. There are other things in Africa besides the
gold of Opar, and when we leave the country there is no reason why
we shouldn’t take something with us that will repay us for our time
and investment.”
“What do you mean?” asked Peebles.
“I have spent a lot of time talking with Owaza,” replied Kraski,
“trying to learn their crazy language, and I have come to find out a
lot about the old villain. He’s as crooked as they make ’em, and if he
were to be hanged for all his murders, he’d have to have more lives

than a cat, but notwithstanding all that, he’s a shrewd old fellow,
and I’ve learned a lot more from him than just his monkey talk—I
have learned enough, in fact, so that I feel safe in saying that if we
stick together we can go out of Africa with a pretty good sized stake.
Personally, I haven’t given up the gold of Opar yet. What we’ve lost,
we’ve lost, but there’s plenty left where that came from, and some
day, after this blows over, I’m coming back to get my share.”
“But how about this other thing?” asked Flora. “How can Owaza
help us?”
“There’s a little bunch of Arabs down here,” explained Kraski,
“stealing slaves and ivory. Owaza knows where they are working and
where their main camp is. There are only a few of them, and their
blacks are nearly all slaves who would turn on them in a minute.
Now the idea is this: we have a big enough party to overpower them
and take their ivory away from them if we can get their slaves to
take our side. We don’t want the slaves; we couldn’t do anything
with them if we had them, so we can promise them their freedom
for their help, and give Owaza and his gang a share in the ivory.”
“How do you know Owaza will help us?” asked Flora.
“The idea is his; that’s the reason I know,” replied Kraski.
“It sounds good to me,” said Peebles; “I ain’t fer goin’ ’ome
empty ’anded.” And in turn the others signified their approval of the
scheme.

A
CHAPTER XI
STRANGE INCENSE BURNS
S Tarzan carried the dead Bolgani from the village of the
Gomangani, he set his steps in the direction of the building he
had seen from the rim of the valley, the curiosity of the man
overcoming the natural caution of the beast. He was traveling up
wind and the odors wafted down to his nostrils told him that he was
approaching the habitat of the Bolgani. Intermingled with the scent
spoor of the gorilla-men was that of Gomangani and the odor of
cooked food, and the suggestion of a heavily sweet scent, which the
ape-man could connect only with burning incense, though it seemed
impossible that such a fragrance could emanate from the dwellings
of the Bolgani. Perhaps it came from the great edifice he had seen—
a building which must have been constructed by human beings, and
in which human beings might still dwell, though never among the
multitudinous odors that assailed his nostrils did he once catch the
faintest suggestion of the man scent of whites.
When he perceived from the increasing strength of their odor,
that he was approaching close to the Bolgani, Tarzan took to the
trees with his burden, that he might thus stand a better chance of
avoiding discovery, and presently, through the foliage ahead, he saw
a lofty wall, and, beyond, the outlines of the weird architecture of a
strange and mysterious pile—outlines that suggested a building of
another world, so unearthly were they, and from beyond the wall
came the odor of the Bolgani and the fragrance of the incense,
intermingled with the scent spoor of Numa, the lion. The jungle was
cleared away for fifty feet outside the wall surrounding the building,
so that there was no tree overhanging the wall, but Tarzan
approached as closely as he could, while still remaining reasonably
well concealed by the foliage. He had chosen a point at a sufficient
height above the ground to permit him to see over the top of the
wall.

The building within the enclosure was of great size, its different
parts appearing to have been constructed at various periods, and
each with utter disregard to uniformity, resulting in a conglomeration
of connecting buildings and towers, no two of which were alike,
though the whole presented a rather pleasing, if somewhat bizarre
appearance. The building stood upon an artificial elevation about ten
feet high, surrounded by a retaining wall of granite, a wide staircase
leading to the ground level below. About the building were
shrubbery and trees, some of the latter appearing to be of great
antiquity, while one enormous tower was almost entirely covered by
ivy. By far the most remarkable feature of the building, however, lay
in its rich and barbaric ornamentation. Set into the polished granite
of which it was composed was an intricate mosaic of gold and
diamonds; glittering stones in countless thousands scintillated from
façades, minarets, domes, and towers.
The enclosure, which comprised some fifteen or twenty acres,
was occupied for the most part by the building. The terrace upon
which it stood was devoted to walks, flowers, shrubs, and
ornamental trees, while that part of the area below, which was
within the range of Tarzan’s vision, seemed to be given over to the
raising of garden truck. In the garden and upon the terrace were
naked blacks, such as he had seen in the village where he had left
La. There were both men and women, and these were occupied with
the care of growing things within the enclosure. Among them were
several of the gorilla-like creatures such as Tarzan had slain in the
village, but these performed no labor, devoting themselves, rather, it
seemed, to directing the work of the blacks, toward whom their
manner was haughty and domineering, sometimes even brutal.
These gorilla-men were trapped in rich ornaments, similar to those
upon the body which now rested in a crotch of the tree behind the
ape-man.
As Tarzan watched with interest the scene below him, two
Bolgani emerged from the main entrance, a huge portal, some thirty
feet in width, and perhaps fifteen feet high. The two wore head-
bands, supporting tall, white feathers. As they emerged they took
post on either side of the entrance, and cupping their hands before

their mouths gave voice to a series of shrill cries that bore a marked
resemblance to trumpet calls. Immediately the blacks ceased work
and hastened to the foot of the stairs descending from the terrace to
the garden. Here they formed lines on either side of the stairway,
and similarly the Bolgani formed two lines upon the terrace from the
main portal to the stairway, forming a living aisle from one to the
other. Presently from the interior of the building came other trumpet-
like calls, and a moment later Tarzan saw the head of a procession
emerging. First came four Bolgani abreast, each bedecked with an
ornate feather headdress, and each carrying a huge bludgeon erect
before him. Behind these came two trumpeters, and twenty feet
behind the trumpeters paced a huge, black-maned lion, held in leash
by four sturdy blacks, two upon either side, holding what appeared
to be golden chains that ran to a scintillant diamond collar about the
beast’s neck. Behind the lion marched twenty more Bolgani, four
abreast. These carried spears, but whether they were for the
purpose of protecting the lion from the people or the people from
the lion Tarzan was at a loss to know.
The attitude of the Bolgani lining either side of the way between
the portal and the stairway indicated extreme deference, for they
bent their bodies from their waists in a profound bow while Numa
was passing between their lines. When the beast reached the top of
the stairway the procession halted, and immediately the Gomangani
ranged below prostrated themselves and placed their foreheads on
the ground. Numa, who was evidently an old lion, stood with lordly
mien surveying the prostrate humans before him. His evil eyes
glared glassily, the while he bared his tusks in a savage grimace, and
from his deep lungs rumbled forth an ominous roar, at the sound of
which the Gomangani trembled in unfeigned terror. The ape-man
knit his brows in thought. Never before had he been called upon to
witness so remarkable a scene of the abasement of man before a
beast. Presently the procession continued upon its way descending
the staircase and turning to the right along a path through the
garden, and when it had passed them the Gomangani and the
Bolgani arose and resumed their interrupted duties.

Tarzan remained in his concealment watching them, trying to
discover some explanation for the strange, paradoxical conditions
that he had witnessed. The lion, with his retinue, had turned the far
corner of the palace and disappeared from sight. What was he to
these people, to these strange creatures? What did he represent?
Why this topsy-turvy arrangement of species? Here man ranked
lower than the half-beast, and above all, from the deference that
had been accorded him, stood a true beast—a savage carnivore.
He had been occupied with his thoughts and his observations for
some fifteen minutes following the disappearance of Numa around
the eastern end of the palace, when his attention was attracted to
the opposite end of the structure by the sound of other shrill
trumpet calls. Turning his eyes in that direction, he saw the
procession emerging again into view, and proceeding toward the
staircase down which they had entered the garden. Immediately the
notes of the shrill call sounded upon their ears the Gomangani and
the Bolgani resumed their original positions from below the foot of
the staircase to the entrance to the palace, and once again was
homage paid to Numa as he made his triumphal entry into the
building.
Tarzan of the Apes ran his fingers through his mass of tousled
hair, but finally he was forced to shake his head in defeat—he could
find no explanation whatsoever for all that he had witnessed. His
curiosity, however, was so keenly piqued that he determined to
investigate the palace and surrounding grounds further before
continuing on his way in search of a trail out of the valley.
Leaving the body of Bolgani where he had cached it, he started
slowly to circle the building that he might examine it from all sides
from the concealing foliage of the surrounding forest. He found the
architecture equally unique upon all sides, and that the garden
extended entirely around the building, though a portion upon the
south side of the palace was given over to corrals and pens in which
were kept numerous goats and a considerable flock of chickens.
Upon this side, also, were several hundred swinging, beehive huts,
such as he had seen in the native village of the Gomangani. These

he took to be the quarters of the black slaves, who performed all the
arduous and menial labor connected with the palace.
The lofty granite wall which surrounded the entire enclosure was
pierced by but a single gate which opened opposite the east end of
the palace. This gate was large and of massive construction,
appearing to have been built to withstand the assault of numerous
and well-armed forces. So strong did it appear that the ape-man
could not but harbor the opinion that it had been constructed to
protect the interior against forces equipped with heavy battering
rams. That such a force had ever existed within the vicinity in
historic times seemed most unlikely, and Tarzan conjectured,
therefore, that the wall and the gate were of almost unthinkable
antiquity, dating, doubtless, from the forgotten age of the Atlantians,
and constructed, perhaps, to protect the builders of the Palace of
Diamonds from the well-armed forces that had come from Atlantis to
work the gold mines of Opar and to colonize central Africa.
While the wall, the gate, and the palace itself, suggested in many
ways almost unbelievable age, yet they were in such an excellent
state of repair that it was evident that they were still inhabited by
rational and intelligent creatures; while upon the south side Tarzan
had seen a new tower in process of construction, where a number of
blacks working under the direction of Bolgani were cutting and
shaping granite blocks and putting them in place.
Tarzan had halted in a tree near the east gate to watch the life
passing in and out of the palace grounds beneath the ancient portal,
and as he watched, a long cavalcade of powerful Gomangani
emerged from the forest and entered the enclosure. Swung in hides
between two poles, this party was carrying rough-hewn blocks of
granite, four men to a block. Two or three Bolgani accompanied the
long line of carriers, which was preceded and followed by a
detachment of black warriors, armed with battle-axes and spears.
The demeanor and attitude of the black porters, as well as of the
Bolgani, suggested to the ape-man nothing more nor less than a
caravan of donkeys, plodding their stupid way at the behest of their
drivers. If one lagged he was prodded with the point of a spear or
struck with its haft. There was no greater brutality shown than in the

ordinary handling of beasts of burden the world around, nor in the
demeanor of the blacks was there any more indication of objection
or revolt than you see depicted upon the faces of a long line of
burden-bearing mules; to all intents and purposes they were dumb,
driven cattle. Slowly they filed through the gateway and disappeared
from sight.
A few moments later another party came out of the forest and
passed into the palace grounds. This consisted of fully fifty armed
Bolgani and twice as many black warriors with spears and axes.
Entirely surrounded by these armed creatures were four brawny
porters, carrying a small litter, upon which was fastened an ornate
chest about two feet wide by four feet long, with a depth of
approximately two feet. The chest itself was of some dark, weather-
worn wood, and was reinforced by bands and corners of what
appeared to be virgin gold in which were set many diamonds. What
the chest contained Tarzan could not, of course, conceive, but that it
was considered of great value was evidenced by the precautions for
safety with which it had been surrounded. The chest was borne
directly into the huge, ivy-covered tower at the northeast corner of
the palace, the entrance to which, Tarzan now first observed, was
secured by doors as large and heavy as the east gate itself.
At the first opportunity that he could seize to accomplish it
undiscovered, Tarzan swung across the jungle trail and continued
through the trees to that one in which he had left the body of the
Bolgani. Throwing this across his shoulder he returned to a point
close above the trail near the east gate, and seizing upon a moment
when there was a lull in the traffic he hurled the body as close to the
portal as possible.
“Now,” thought the ape-man, “let them guess who slew their
fellow if they can.”
Making his way toward the southeast, Tarzan approached the
mountains which lie back of the Valley of the Palace of Diamonds.
He had often to make detours to avoid native villages and to keep
out of sight of the numerous parties of Bolgani that seemed to be
moving in all directions through the forest. Late in the afternoon he
came out of the hills into full view of the mountains beyond—rough,

granite hills they were, whose precipitous peaks arose far above the
timber line. Directly before him a well-marked trail led into a canyon,
which he could see wound far upward toward the summit. This,
then, would be as good a place to commence his investigations as
another. And so, seeing that the coast was clear, the ape-man
descended from the trees, and taking advantage of the underbrush
bordering the trail, made his way silently, yet swiftly, into the hills.
For the most part he was compelled to worm his way through
thickets, for the trail was in constant use by Gomangani and Bolgani,
parties passing up it empty-handed and, returning, bearing blocks of
granite. As he advanced more deeply into the hills the heavy
underbrush gave way to a lighter growth of scrub, through which he
could pass with far greater ease though with considerable more risk
of discovery. However, the instinct of the beast that dominated
Tarzan’s jungle craft permitted him to find cover where another
would have been in full view of every enemy. Half way up the
mountain the trail passed through a narrow gorge, not more than
twenty feet wide and eroded from solid granite cliffs. Here there was
no concealment whatsoever, and the ape-man realized that to enter
it would mean almost immediate discovery. Glancing about, he saw
that by making a slight detour he could reach the summit of the
gorge, where, amid tumbled, granite boulders and stunted trees and
shrubs, he knew that he could find sufficient concealment, and
perhaps a plainer view of the trail beyond.
Nor was he mistaken, for, when he had reached a vantage point
far above the trail, he saw ahead an open pocket in the mountain,
the cliffs surrounding which were honeycombed with numerous
openings, which, it seemed to Tarzan, could be naught else than the
mouths of tunnels. Rough wooden ladders reached to some of them,
closer to the base of the cliffs, while from others knotted ropes
dangled to the ground below. Out of these tunnels emerged men
carrying little sacks of earth, which they dumped in a common pile
beside a rivulet which ran through the gorge. Here other blacks,
supervised by Bolgani, were engaged in washing the dirt, but what
they hoped to find or what they did find, Tarzan could not guess.

Along one side of the rocky basin many other blacks were
engaged in quarrying the granite from the cliffs, which had been cut
away through similar operations into a series of terraces running
from the floor of the basin to the summit of the cliff. Here naked
blacks toiled with primitive tools under the supervision of savage
Bolgani. The activities of the quarrymen were obvious enough, but
what the others were bringing from the mouths of the tunnels
Tarzan could not be positive, though the natural assumption was
that it was gold. Where, then, did they obtain their diamonds?
Certainly not from these solid granite cliffs.
A few minutes’ observation convinced Tarzan that the trail he had
followed from the forest ended in this little cul-de-sac, and so he
sought a way upward and around it, in search of a pass across the
range.
The balance of that day and nearly all the next he devoted to his
efforts in this direction, only in the end to be forced to admit that
there was no egress from the valley upon this side. To points far
above the timber line he made his way, but there, always, he came
face to face with sheer, perpendicular cliffs of granite towering high
above him, upon the face of which not even the ape-man could find
foothold. Along the southern and eastern sides of the basin he
carried his investigation, but with similar disappointing results, and
then at last he turned his steps back toward the forest with the
intention of seeking a way out through the valley of Opar with La,
after darkness had fallen.
The sun had just risen when Tarzan arrived at the native village
in which he had left La, and no sooner did his eyes rest upon it than
he became apprehensive that something was amiss, for, not only
was the gate wide open but there was no sign of life within the
palisade, nor was there any movement of the swinging huts that
would indicate that they were occupied. Always wary of ambush,
Tarzan reconnoitered carefully before descending into the village. To
his trained observation it became evident that the village had been
deserted for at least twenty-four hours. Running to the hut in which
La had been hidden he hastily ascended the rope and examined the
interior—it was vacant, nor was there any sign of the High Priestess.

Descending to the ground, the ape-man started to make a thorough
investigation of the village in search of clews to the fate of its
inhabitants and of La. He had examined the interiors of several huts
when his keen eyes noted a slight movement of one of the swinging,
cage-like habitations some distance from him. Quickly he crossed the
intervening space, and as he approached the hut he saw that no
rope trailed from its doorway. Halting beneath, Tarzan raised his face
to the aperture, through which nothing but the roof of the hut was
visible.
“Gomangani,” he cried, “it is I, Tarzan of the Apes. Come to the
opening and tell me what has become of your fellows and of my
mate, whom I left here under the protection of your warriors.”
There was no answer, and again Tarzan called, for he was
positive that someone was hiding in the hut.
“Come down,” he called again, “or I will come up after you.”
Still there was no reply. A grim smile touched the ape-man’s lips
as he drew his hunting knife from its sheath and placed it between
his teeth, and then, with a cat-like spring, leaped for the opening,
and catching its sides, drew his body up into the interior of the hut.
If he had expected opposition, he met with none, nor in the
dimly lighted interior could he at first distinguish any presence,
though, when his eyes became accustomed to the semi-darkness, he
descried a bundle of leaves and grasses lying against the opposite
wall of the structure. Crossing to these he tore them aside revealing
the huddled form of a terrified woman. Seizing her by a shoulder he
drew her to a sitting position.
“What has happened?” he demanded. “Where are the villagers?
Where is my mate?”
“Do not kill me! Do not kill me!” she cried. “It was not I. It was
not my fault.”
“I do not intend to kill you,” replied Tarzan. “Tell me the truth and
you shall be safe.”
“The Bolgani have taken them away,” cried the woman. “They
came when the sun was low upon the day that you arrived, and they
were very angry, for they had found the body of their fellow outside
the gate of the Palace of Diamonds. They knew that he had come

here to our village, and no one had seen him alive since he had
departed from the palace. They came, then, and threatened and
tortured our people, until at last the warriors told them all. I hid. I
do not know why they did not find me. But at last they went away,
taking all the others with them; taking your mate, too. They will
never come back.”
“You think that the Bolgani will kill them?” asked Tarzan.
“Yes,” she replied, “they kill all who displease them.”
Alone, now, and relieved of the responsibility of La, Tarzan might
easily make his way by night through the valley of Opar and to
safety beyond the barrier. But perhaps such a thought never entered
his head. Gratitude and loyalty were marked characteristics of the
ape-man. La had saved him from the fanaticism and intrigue of her
people. She had saved him at a cost of all that was most dear to her,
power and position, peace and safety. She had jeopardized her life
for him, and become an exile from her own country. The mere fact
then that the Bolgani had taken her with the possible intention of
slaying her, was not sufficient for the ape-man. He must know
whether or not she lived, and if she lived he must devote his every
energy to winning her release and her eventual escape from the
dangers of this valley.
Tarzan spent the day reconnoitering outside the palace grounds,
seeking an opportunity of gaining entrance without detection, but
this he found impossible inasmuch as there was never a moment
that there were not Gomangani or Bolgani in the outer garden. But
with the approach of darkness the great east gate was closed, and
the inmates of the huts and palace withdrew within their walls,
leaving not even a single sentinel without—a fact that indicated
clearly that the Bolgani had no reason to apprehend an attack. The
subjugation of the Gomangani, then, was apparently complete, and
so the towering wall surrounding their palace, which was more than
sufficient to protect them from the inroads of lions, was but the
reminder of an ancient day when a once-powerful, but now
vanished, enemy threatened their peace and safety.
When darkness had finally settled Tarzan approached the gate,
and throwing the noose of his grass rope over one of the carved

lions that capped the gate posts, ascended quickly to the summit of
the wall, from where he dropped lightly into the garden below. To
insure an avenue for quick escape in the event that he found La, he
unlatched the heavy gates and swung them open. Then he crept
stealthily toward the ivy-covered east tower, which he had chosen
after a day of investigation as offering easiest ingress to the palace.
The success of his plan hinged largely upon the age and strength of
the ivy which grew almost to the summit of the tower, and, to his
immense relief, he found that it would easily support his weight.
Far above the ground, near the summit of the tower, he had seen
from the trees surrounding the palace an open window, which,
unlike the balance of those in this part of the palace, was without
bars. Dim lights shone from several of the tower windows, as from
those of other parts of the palace. Avoiding these lighted apertures,
Tarzan ascended quickly, though carefully, toward the unbarred
window above, and as he reached it and cautiously raised his eyes
above the level of the sill, he was delighted to find that it opened
into an unlighted chamber, the interior of which, however, was so
shrouded in darkness that he could discern nothing within. Drawing
himself carefully to the level of the sill he crept quietly into the
apartment beyond. Groping through the blackness, he cautiously
made the rounds of the room, which he found to contain a carved
bedstead of peculiar design, a table, and a couple of benches. Upon
the bedstead were stuffs of woven material, thrown over the softly
tanned pelts of antelopes and leopards.
Opposite the window through which he had entered was a closed
door. This he opened slowly and silently, until, through a tiny
aperture he could look out upon a dimly lighted corridor or circular
hallway, in the center of which was an opening about four feet in
diameter, passing through which and disappearing beyond a similar
opening in the ceiling directly above was a straight pole with short
crosspieces fastened to it at intervals of about a foot—quite
evidently the primitive staircase which gave communication between
the various floors of the tower. Three upright columns, set at equal
intervals about the circumference of the circular opening in the
center of the floor helped to support the ceiling above. Around the

outside of this circular hallway there were other doors, similar to that
opening into the apartment in which he was.
Hearing no noise and seeing no evidence of another than himself,
Tarzan opened the door and stepped into the hallway. His nostrils
were now assailed strongly by the same heavy fragrance of incense
that had first greeted him upon his approach to the palace several
days before. In the interior of the tower, however, it was much more
powerful, practically obliterating all other odors, and placing upon
the ape-man an almost prohibitive handicap in his search for La. In
fact as he viewed the doors upon this single stage of the tower, he
was filled with consternation at the prospect of the well-nigh
impossible task that confronted him. To search this great tower
alone, without any assistance whatever from his keen sense of
scent, seemed impossible of accomplishment, if he were to take
even the most ordinary precautions against detection.
The ape-man’s self-confidence was in no measure blundering
egotism. Knowing his limitations, he knew that he would have little
or no chance against even a few Bolgani were he to be discovered
within their palace, where all was familiar to them and strange to
him. Behind him was the open window, and the silent jungle night,
and freedom. Ahead danger, predestined failure; and, quite likely,
death. Which should he choose? For a moment he stood in silent
thought, and then, raising his head and squaring his great shoulders,
he shook his black locks defiantly and stepped boldly toward the
nearest door. Room after room he had investigated until he had
made the entire circle of the landing, but in so far as La or any clew
to her were concerned his search was fruitless. He found quaint
furniture and rugs and tapestries, and ornaments of gold and
diamonds, and in one dimly lighted chamber he came upon a
sleeping Bolgani, but so silent were the movements of the ape-man
that the sleeper slept on undisturbed, even though Tarzan passed
entirely around his bed, which was set in the center of the chamber,
and investigated a curtained alcove beyond.
Having completed the rounds of this floor, Tarzan determined to
work upward first and then, returning, investigate the lower stages
later. Pursuant to this plan, therefore, he ascended the strange

stairway. Three landings he passed before he reached the upper
floor of the tower. Circling each floor was a ring of doors, all of which
were closed, while dimly lighting each landing were feebly burning
cressets—shallow, golden bowls—containing what appeared to be
tallow, in which floated a tow-like wick.
Upon the upper landing there were but three doors, all of which
were closed. The ceiling of this hallway was the dome-like roof of
the tower, in the center of which was another circular opening,
through which the stairway protruded into the darkness of the night
above.
As Tarzan opened the door nearest him it creaked upon its
hinges, giving forth the first audible sound that had resulted from his
investigations up to this point. The interior of the apartment before
him was unlighted, and as Tarzan stood there in the entrance in
statuesque silence for a few seconds following the creaking of the
hinge, he was suddenly aware of movement—of the faintest shadow
of a sound—behind him. Wheeling quickly he saw the figure of a
man standing in an open doorway upon the opposite side of the
landing.

E
CHAPTER XII
THE GOLDEN INGOTS
STEBAN MIRANDA had played the rôle of Tarzan of the Apes with
the Waziri as his audience for less than twenty-four hours when
he began to realize that, even with the lee-way that his supposedly
injured brain gave him, it was going to be a very difficult thing to
carry on the deception indefinitely. In the first place Usula did not
seem at all pleased at the idea of merely taking the gold away from
the intruders and then running from them. Nor did his fellow
warriors seem any more enthusiastic over the plan than he. As a
matter of fact they could not conceive that any number of bumps
upon the head could render their Tarzan of the Apes a coward, and
to run away from these west coast blacks and a handful of
inexperienced whites seemed nothing less than cowardly.
Following all this, there had occurred in the afternoon that which
finally decided the Spaniard that he was building for himself
anything other than a bed of roses, and that the sooner he found an
excuse for quitting the company of the Waziri the greater would be
his life expectancy.
They were passing through rather open jungle at the time. The
brush was not particularly heavy and the trees were at considerable
distances apart, when suddenly, without warning, a rhinoceros
charged them. To the consternation of the Waziri, Tarzan of the Apes
turned and fled for the nearest tree the instant his eyes alighted
upon charging Buto. In his haste Esteban tripped and fell, and when
at last he reached the tree instead of leaping agilely into the lower
branches, he attempted to shin up the huge bole as a schoolboy
shins up a telegraph pole, only to slip and fall back again to the
ground.
In the meantime Buto, who charges either by scent or hearing,
rather than by eyesight, his powers of which are extremely poor, had
been distracted from his original direction by one of the Waziri, and

after missing the fellow had gone blundering on to disappear in the
underbrush beyond.
When Esteban finally arose and discovered that the rhinoceros
was gone, he saw surrounding him a semi-circle of huge blacks,
upon whose faces were written expressions of pity and sorrow, not
unmingled, in some instances, with a tinge of contempt. The
Spaniard saw that he had been terrified into a practically irreparable
blunder, yet he seized despairingly upon the only excuse he could
conjure up.
“My poor head,” he cried, pressing both palms to his temples.
“The blow was upon your head, Bwana,” said Usula, “and your
faithful Waziri thought that it was the heart of their master that
knew no fear.”
Esteban made no reply, and in silence they resumed their march.
In silence they continued until they made camp before dark upon
the bank of the river just above a waterfall. During the afternoon
Esteban had evolved a plan of escape from his dilemma, and no
sooner had he made camp than he ordered the Waziri to bury the
treasure.
“We shall leave it here,” he said, “and tomorrow we shall set
forth in search of the thieves, for I have decided to punish them.
They must be taught that they may not come into the jungle of
Tarzan with impunity. It was only the injury to my head that
prevented me from slaying them immediately I discovered their
perfidy.”
This attitude pleased the Waziri better. They commenced to see a
ray of hope. Once again was Tarzan of the Apes becoming Tarzan.
And so it was that with lighter hearts and a new cheerfulness they
set forth the next morning in search of the camp of the Englishmen,
and by shrewd guessing on Usula’s part they cut across the jungle to
intercept the probable line of march of the Europeans to such
advantage that they came upon them just as they were making
camp that night. Long before they reached them they smelled the
smoke of their fires and heard the songs and chatter of the west
coast carriers.

Then it was that Esteban gathered the Waziri about him. “My
children,” he said, addressing Usula in English, “these strangers have
come here to wrong Tarzan. To Tarzan, then, belongs the
vengeance. Go, therefore, and leave me to punish my enemies alone
and in my own way. Return home, leave the gold where it is, for it
will be a long time before I shall need it.”
The Waziri were disappointed, for this new plan did not at all
accord with their desires, which contemplated a cheerful massacre of
the west coast blacks. But as yet the man before them was Tarzan,
their big Bwana, to whom they had never failed in implicit
obedience. For a few moments following Esteban’s declaration of his
intention, they stood in silence shifting uneasily, and then at last
they commenced to speak to one another in Waziri. What they said
the Spaniard did not know, but evidently they were urging
something upon Usula, who presently turned toward him.
“Oh, Bwana,” cried the black. “How can we return home to the
Lady Jane and tell her that we left you injured and alone to face the
rifles of the white men and their askari? Do not ask us to do it,
Bwana. If you were yourself we should not fear for your safety, but
since the injury to your head you have not been the same, and we
fear to leave you alone in the jungle. Let us, then, your faithful
Waziri, punish these people, after which we will take you home in
safety, where you may be cured of the evils that have fallen upon
you.”
The Spaniard laughed. “I am entirely recovered,” he said, “and I
am in no more danger alone than I would be with you,” which he
knew, even better than they, was but a mild statement of the facts.
“You will obey my wishes,” he continued sternly. “Go back at once
the way that we have come. After you have gone at least two miles
you may make camp for the night, and in the morning start out
again for home. Make no noise, I do not want them to know that I
am here. Do not worry about me. I am all right, and I shall probably
overtake you before you reach home. Go!”
Sorrowfully the Waziri turned back upon the trail they had just
covered and a moment later the last of them disappeared from the
sight of the Spaniard.

With a sigh of relief Esteban Miranda turned toward the camp of
his own people. Fearing that to surprise them suddenly might invite
a volley of shots from the askari he whistled, and then called aloud
as he approached.
“It is Tarzan!” cried the first of the blacks who saw him. “Now
indeed shall we all be killed.”
Esteban saw the growing excitement among the carriers and
askari—he saw the latter seize their rifles and that they were
fingering the triggers nervously.
“It is I, Esteban Miranda,” he called aloud. “Flora! Flora, tell those
fools to lay aside their rifles.”
The whites, too, were standing watching him, and at the sound
of his voice Flora turned toward the blacks. “It is all right,” she said,
“that is not Tarzan. Lay aside your rifles.”
Esteban entered the camp, smiling. “Here I am,” he said.
“We thought that you were dead,” said Kraski. “Some of these
fellows said that Tarzan said that he had killed you.”
“He captured me,” said Esteban, “but as you see he did not kill
me. I thought that he was going to, but he did not, and finally he
turned me loose in the jungle. He may have thought that I could not
survive and that he would accomplish his end just as surely without
having my blood upon his hands.”
“ ’E must have knowed you,” said Peebles. “You’d die, all right, if
you were left alone very long in the jungle—you’d starve to death.”
Esteban made no reply to the sally but turned toward Flora. “Are
you not glad to see me, Flora?” he asked.
The girl shrugged her shoulders. “What is the difference?” she
asked. “Our expedition is a failure. Some of them think you were
largely to blame.” She nodded her head in the general direction of
the other whites.
The Spaniard scowled. None of them cared very much to see
him. He did not care about the others, but he had hoped that Flora
would show some enthusiasm about his return. Well, if she had
known what he had in his mind, she might have been happier to see
him, and only too glad to show some kind of affection. But she did
not know. She did not know that Esteban Miranda had hidden the

golden ingots where he might go another day and get them. It had
been his intention to persuade her to desert the others, and then,
later, the two would return and recover the treasure, but now he
was piqued and offended—none of them should have a shilling of it
—he would wait until they left Africa and then he would return and
take it all for himself. The only fly in the ointment was the thought
that the Waziri knew the location of the treasure, and that, sooner or
later, they would return with Tarzan and get it. This weak spot in his
calculations must be strengthened, and to strengthen it he must
have assistance which would mean sharing his secret with another,
but whom?
Outwardly oblivious of the sullen glances of his companions he
took his place among them. It was evident to him that they were far
from being glad to see him, but just why he did not know, for he
had not heard of the plan that Kraski and Owaza had hatched to
steal the loot of the ivory raiders, and that their main objection to
his presence was the fear that they would be compelled to share the
loot with him. It was Kraski who first voiced the thought that was in
the minds of all but Esteban.
“Miranda,” he said, “it is the consensus of opinion that you and
Bluber are largely responsible for the failure of our venture. We are
not finding fault. I just mention it as a fact. But since you have been
away we have struck upon a plan to take something out of Africa
that will partially recompense us for the loss of the gold. We have
worked the thing all out carefully and made our plans. We don’t
need you to carry them out. We have no objection to your coming
along with us, if you want to, for company, but we want to have it
understood from the beginning that you are not to share in anything
that we get out of this.”
The Spaniard smiled and waved a gesture of unconcern. “It is
perfectly all right,” he said. “I shall ask for nothing. I would not wish
to take anything from any of you.” And he grinned inwardly as he
thought of the more than quarter of a million pounds in gold which
he would one day take out of Africa for himself, alone.
At this unexpected attitude of acquiescence upon Esteban’s part
the others were greatly relieved, and immediately the entire

atmosphere of constraint was removed.
“You’re a good fellow, Esteban,” said Peebles. “I’ve been sayin’
right along that you’d want to do the right thing, and I want to say
that I’m mighty glad to see you back here safe an’ sound. I felt
terrible when I ’eard you was croaked, that I did.”
“Yes,” said Bluber, “John he feel so bad he cry himself to sleep
every night, ain’t it, John?”
“Don’t try to start nothin’, Bluber,” growled Peebles, glaring at the
Jew.
“I vasn’t commencing to start nodding,” replied Adolph, seeing
that the big Englishman was angry; “of course ve vere all sorry dat
ve t’ought Esteban was killed und ve is all glad dot he is back.”
“And that he don’t want any of the swag,” added Throck.
“Don’t worry,” said Esteban, “If I get back to London I’ll be happy
enough—I’ve had enough of Africa to last me all the rest of my life.”
Before he could get to sleep that night, the Spaniard spent a
wakeful hour or two trying to evolve a plan whereby he might secure
the gold absolutely to himself, without fear of its being removed by
the Waziri later. He knew that he could easily find the spot where he
had buried it and remove it to another close by, provided that he
could return immediately over the trail along which Usula had led
them that day, and he could do this alone, insuring that no one but
himself would know the new location of the hiding place of the gold,
but he was equally positive that he could never again return later
from the coast and find where he had hidden it. This meant that he
must share his secret with another—one familiar with the country
who could find the spot again at any time and from any direction.
But who was there whom he might trust! In his mind he went
carefully over the entire personnel of their safari, and continually his
mind reverted to a single individual—Owaza. He had no confidence
in the wily old scoundrel’s integrity, but there was no other who
suited his purpose as well, and finally he was forced to the
conclusion that he must share his secret with this black, and depend
upon avarice rather than honor for his protection. He could repay the
fellow well—make him rich beyond his wildest dreams, and this the
Spaniard could well afford to do in view of the tremendous fortune

at stake. And so he fell asleep dreaming of what gold, to the value
of over a quarter of a million pounds sterling, would accomplish in
the gay capitals of the world.
The following morning while they were breakfasting Esteban
mentioned casually that he had passed a large herd of antelope not
far from their camp the previous day, and suggested that he take
four or five men and do a little hunting, joining the balance of the
party at camp that night. No one raised any objection, possibly for
the reason that they assumed that the more he hunted and the
further from the safari he went the greater the chances of his being
killed, a contingency that none of them would have regretted, since
at heart they had neither liking nor trust for him.
“I will take Owaza,” he said. “He is the cleverest hunter of them
all, and five or six men of his choosing.” But later, when he
approached Owaza, the black interposed objections to the hunt.
“We have plenty of meat for two days,” he said. “Let us go on as
fast as we can, away from the land of the Waziri and Tarzan. I can
find plenty of game anywhere between here and the coast. March
for two days, and then I will hunt with you.”
“Listen,” said Esteban, in a whisper. “It is more than antelope
that I would hunt. I cannot tell you here in camp, but when we have
left the others I will explain. It will pay you better to come with me
today than all the ivory you can hope to get from the raiders.”
Owaza cocked an attentive ear and scratched his woolly head.
“It is a good day to hunt, Bwana,” he said. “I will come with you
and bring five boys.”
After Owaza had planned the march for the main party and
arranged for the camping place for the night, so that he and the
Spaniard could find them again, the hunting party set out upon the
trail that Usula had followed from the buried treasure the preceding
day. They had not gone far before Owaza discovered the fresh spoor
of the Waziri.
“Many men passed here late yesterday,” he said to Esteban,
eyeing the Spaniard quizzically.
“I saw nothing of them,” replied the latter. “They must have
come this way after I passed.”

“They came almost to our camp, and then they turned about and
went away again,” said Owaza. “Listen, Bwana, I carry a rifle and
you shall march ahead of me. If these tracks were made by your
people, and you are leading me into ambush, you shall be the first
to die.”
“Listen, Owaza,” said Esteban, “we are far enough from camp
now so that I may tell you all. These tracks were made by the Waziri
of Tarzan of the Apes, who buried the gold for me a day’s march
from here. I have sent them home, and I wish you to go back with
me and move the gold to another hiding place. After these others
have gotten their ivory and returned to England, you and I will come
back and get the gold, and then, indeed, shall you be well
rewarded.”
“Who are you, then?” asked Owaza. “Often have I doubted that
you are Tarzan of the Apes. The day that we left the camp outside of
Opar one of my men told me that you had been poisoned by your
own people and left in the camp. He said that he saw it with his own
eyes—your body lying hidden behind some bushes—and yet you
were with us upon the march that day. I thought that he lied to me,
but I saw the consternation in his face when he saw you, and so I
have often wondered if there were two Tarzans of the Apes.”
“I am not Tarzan of the Apes,” said Esteban. “It was Tarzan of
the Apes who was poisoned in our camp by the others. But they only
gave him something that would put him to sleep for a long time,
possibly with the hope that he would be killed by wild animals before
he awoke. Whether or not he still lives we do not know. Therefore
you have nothing to fear from the Waziri or Tarzan on my account,
Owaza, for I want to keep out of their way even more than you.”
The black nodded. “Perhaps you speak the truth,” he said, but
still he walked behind, with his rifle always ready in his hand.
They went warily, for fear of overtaking the Waziri, but shortly
after passing the spot where the latter had camped they saw that
they had taken another route and that there was now no danger of
coming in contact with them.
When they had reached a point within about a mile of the spot
where the gold had been buried, Esteban told Owaza to have his

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Complex Analysis And Geometry Proceedings Of A Conference At The Ohio State University June 36 1999 Reprint 2017 Jeffery D Mcneal Editor

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Complex Analysis And Geometry Proceedings Of A Conference At The Ohio State University June 36 1999 Reprint 2017 Jeffery D Mcneal Editor

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