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COMPLEX MANIFOLDS
JAMES MORROW
KUNIHIKO I(oDAIRA
AMS CHELSEA PUBLISHING
American Mathematical Society· Providence, Rhode Island

2000 Mathematics Subject Classification. Primary 32Qxx.
Library of Congress Cataloging-in-Publication Data
Morrow, James A., 1941-
Complex manifolds /
James Morrow, Kunihiko Kodaira.
p. cm.
Originally published: New York: Holt, Rinehart and Winston, 1971.
Includes bibliographical references
and index.
ISBN 0-8218-4055-X (alk. paper)
1. Complex manifolds. I. Kodaira, Kunihiko, 1915-II. Title.
QA331.M82 2005
515'.946---dc22
© 1971 held by the American Mathematical Society.
20051
Reprinted with errata by the American Mathematical Society, 2006
Printed in the United States of America.
§ The paper used in this book is acid-free and falls within the guidelines
established
to ensure permanence and durability.
Visit
the
AMS home page at http://www.ams.org/
10987654321 11 10 09 08 07 06

Preface
The study of algebraic curves and surfaces is very classical. Included
among the principal investigators are Riemann, Picard, Lefschetz, Enriques,
Severi, and Zariski. Beginning in the late
1940s, the study of abstract (not
necessarily algebraic) complex manifolds began to interest many mathe
maticians. The restricted class
of Kahler manifolds called Hodge manifolds
turned out to be algebraic. The proof
of this fact is sometimes called the
Kodaira embedding theorem, and its proof relies on the use
of the vanishing
theorems for certain cohomology groups on Kahler manifolds with positive
lines fundles proved somewhat earlier
by Kodaira. This theorem is analogous
to the theorem
of Riemann that a compact Riemann surface is algebraic.
This book
is a revision and organization of a set of notes taken from the
lectures
of Kodaira at Stanford University in 1965-1966.
One of the main
points was to give the original proof
of the Kodaira embedding theorem.
There
is a generalization of this theorem by Grauert. Its proof is not included
here.
Beginning in the mid-1950s Kodaira and Spencer began the study
of
deformations of complex manifolds. A great deal of this book is devoted to
the study
of deformations. Included are the semicontinuity theorems and the
local completeness theorem
of Kuranishi. There has also been a great deal
accomplished on the classification
of complex surfaces (complex dimension
2). That material
is not included here.
The outline
is roughly as follows. Chapter I includes some of the basic
ideas such
as surgery, quadric transformations, infinitesimal deformations,
deformations. In Chapter
2, sheaf cohomology is defined and some of the
completeness theorems are proved by power series methods. The de Rham
and Dolbeault theorems are also proved. In Chapter 3 Kahler manifolds
are studied and the vanishing and embedding theorems are proved. In Chapter
4 the theory
of elliptic partial differential equations is used to study the
semi-continuity theorems and Kuranishi's theorem.
It will help the reader if he knows some algebraic topology. Some results
from elliptic partial differential equations are used for which complete
references are given. The sheaf theory
is self-contained.
We wish to thank the publisher for patience shown to the authors and
Nancy Monroe for her excellent typing.
Seattle, Washington
January 1971
v
James A. Morrow
Kunihiko Kodaira

Contents
Preface v
Chapter 1. Definitions and Examples of Complex Manifolds 1
1. Holomorphic Functions 1
2. Complex Manifolds and Pseudogroup Structures 7
3. Some Examples of Construction (or Description) of
Compact Complex Manifolds 11
4. Analytic Families; Deformations 18
Chapter 2. Sheaves and Cohomology 27
1. Germs of Functions 27
2. Cohomology Groups 30
3. Infinitesimal Deformations 35
4. Exact Sequences 56
5. Vector Bundles 62
6. A Theorem of Dolbeault (A fine resolution of (I)) 73
Chapter 3. Geometry of Complex Maoifolds 83
1. Hermitian Metrics; Kahler Structures 83
2. Norms and Dual Forms 92
3. Norms for Holomorphic Vector Bundles 100
4. Applications of Results on Elliptic Operators 102
5. Covariant Differentiation on Kahler Manifolds 106
6. Curvatures on Kahler Manifolds 116
7. Vanishing Theorems 125
8. Hodge Manifolds 134
Chapter 4. Applications of Elliptic Partial Differential Equations to
Deformations
147
1. Infinitesimal Deformations 147
2. An Existence Theorem for Deformations
I.
(No Obstructions) 155
3. An Existence Theorem for Deformations II. (Kuranishi's
Theorem)
165
4. Stability Theorem 173
Bibliography 186
Index 189
Errata 193 vii

Complex Manifolds

[1]
Definitions and Examples
of Complex Manifolds
I. Holomorphic Functions
The facts of this section must be well known to the reader. We review
them briefly.
DEFINITION 1.1. A complex-valued function J(z) defined on a connected
open domain
W
s;;; en is called hoiomorphic, if for each a = (a1> "', an) e W,
J(z) can be represented as a convergent power series
+00
L ek, ... k
n(Z1 -a1)k, ... (zn -a,,)k"
k,~O.kn~O
in some neighborhood of a.
REMARK. If p(z) = LCk ... k
n (Z1 -a1)k, •.• (z" -an)k" converges at z = w, then
p(z) converges for any z such that IZk -akl < IWk -akl for 1 :S k :S n.
Proof We may assume a = O. Then there is a constant C> 0 such that
for all coefficients Ck .... kn'
Ie W"l .•• wknl < C
k, .. ·kn 1 ,,_.
Hence
Ie zk, ... zknl < C 2 '" 2
I
Z Ik' I Z Ik" k, ... kn 1 ,,- •
W1 W"
(1)
If Izdwil < 1 for 1 :S i:S n, (1) gives
L Ie", "'knZ~' '" zktl :S C. n ( 1 I) < + 00.
1=1 Zi
1--
Wi
Q.E.D.
1

2 DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS
We have the following picture:
Figure I
n is the region {zllzil < Iwd i ~ n}.
For convenience, we let
P(a,r) = {zllz. -a.1 < r., v = 1, "', n}.
Sometimes we call Pea, r) a po/ydisc or po/ycylinder. A complex-valued func
tion/(z) = /(x1 + iYI, ... , Xn + iYn), where i = J -1 can be considered as a
function of
2n real variables. Then:
DEFINITION 1.2. A complex-valued function of n complex variables is con
tinuous or differentiable if it
is continuous or differentiable when considered
as a function
of 2n real variables.
We have:
THEOREM 1.1. (Osgood) If fez) = /(Zl' "', Zn) is a continuous function
on a domain
W
£ en, and if / is holomorphic with respect to each z" when
the other variables Zi are fixed, then/is holomorphic in W.
Proof Take any a E Wand choose r so that pea, r) ~ W. We use the
Cauchy integral theorem for the representation for Z E Pea, r)
f(
. . . ) -_1 f f(wl, z2, ... , Zn) d
ZI, , Z" - • J, wI>
2Xl Iw,-lId=r, WI -Zl
f( ... )-1 f. f(wl,W2,Z3,···,z")d
WI> Z2' ,Z" --. W2,
2x! Iwz-lIzl=rz W2 -Z2
and so on.

1. HOLOMORPHIC FUNCTIONS 3
Substituting we get
We are assuming
I
z. -a'l < 1.
w. -a.
Hence the series
1 1 [ 1 ] 1
w. -Z. = (w. -a.) + (a. -z.) = 1 -(Zy -ay/w. -aJ w. -a.
(
1 ) 00 (Z -a )k = L v v
w. -a. k=O w. -a.
converges absolutely in P(a, r). Integrating term by term we get
00
J(z) = L ct
! ••• kn(zt -a 1)k! ••• (zn -an)kn, (2)
n=O
where
Then
where M = sup{IJ(w)llw E P(a, r)}. It follows that the representation (2) for
J(z) is valid for Z E P(a, r) and hence the theorem is true.
We now introduce the Cauchy-Riemann equations. Let/(z) be a differen
tiable function
on domain
n f; en.
DEFINITION 1.3. The operators a/azy, a/oz., 1 ~ v ~ n are defined by
af 1 (aJ . Of)
o~. = 2 ax. -I oY. '
af 1 (af . OJ)
oz. = 2 OXy + I Oy. '
where z. = Xy + iy. as usual.

4 DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS
Let f(z) = u(x, y) + h'(x, y). Then
of = ~ [au + i av + i(au + i av)]
az 2 ox ax ay oy
= ~ [OU _ ov + i(OV + aU)].
2 ox oy ox oy
So, af/oz = ° if and only if ou/ax = oll/ay and or/ax = -ou/ay (the Cauchy
Riemann equations).
REMARK. If
of/oz = 0, then df/dx = of/oz, where df/dx = ou/ox + i(ov/ox).
The following calculation verifies this:
of = ~ [au + i ov _ i(OU + i Ov)]
OZ 2 ax ax ay ay
= ~ [OU + i OV + i (av _ i au)] .
2 ax ax ax ax
THEOREM 1.2. Let fez) be a (continuously) differentiable function on the
open set Q s;;; en. Thenf(z) is holomorphic if and only if of/oz. = 0, i :s v :S n.
Proof This follows easily from Osgood's theorem and the classical
fact for functions
of one complex variable. We need another simple calcula
tion. From
now on differentiable will mean having continuous derivatives
of all orders
(C"").
PROPOSITION 1.1. Suppose few) =f(w1, ... , wm) and 9 .. (Z) I:s A.:s mare
differentiable and such that the domain off contains the range of (91' ... , 9 .. )
= 9. Then f[91(Z), .. " 9m(Z)] is differentiable and if w;.(z) = 9;.(z),
of = f (Of ow;. + !L ow;.)
oz. ..= lOW). oz. ow). oz. '
(3)
(4)
Proof All statements follow trivially from the chain rule of calculus.
For punishment we calculate (3). Let 11'). = U). + iv). = 9.(z). Then

I. HOLOMORPHIC FUNCTIONS 5
Making the substitutions,
1
U A = 2 (g A + 9 A),
we get
oj[g(Z)] = f {OJ! (09A + 09A)
OZ, A= I oUA 2 oz. oz.
oj (1 )(09A 09A)}
+ OVA 2i OZ. -OZ.
f {I (OJ . Of) og A
= A= I 2 OUA -I OVA oz.
I (oj . of) 09 A}
+--+1--
2 OUA OVA OZ.'
which gives (2).
COROLLARY 1. If f(w) is holomorphic in wand if w = g(z) = [gl(z), "',
gm(z)] where each g;.(z) is holomorphic in z, thenf[g(z)] is holomorphic in z.
COROLLARY 2. The set ()n of all functions holomorphic on n forms a ring.
In order
to study complex manifolds we must consider holomorphic
maps. Let
U be a domain in en and letfbe a map from U into em,
f(Zl' '.', zn) = [ft(z), ... ,fm(z)].
DEFINITION 1.4. f is holomorphic if each f;. is holomorphic. The matrix
ojl ojm
OZI OZI
= (iz:);.=I ..... m
ojl ojm
v= 1, ...• n
OZn OZn
is called the Jacobian matrix. If m = n, the determinant, det(of;./ozv) is called
the Jacobian. Writing
out the real and imaginary parts
W;. = U;. + iv;. = f;.,
z. = x. + iy., we have 2n functions U;., V;. of 2n real variables x., y •. We
write briefly

6 DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS
REMARK. If I is holomorphic, a(u, v)/a(x, y) = Idet(al.,jaz.)I
Z ~ o.
Proof We write it out for n = 2 and leave the general case to the reader.
We use the Cauchy-Riemann equations
and set
a.A = aUA/aX. = aVA/ay.,
bVA = aVA/aX. = -au}../ay •. Then
au, av, oUz avz
all bll al2 bl2
ax, ax, ax, ax,
av,
=
-b'l all -bJ2 al2
aUI aU2 avz
ay,
aYI ay, aYI
a21 b21 a22 b22
-b21 a2'
-b22 a22
We perform the following sequence of operations: Multiply column 2 by i and
add it to column I ; do the same with columns 4 and 3. Then multiply row 1
by i and subtract it from row 2; do the same with rows 3 and 4. Making use
of the fact that B.A = aIA/aZ. = a.A + ib.A, we get
gil gl2
* *
a(u, v) gZI g22
* * = Idet(g.A)1
2 --=
0 0 gl2 o(x, y) gil
0 0 gz, gZ2
by interchanging columns 2 and 3 and rows 2 and 3. Q.E.D.
THEOREM 1.3. (Inverse Mapping Theorem) Let/: V -+ en be a holomor
phic map.
If
det(oJ,./oz.)lz= .. :F-0, then for a sufficiently small neighborhood N
of a,Jis a bijective map N -+ I(N);J(N) is open and/-'I/(N) is holomorphic
on/(N).
Proof The remark gives o(u, L,)/a(X, y) :F-0 at a. We then use the inverse
mapping theorem for differentiable (real variable) functions
to conclude that
I(N) is open,
I is bijective, and I-I is differentiable on I(N). Set qJ(w) =
/-I(W); then z" = cp,,[J(z)]. Computing,
o = a~1l = ± aCPIl a~A + a~" a~A
az. A=I aw}.az. awAaz.
But det(alA/az.) = det(a/A/az.) :F-O. So by linear algebra, aqJ,,/aWA = 0 and
qJ =/-1 is holomorphic. Q.E.D.

2. COMPLEX MANIFOLDS AND PSEUDOGROUP STRUCTURES 7
COROLLARY. (Implicit Mapping Theorem) Letf)., A. = I, ... , m be holo
morphic on V ~ en. Let rank (fJf)./fJz.) = r at each point z of V and suppose
in fact
that
det(iJf;./fJzvhsr :# o. If f;.(a) = 0 for A S; m for some a E V, then in
vsr
a small neighborhood of a, the simultaneous equations,
have unique holomorphic solutions
AS; r.
F or more details in this section one may consult Dieudonne (1960).
2. Complex Manifolds and Pseudogroup Structures
We assume given a paracompact Hausdorff space X which will also
generally be
assumed connected. We want to define what we mean by a com
plex structure
on X (or structure of a complex manifold) which will be an
obvious generalization of the concept of a Riemann surface. First we want
to assume X is locally homeomorphic to a piece of
C".
DEFINITION 2.1. By a local complex coordinate on X we mean a topological
homeomorphism z:p -+ z(p) E C" ofa domain U ~ X. z(p) = [Zl(p), ... , z"(p)]
are the local coordinates of X.
DEFINITION 2.2. By a system of local complex analytic coordinates on X
we mean a collection {Zj}jEI (for some index set I) of local complex co
ordinates Zj: Vj -+ C" such that:
(I) X=UUJ•
JEI
(2) The maps fjk: Zk(P) -+ Zj(p) are biholomorphic [that is, Zj 0 Zk-1 =
fjk and r;,/ = Zk 0 zj I are holomorphic maps from Zk( Vj n Vk) onto
Zj(Vj n Vk)] for each pair of indices (j, k) with Vj n Vk :# ljJ.
DEfiNITION 2.3. Two systems {Zj}jd' {II').}).'A are equivalent if the maps
Zj(p) -+ w).(p) are biholomorphic when and where defined.
DEfiNITION 2.4. By a complex structure on X we mean an equivalence class
of systems of local complex (analytic) coordinates on X. Bya complex mani
fold
M we mean a paracompact Hausdorff space X together with a complex
structure defined on X.

8 DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS
EXAMPLE, Complex projective space lPn, This is constructed from
en+
1
_ {O} by identifying (p '" q)p = (pO, pI, ... , pn) and q = (qO, ... , qn) if
and only
if
pA = cqA for some nonzero c E C, for 0 ~ A. ~ n. Then IPn = en + I -
{O}/'" is a compact Hausdorff space and one can construct a system of com
plex coordinates as follows:
We let
Vj = {p E IPnlpj ¥-O}. Then {Vj}jsn is an
, f rrM 0 V th ( OJ-I j+ I n) h
opencovenngo 10. n j emapzj= Zj,"',Zj ,Zj ,,,·,zj,were
z/ = pA/pj gives a local coordinate on Vj; in fact, Zj(V) = en. Then
fjk: Zk --+ Zj is given by zj = z:/zt for A =F k, z~ = I/z{. (One simply multi
plies by
pk/p
j
,) Thus we see that
{Vj, zJ is a complex analytic system defining
a complex structure on IPn.
Generalizing this procedure we introduce the idea of a pseudogroup
structure. All spaces
will be Hausdorff in what follows.
DEFINITION 2.5. A local homeomorphism f between two spaces X and
Y is a
homeomorphism
of an open set
V in X to an open setf(V) in Y. One has a
similar definition
of local diffeomorphism. A local homeomorphism (diffeo
morphism)
of X is such a map with X =
Y.
Let 9 be a domain of IRn or en. Letfand 9 be local diffeomorphisms of 9.
If open W £:; 9, fl W denotes f restricted to W which is the restriction off to
domain
(f) n W. If W is some open set such that
9 is defined on Wand
W nf(V) ¥-4l. then 9 of is defined onf-I[W nf(V)],
feU)
Figure 2
DEFINITION 2.6. A pseudogroup of transformations in 9 is a set r of local
diffeomorphisms
of
8 such that
(I) fEf=:.I-
IEr.
(2) fE r, 9 E r = go IE r where defined.
(3) fE r=/1 WE r for any open W£:; 8.
(4) The identity map id E r.
(5) (completeness) Let I be any local diffeomorphism of 9. If [} = u Vj
andll Vj E r for eachj, thenfE r.

2. COMPLEX MANIFOLDS AND PSEUDOGROUP STRUCTURES 9
DEFINITION 2.7. Let r (a pseudogroup on 9) and X (a paracompact Haus
dorff space) be given. By a system of local r-coordinates we mean a set
{ZjLd of local topological homeomorphisms Zj of X into 9 such that
Zj a Z;;l E r whenever it is defined. {w;.} and {Zj} are equivalent (f-equivalent)
if W;. a zj' E r when defined. A r-structure on X is an equivalence class of
systems of local r-coordinates on X. A r-manifold is a paracompact Haus
dorff space X together with a r-structure on X.
EXAMPLES
1. 9 = en, re = (all local biholomorphic maps of e").Thenarc-struc
ture is a complex structure, and a re-manifold is a complex manifold.
2. 9 = ~", fd = (all local diffeomorphisms of ~n). Then a fd-structure
is a differentiable structure and a fd-manifold is a differentiable manifold.
3. Let r be the set of a local diffeomorphism / of ~2" satisfying the
following condition.
The matrix
(e;..) will be defined to be
0 -1
1 0
0
0 -1
0
0 0 -1
1 0
where the blocks (? -~) occur on the diagonal and the rest of the entries are
zeros. If x = (x', ... , x2n) E ~2n,f(x) = [!t(x), ... ./2ix)] then the derivatives
of / should satisfy
A system satisfying Example I
is called a Hamiltonian dynamical system,
and such an / is a canonical trans/ormation. In this case a f-structure is
called a canonical structure.
4. Let r = (local affine transformations of
~"). These transformations
have the form
n
/A(X) = L a! x
Y + b
Y= ,
where the a~, b;' are constants and the matrix (a~) is nonsingular. In this case
a f -structure is called flat affine structure.
If pseudogroup structures f, and f 2 are such that f, c f 2' then every
system
of local
f, coordinates is a system of local f2 coordinates, and f,
equivalence implies r 2 equivalence. Hence, every f,-structure determines a

10 DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS
f2-structure. By assumption f c fd for all f. So every f-structure on X
determines a differentiable structure on X and every f-manifold is a differen
tiable structure on X and every f-manifold is a differentiable manifold. The
f-structure M is defined on the differentiable manifold X.
The problem of determining the f-structures on a given differentiable
manifold
M for given
f is one of the most important (and difficult) problems
in geometry.
It is known, for example, that if
X is a compact orientable
differentiable surface (real dimension 2), then the only complex structures
on X are those of the classical Riemann surfaces. I n case X = S2 (as a differen
tiable manifold), then X = pI complex analytically (this is a classical fact).
If the underlying differentiable manifold X is diffeomorphic to pn, then one
conjectures that X = pn complex analytically [see Hirzebruch and Kodaira
(1957)J, and Kodaira and Spencer (1958). If S211 is the sphere with its usual
differentiable structure, it can be shown [Borel
and Serre (1953) and Wu
(1952)] that s2n for n
=/; 1,3 has no complex structure
1
2n + I
[s2n = {(Xl' ••• , X2n+l) i~2 xf, (Xl'···' X2n+I) E 1R2n+I}J.
For S2 there is the usual complex structure. It has been recently proved by
A. Adler (1969)
that
S6 has no complex structure. As a final example, let M
be a compact surface and let f+ be the pseudogroup of all local affine
transformations,
v = 1,2
such that
We have:
THEOREM 2.1. [Benzecri (1959)] If a f+ -structure exists on M, then the
genus
of M is I. If M is not a torus, then M cannot be covered by any system {(x), X])} of local coordinates such that lax~/ax;;1 is constant on Uj n Uk
for each pair of indices (j, k).
The proof will not be given here.
We continue with
the definitions. Let M be a complex manifold, Wan
open set in M, and
{Zj} a coordinate system. Then a mapping I: W ~ C
l is
holomorphic (difJerentiable, and so on) if I 0 zj I is holomorphic (d(fJerentiable,
and so on) for eachj where defined. Let N be another complex manifold with
coordinates {II";.} and I: W -. N. Then I is holomorphic (differentiable, and so
on) if lI"A 0 I 0 zj I is holomorphic where defined.
DEFINITION 2.8. A subset SsM of a complex manifold is a (complex)
analytic subvariety if, for each
S E
S, there are holomorphic functions IA(P)
defined on a neighborhood lJ 3 S, 1 :::;; A. :::;; r, such that S n U = {p I/ip) = 0,

3. CONSTRUCTION OF COMPACT COMPLEX MANIFOLDS 11
I ~ A. ~ r}. Then fA = 0, I ~ A. ~ n, are the local equations defining S at s.
The subvariety S is called a submanifold if S is defined at each s E S by local
equationsf. = 0 such that
k [
iJf;.(P)] .. d d f
ran --= r IS 10 epen ent 0 s.
azj(p)
Suppose det(afA/azj)1 ~A:Sr =F O. Then letting
Isv,;,
w7(p) = lip),
w;(p) = z;(p),
for A = I, "', r
for A = r + 1, ... , n,
we have a local coordinate li'i = (wJ, "', wi» such that S: wJ = 11'] = ...
= wj = 0 (is defined by). Let (;(p) = IV'/A(p) = zj+A(p) for PES n Vj• Then
S is a complex manifold with local coordinates gj}'
We want to introduce meromorphic functions on a complex manifold.
They should be those functions which are locally quotients
of holomorphic
functions. More precisely:
DEFINITION 2.9. A meromorphic function f on M is a complex-valued
func
tion defined outside of some proper subvariety S of M (S =F M) and such that
given
q EM, there is a neighborhood V of q in M and local holomorphic
functions
g, h on V such thatf(p) = g(p)/h(p) for p E V -
S.
EXAMPLES
l. Any holomorphic mapf:M ..... [pI1 = C U {co}, [S =f-I(oo)].
2. In C2,j(Zl> Z2) = ZI/ZZ or f(zl' Z2) = P(ZI' Z2)/Q(Zl' Z2)' where P and
Q are polynomials.
3. Some Examples of Construction (or Description)
of Compact Complex Manifolds
First we have submanifolds of known manifolds ([pi", [pili X [pi", and so on).
Let [pi" have homogeneous coordinates «(0' "', (,,). Let fi 0, I ~ A ~ m be
homogeneous polynomials and define
M =
{( IfiO = 0, I ~ A ~ m}. Suchan
M is called a projective algebraic (or simply algebraic) variety. If the rank of
(afA/a(.>c is independent of (E M, then M is a complex manifold. These are
exactly the classical algebraic (projective algebraic) manifolds. In some cases
the equationsfA
=
0 give some easily read information about M. For instance,
if f is homogeneous of degree d, then Md = {Clfm = O} is called a hyper
surface in [pi" of order d. If at least one of (ofliJ().)«) =F 0, I ~ A. ~ n, for each
( E M d, then M d is nonsingular.

12 DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS
EXAMPLES
1. Md S;; 1FD2 a nonsingular plane curve of order d is a Riemann surface
of genus 9 = td(d -3) + I.
2. A nonsingular M d £; 1FD3. M d is simply connected and the Euler num
ber X(Md) = d(d
2
-4d + 6). [The formulas in Examples 1 and 2 can be ob
tained from Hirzebruch (1962), p. 91, Equation (5). They are well-known
classical formulas. The simple connectivity
is also well known and it follows
from the Lefschetz theorems
on hyperplane sections-see Milnor (I963),
p.
41.]
3. Let M
£; 1FD3 be defined by
M = {((.(2 -(0(3 = 0, (0(2 -(~= 0, (~-(1(3 = O}.
We claim that M is complex analytically homeomorphic to pl. One can easily
check
that the map
fJ: IFDI --.IFD
3 defined by fJ(t) = (t5, t~tl' tot~, tn where
t = (to , t,) E IFD I, is a biholomorphic map of IFD I onto M.
We remark
that in the cases of complex or differentiable structures,
sub
manifolds give many examples; but for general i-structures one does not
usually get sub i-structures.
Second we get quotient spaces.
DEFINITION 3.1. An analytic automorphism of M is a biholomorphic map
of M onto M. The set of all analytic automorphisms of M forms a group 9
with respect to composition. Let G £; 9 be a subgroup.
DEFINITION 3.2. G is called a properly discontinuous group of analytic auto
morphisms of M if for any pair of compact subsets K" K2 £; M, the set
{g E G I gK, n K2 =t= <p} is finite.
DEFINITION 3.3. G has no fixed points if for all 9 E G, 9 =t= 1, 9 has no fixed
points.
THEOREM 3.1. If G is properly discontinuous and has no fixed point, then
the quotient space
MIG is a complex manifold in an obvious natural manner.
Proof We shall assume that M is connected (or a countable union of
connected manifolds) and paracompact. Hence, M is
u-compact (a countable
union
of compact sets). Let MIG = {Gp P EM}, where G p = {g(p) pEG}
are the orbits of p E M. As notation set MIG = M*, Gp = p*. We shall show
that given q E M we can choose a neighborhood
V of q such that PI' P2 E V,
PI =t= P2 gives P; =t= p;. In fact, there is V 3 q such that gV (" V = <p for all
9 E G, 9 =t= 1. M is locally compact so let VI => V2 => V3 ... be a base of rela-

3. CONSTRUCTION OF COMPACT COMPLEX MANIFOLDS 13
tively compact neighborhoods at q. Then Fm = {g IgVm n Vm =F cp} is a finite
subset
of G and
Fm;;2 Fm +S' ;;2 •••• If 3gm E Fm, gm =F 1 for all m, then since
each
Fm is finite,
n Fm 3 g, 9 =F 1. Therefore, gVm n Vm =F cp, for all m and
Vm ~ q, gives g(q) = q, contradicting the nonexistence of fixed points. Hence
we cover M with open sets Vj such that PI' P2 E Vj implies pi =F p~ and
thus, Vj ~ V; = {p* I P E Vj} is I -1. We give V; the complex structure
that Vj has. That is, if Zj: P ~ Zj(p) is a local coordinate on Vj, then
zj: P* ~ zj(p*) = Zj(P) gives a local coordinate on M*. The system {zj}
then defines a complex structure on M* and the topology of M* is just the
quotient topology for the map
M
~ M*. Q.E.D.
EXAMPLES
1. Complex tori. Let M = cn. Take 2n vectors {WI' .", w2n}, Wk =
(Wkl' "., wkn) E C
n so that the Wj are linearly independent over lit Let
2n
G={glg:z~g(z)=z+ Lmkwk,mkEZ}.
k=1
Tn = en/G isa (complex) torus of complex dimension n. Let n = I and arrange
it so
that
WI = I, Wz = w, where the imaginary part of W is positive. Then
T=CI/G.
Figure 3
exp 2,,/ 2 I
We have a map C - C*, z~ w = e "z where C* = {zlz =F O}.lfwe first
take
g(z) = z + mlw + m2 and then exponentiate, we get
e2Iti(z+mlwl. So
exp 21ti 0 9 = oe
ml
• exp 21ti where oe = e2
"iw and g(z) = z + mlw + m2' and
0< lexl < 1 since Im(w) > O. Looking a little closer we see we have the diagram
C~C*
·1 n".' I~'
C-C
which commutes. Hence, if we let G* = {g* I g*: w ~ exmw, me Z}, we see
T = C/G = C*/G*.

14 DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS
Figure 4
2. Hop! manifolds. Let W = eN -{O} and G = {gIn I m E oZ, g(Wl' ... ,
wN) = «Xl 11'1' ••• , (XN wN), where ° < l(Xvl < I}. Then WIG is a compact com
plex manifold since it is easy to see that G is properly discontinuous and has
no fixed points on W. It is also easy to see that WIG is diffeomorphic to
Sl x S2N-l.
3. Let M be the algebraic surface (complex dimension 2) defined:
M = {( I a + (i + (i + ,~ = o} £ p3.
Let
G = {gm 1m = 0, 1,2,3,4 where g«(o, ... , (3)
= (p(o, p2( 1, p3(2' p4(3) and p = e21ti/5}.
Then 9 is a biholomorphic map p3 ~ p3 and g5 = 1. Consider the fixed
points
of gm on
p3. They satisfy (0 = v ~ 3), (p,"(v+l) -c) (. = ° and the
fixed points
are (l,
0,0,0), (0, I, 0, 0), (0,0, I, 0), and (0, 0, 0, I). These
points are not on M so there are no fixed points on M and M /G is a complex
mamfold. We saw before
that M is simply connected and
X(M) = d(d
2
-
4d + 6) where d = 5. Therefore, the Euler number of M is 55. Then the fun
damental group 1C1(M/G)
~ G and x(MIG) = II.
4. Last we have the classical examples of Riemann surfaces and their
universal covering surfaces.
If
S is a compact Riemann surface of genus 9 ~ 2,
the universal covering surface of S is the unit disk D = {z E e11lzl < I}. Then
S = D/G where each element of G is an automorphism of D and hence of
the form
. z -(X
g(z) = e
l8
--,
(Xz -I
I(XI < l.

3. CONSTRUCTION OF COMPACT COMPLEX MANIFOLDS 15
Finally we consider surgeries. Given a complex manifold M and a com
pact submanifold (subvariety) ScM, suppose we also have a neighborhood
W => S and manifolds S* c W* with W* a neighborhood of S*. Suppose
j: W* -S* -.... W -S is a biholomorphic map onto W -S. Then we can re
place
W by W* and obtain a new manifold M* = (M -W) u W*. More
precisely, M*
= (M -
S) u W* where each point z* e W* -S* is identified
with z
= j(z*).
-
f
[-~J
Figure 5
EXAMPLE I. Hirzebruch (1951) Let M = 1Jl>' X 1Jl>'. In homogeneous
coordinates, 1Jl>' = {C/ ( = «(0' ~,)}; = {C u {(Xl in inhomogeneous coordin
ates, (= (d(o e C u too}. M = 1Jl>' X 1Jl>' = {(z, 0 I z e 1Jl>', (e 1Jl>1} contains
S = to} X IJl>I and W = D X 1Jl>' where D = {zllzl < e} is a neighborhood of
Sin M. Let W* = D X 1Jl>'* = {(z, (*) I zeD, (* e 1Jl>'*} and S* = to} x 1Jl>*.
Fix an integer m > 0 and define j: W* -S* -.... W -S as follows:
j(z, (*) -.... (z, 0 = [z,«(*/z'")] where 0 < Izl < B.
Then j is biholomorphic on W* -S* and let M! = (M -S) u W* where
(z, 0 = (z, (*) if (* = zln(, 0 < Izl < f:.
REMARK. M and M! are topologically different if m is odd.
Proof (for m = I). M = 1Jl>' X IJl>I is homeomorphic to S2 x S2. We
show that the homology intersection properties
of M and
Mi are distinct,
hence, proving that they are topologically different. A base for HiM, Z) is
given by {SI' S2} where SI = to} X 1Jl>1, S2 = IJl>I X to}. Hence, any 2-cycIe C
is homologous ("') to as, + bS2, a, b e Z. The intersection multiplicity
I(C, C) = J(aS, + bS2, aSI + bS2) = a
2
[(SI' SI) + b
2
[(S2 ,S2) + 2abl(SI' S2).
Since St. S2 occur as fibres in IJl>I x 1Jl>1, [(SI, S,) = [(S2' S2) = o. Hence,
I(C, C) = 2ab = 0 (mod 2). (1)

16 DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS
In M we have the following picture:
w w'
M
./""
~
I--V
s, s .
Figure 6
where At" is the submanifold of M~ defined by C = c and C* = zc with the
coordinates explained before.
Then
At" is a 2-cycle and Ao '" Ac. Hence
/(Ao, Ao) = /(Ao, Ac) = 1. Since for any 2-cycle Z on M, /(Z, Z) == 0 (mod 2)
we see M::f: Mr.
REMARKS
1. M!::f: M:(m ::f: n) as complex manifolds.
2. M~m = M topologically.
3. M~m+1 = M~ topologically.
These facts are proved in Hirzebruch
(1951).
EXAMPLE 2. (Logarithmic Transformation) LetM = T x Pl,T = C/G,
G = {mw + n I m, n E 7L, 1m w > O} where T is a torus of complex dimension 1.
For any C E C, we denote the class in C/G = T by [C]' We perform surgery on
M as follows: Let S = {O} x T, W = D x T where p1 = C u {<X)} and
OED = {z E Clizi < e}.
w
T
s
Figure 7
Then set W* = D x T = {z, [(*] I zED, [(*] E T} and S* = {OJ x T £ D x
T. Define/: W* -S*~ W -S as follows:
/: (z, [(*]) ~ {z, [(* + (l/2ni) log z]},
where 0 < Izi < e.

3. CONSTRUCTION OF COMPACT COMPLEX MANIFOLDS 17
Then f is biholomorphic and we can form M* = (M -S) u W*, where
(z, [C]) = (z, [(*]) if [C] = [(* + (1/2ni) log z], 0 < Izl < 6.
REMARK. For the first Betti numbers bl we have b2(M) = bz(T) = 2, but
bl (M*) = I. In fact, M* is topologically homeomorphic to S3 x sl.
Proof H2(M, Z) ~ 7L ffi 7L is clear by the Kunneth theorem. To study
M*, first we notice that M -W = (l?1 -D) x T is homeomorphic to a x T
where a is a closed disk, and T is homeomorphic to SI x Sl. If ( = x + yw,
we can identify [C] with (x, y), where x + 1 is identified with x, y + 1 with y,
where x and yare real (E IR). Therefore W* = D X Sl X Sl, M -W = a x
Sl x Sl. Since we are only interested for the moment in the topological type
of M* we may as well assume that D is the unit disk and that the identification
in
the definition of surgery takes place on the boundary of D =
{e
i91 0 $ () $
2n}. Then we identify (w, x*, y*) and (w, x, y) if x = x* + «(}/2n), y = y*.
Hence, M* = B X SI where B is a circle bundle over S2 ; and in fact, we easily
see
by the transition function that this is the Hopf bundle
S3 -+ S2. Hence
B = S3. This proves that M* = S3 X Sl; b1 (M*) = 1 follows.
EXAMPLE 3. We mention also the classical quadric transformation
(blowing up, u-process). First we discuss the case where M has complex di
mension 2. Let S = p be any point on M, and let S* = pI be a copy of the
Riemann sphere. We define M* = (M -p) u pi as follows: Choose a co
ordinate patch W = {(ZI' z2)llz11 < 6, IZ21 < 6} in a neighborhood of p so that
Zl(P) = zz{p) = o. We define a submanifold W* of W x pI as follows:
W* = {(ZI' Z2; (I' (2) E W x pi I ZI(2 -Z2 (I = O},
where ((1(2) are homogeneous coordinates on pl. W* is a submanifold since
(aflaz1) = (2' (af/azz) = -(I iff= ZtC2 -ZZ(l' and hence [(af/az1 ), (af/azz)]
:F (0, 0). Letf*: W* -+ W be the restriction of the projection map W x pi -+
W to W*. Then W* 20 X pi = S*, f*: S* -+p = (0,0), andf*: W* -S*-+
W -pis biholomorphic. The first two statements are obvious. For the proof
of the last, let (ZI' Zz; (10 (z) f/ S*. Then at least one of Zi:f: 0 and hence
«(I' (z) is determined by (Zl' ZZ)f*-I: (Zlo Z2) -+ (Zl' Zz; Zl, zz). By surgery we
obtain M* = (M -p) u pl. We make the following definition:
DEFINITION 3.4. The quadric transformation Qp with center p is the mani
fold
Qp(M) = M*.
REMARK.
QPm··· Qp,(pZ) can be complicated! For example,
Qp6 ... Qp,(pz) = {( I(~ + ci + ,~ + (~ = O} S; p3.

18 DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS
For manifolds M of dimension ~ 2 we proceed analogously. If
dime M = n, let (Zl •... , zn)
be coordinates centered at pEp = (0, ···,0)]. If W = {(Zl' ... , zn)llz,,1 <
E, I $ C( $ n}, we set W* = {(z, c) I Z;. C. -Z. C ... = O,i $ A, v $ n} s;;; W x pn-l.
Again W* is a manifold, projection onto W defines a biholomorphic map
W* -pn-l -+ W -p, by (z, 0 -+ z. We form M* = (M -p) u W* =
(M -p) U pn-l and call M* = QP(M) the quadric transform of M with
center p.
4. Analytic Families; Deformations
Consider a torus Tro = CfG, G = {mw + n I m, n E 71. 1m w -+ O}. We have
a family
of tori depending on the parameter w. Many examples of compact
complex manifolds depend on parameters built into their definitions. We also
have the examples
of hypersurfaces of degree d in
pn. Each such surface
Md = {C 1/<0 = O} is defined by a function I of the form 1= Lka+'.'+kn=d
aka ••. kn 'io ... C~n. In a sense to be made precise Md depends" analytically"
on the coefficients aka •.• k
n off We make the following definition:
DEFINITION 4.1. Let B be a (connected) complex manifold and let
{M,l t E B} be a set of compact complex manifolds depending on t E B. We say
that M, depends holomorphically (or complex analytically) on t and that
{M,l t E B} forms a complex analyticlamily if there is a complex manifold .It
and a holomorphic map (jj onto B such that
(I) (jj-l(t) = M, for each t E B, and
(2) the rank of the Jacobian of (jj is equal to the complex dimension of
B at each point of .It.
We note that (2) implies M, is a complex submanifold of .It.
Now for some examples. As before, we denote Tw = CfG,
G = {n + mw I n, m E 71., 1m w > OJ.
Let B = {w 11m w > O} c: C. Let f§ = {9mn 19mn: (w, z) -+ (w, Z + mw + n)}.
Then f§ is a properly discontinuous group of transformations on B x C with
out fixed point. Hence, .It = B x C/f§ is a complex manifold. The projection
map B x C -+ B induces a hoi om orphic map .It ..!! B, and (jj-I(W) = Tw. It is
easy to see that the Jacobian condition is satisfied so {T", I wEB} forms a
complex analytic family.
But suppose we proceed as follows: Again Tw = CfG and the map
C -+ CfG is written Z -+ [z]. Let D = unit disk = {tlltl < I}. On D x Tro con
sider the group f§ = {I, 9} where 9: (t, [z]) -+ (-t, [z + !]) is of order 2.

4. ANALYTIC FAMILIES; DEFORMATIONS 19
Then I'§ is properly discontinuous and has no fixed points so D x T(J)/I'§ is a
complex manifold. Let
1t: D x
T(J) -+ D be defined by (t, [z]) -+'t = t
2
• Then
the diagram
(t, [z])
~ (-t, [z + t])
.j " j.
t2 __ t2
commutes so 1t defines a holomorphic map on .;It. The Jacobian condition
is not satisfied by 1t, since (j'J't/ot) = 2t = 0 at t = o. We notice that 1t-
1('t) =
T (J) if't =1= 0, but 1t -1(0) = T*, a torus of period w/2.
DEFINITION 4.2. Let M, N be compact complex manifolds. M is a deforma
tion
of N if there is a complex analytic family such that M, N
s;;; {Mt It E B},
that is, Mta = M, Mtl = N.
We have the following sequence of problems to guide our work:
PROBLEM. Determine all complex structures on a given X.
PROBLEM. Determine all deformations of a given compact manifold
M.
PROBLEM. (easier?) Determine all "sufficiently small" deformations
of a given M.
DEFINITION 4.3. We say that all sufficiently small deformations have a cer
tain property f!jJ if, for any complex analytic family {Mt I t E B} such that
Mta = M, we can find a neighborhood N, to ENe B such that Mt has f!jJ for
each
tEN.
By standard techniques in differential topology we prove the following
theorem:
THEOREM 4.1. Let Mt be a complex analytic family of complex manifolds
Mt• Then M t and Mto are diffeomorphic for any t, to E B.
Proof The reader will notice that we really only use the differentiability
of the map 1t: .;It -+ B, analyticity is not needed. In fact, we prove: Let .;It be
a differentiable family
of compact differentiable manifolds such that the
dif
ferentiable map 1t: .;It -+ B has maximal rank (.;It and B are differentiable
manifolds). Then
Mt is diffeomorphic to
Mta•

20 DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS
First we construct a Coo vector field 0 on a neighborhood of M,o in ..it
such that 1t induces 7t*(0) = a/as, where s is a member of a coordinate system
(s, x
2
, ••• ,xm) in a neighborhood of the point 10 e B chosen as follows:
Figure 8
We connect 10 and t by an embedded arcy: (-e,1 + e) -+ {yes) I s e( -e,1 + e)}.
A compactness argument shows that we can assume that I and 10 lie in the
same coordinate patch
and since
y is an embedding we can find a chart with
coordinate
(s,
12 , ••. ,1m) around lo(to = (0, ... , 0), I = (s, 0, ... , 0». Because
of the rank condition, 1t-
1(y) = 7t-
I{(s, 0, ···,0) I -e < s < 1 + e}, is a
submanifold
of
..it, and we can assume that (s, xf, ... , xj) are coordinates
of ..it for a given point of 7t -I (y) in some neighborhood qJ J of the point. Then
the vector field (a/as)j on qJj satisfies 1t.(a/as)j = a/as. Then if {Pj} is a parti
tion
of unity subordinate to
{OIl j} (uOU j is a neighborhood of M,o)' the vector
field 0 = LJ pj(a/as)j satisfies our requirements.
For the second part of the proof we seek a solution of the differential
equation
d
ds
xj(r) = 0j[x(r)], (1)
where 0j is the a-component of 0 in the coordinate patch qJ j' with initial
conditions xj(O) = y", where (0, y2, ... , y") is some point close to (0, ···,0).
If s is small enough and Iyl is small enough, Equation (1) has a unique solution
xj(r, y) on some small interval. By compactness, we can assume that M,o c:
U jqJj' a finite union of such patches, and that in each qJj' (1) is satisfied for
Irl < jJ. where jJ. is independent ofj. If xj(r, y) is such a solution, let Xj = jjk(Xk)
and define ef[r,hiO, y)] uniquely on qJj ( qJk by
(2)
Then dxj(r, y) = L axj aef[r,jkj(O, y)]
dr /I axf or '

4. ANALYTIC FAMILIES; DEFORMATIONS 21
and by the uniqueness of the solution to (I)
x/I:, y) = jjk(Xk[T,f,.iO, y)]). (3)
Equation (3) implies that xC-t", y), ITI < p, y E M,O is a well-defined differen
tiable map defined on M,O for each T, It I < p., and x(O, y) = y. Let cpiy) =
x(t, y); then CPo = id (on M,o)' It is also easy to check that 1t[cpt(Y)] = yet)
since 1t.(0) = dlds. Hence, CPt maps M,O into My(t) (for small t). We can re
peat this argument for My(t) and define t/lv: My(t) -+ My(t+v) and by uniqueness
get t/I -t 0 CPt = id, CPt 0 t/I _to = id. Since everything is differentiable, the
theorem is proved. Q.E.D.
REMARK. This argument is very old. For a treatment from the point of
view of Morse theory, see Milnor (1963). Sometimes this theorem is attributed
to Ehresmann (1947).
We consider some more examples
of complex analytic families. The de
pendence
of the complex structure of M
I on t E B can be complicated as we
shall see.
EXAMPLE I. Consider again the family of tori {Teo I w E H} where
H = {w 11m w > O} and Teo = CjG, G = {mw + n I m, n E Z}. From the clas
sical theory of Riemann surfaces we see that Teo and Teo' are conformally
equivalent if Wi = (aw + blew + d) where a, b, e, dE Z, and ad -be = l.
Let r§ be the group of transformations acting on H which have the form
aw + b
w-+ ,
ew + d
a, b, e, dE Z, ad -be = l.
Then it is easily seen that r§ is properly discontinuous on H. A fundamental
region IF for r§(ug§ = H, g§ n IF = cP if g ::f id) is given by the shaded
region in the figure below, hence Teo ::f Teo" if w ::f Wi and w, Wi E IF.
1
Figure 9

22 DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS
The elliptic modular function J defines a conformal map J: H/~ -+ C. So
Tw = Tw' if J(w) = J(w').
EXAMPLE 2. (n-dimensional tori) We give an outline of some of the
facts. A torus
Tn =
Cn/G where G = {L7~ 1 miwi I mi eZ} where WI'···' W211
are complex n-vectors linearly independent over R
(a) We can replace {wi} by any other linearly independent basis of G.
That is,
2n
wi). = L aikwU,
k=l
(4)
where
aik e
Z, det (aik) = 1 are also permissible generators of the lattice
(group)
G.
(b) We may also introduce new coordinates in c
n so
Z). -+ 2)., where
n
2l = L ZvYvl, Yvl e C,
v=1
Then,
(5)
The resulting change from Equations
(4) and (5) becomes
(6)
We may assume that
Wn+1, ••• , W2n are C-linearly independent. Hence by
some change
of coordinates
(Yv).), we can obtain
... w) (6)11
. • . W In (Yv).) = 6>nl
2nn
where I is the n x n identity matrix.
I
Q)ln)
Wnn , (7)
So
we may assume (wij) =
(~), where n = (wij) 1 ~ i, j ~ n and 1=
(b···~)·
(c) We can also break (aik) into pieces:
Then (4) takes the form
wi). = (ajk)(~) = (gD, n~ = An + B, n~ = cn + D.

4. ANALYTIC FAMILIES; DEFORMATIONS
If one assumes that 0; is invertible, then (~D(O;)-I = (~') where
0' = (AO + B)(CO + D)-I,
det(~ ~) = 1.
23
(8)
The following treatment will be a bit sketchy; for more details
con
sult Kodaira-Spencer II (1958). The fact that WI' "', Wn, (l, 0, "', 0),
(0, I, 0, 0 ,), ... (0, ... , 0, I) are real linearly independent implies
det(~ ~) t= 0,
which is the same as (2it det [Im(wJ).)] t= o. Consider the space H = {O
det(Im 0) > O} [some sort of a generalization of 1m W > 0 in Example (1)].
Let C§ = the set of all transformations
0-+ (An + B)(CO + D)-I = n',
where (~ ~) E SL(n, I), the invertible integral matrices of determinant + I.
This group does not really act on H since it is possible for CO + D to be
singular; one should consult Kodaira-Spencer for more details.
H should be
extended to something more general on which
SL(n,
I) acts. In any case,
Tn= Tn-, if 0' = gO, 9 E C§.
We would like to form H/C§. But it turns out that C§ is not discontinuous. In
fact, for any open set
U
c H, there is a point n E U such that {gO Ig E 'Y} n U
is infinite. Hence, the topologial space H/C§ with the quotient topology is not
Hausdorff and hence certainly not even a topological manifold by the usual
definition.
We next give some examples
of families {M t
1 t E B} such that Mt = M
for
t
t= to and Mto t= M.
EXAMPLE 3. A Hop! surface is a compact complex manifold of complex
dimension two which has
W =
(:2 -{(O, O)} as universal covering surface.
More precisely, the
Hopf surface M t is defined by M t =
WI Gt where G t =
{gm I mEl} and g: (ZI' Z2) -+ (azl + tz2, aZ2)' that is, (::) -+ (~ :)(:~),
where 0 < lal < I and t E C. Then M t is a compact complex manifold.
LEMMA 4.1. {M t It E q is a complex analytic family.

24 DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS
Proof = {Mt I t E C} = C x W/f, where f = {ym I mE Z}, and
y(D = (~ i DeJ Q.E.D.
We claim
(1)
M, = Ml (complex analytically) for t
#= O.
(2) Mo #= MI'
Proof of (1). We make the following change of coordinates:
Then the equation
implies
that M 1 = M t when t
#= O.
Proof of (2). First we prove a special case of Hartog's lemma.
LEMMA 4.2. Any holomorphic function defined on W = C
2
-{CO, On can
be extended to a unique holomorphic function on C
2
•
Proof Let f(zl' Z2) be the function on W. Pick a number r > 0, and
define the function
1 1. few, Z2)
F(Zl' Z2) = -. j dw,
2m Iwl=r w -ZI
for Izd < rand Z2 arbitrary. Then F(zl' Z2) is an analytic function in its cylin
der
of definition which is a neighborhood of
(0, 0). If we can prove f = F
where
both are defined, we will be finished. We know thatf(w,
Z2) is holo
morphic
if
Z2 #= O. So Cauchy's theorem gives
Fix Zl' 0 < Izd < r. Then F(ZI' Z2) = f(zl' Z2) for Z2 #= O. Both are analytic
in Z2; therefore,
F(zl' 0) = f(zi' 0).
Hence they agree where defined, proving the lemma.
Now let us suppose M, = Mo. I oF O. Then there is a biholomorphic map
f: M, -+ Mo. W is the universal covering manifold of M, and Mo. sofinduces

4. ANALYTIC FAMILIES; DEFORMATIONS
a map I: W --t W which is biholomorphic, such that
W~W
G'l f I Go
Mt---+Mo
commutes.
It follows that Gt = 1-' Go! Hence for generator 9t of Gt,
9, =1-'9"5' f.
Write the map I in coordinates as
I(z" zz) = [J,(z" zz)'/z(z" Z2)].
25
(9)
Then by Hartog's lemma extend liz" zz) to a holomorphic function F;.
(z" zz) on C
Z
• Then F maps C
Z into C
Z [F = (F1, F2)], and F(O) = O. For if
not, extend
1-1 to
F which satisfies F[F(z)] = z on Wand by continuity,
F[F(O)] = O. But if F(O) =F 0, £[F(O)] = I-I [F(O)] =F O. This contradiction
gives the result. Now expand F)"
F;,(ZI' zz) = F)"z, + F).,zz + F;'3ZT + F) .• ZIZ2 + ....
We know thatf[9,(z)] = 9"5 I [f(z)] so
Rewriting this gives
(
O)±I
F[g,(z)J = ~ rx F(z).
F,(rxz, + lZ2, rxZ2) = rx±IF,(ZI' zz),
Fz
«(1.z1
+ lz2, cxzz) = rx±IP1(ZI, zz)·
Expanding these and taking the linear terms yields
(P
Il P'z) (rx t) = (rx 0) ± 1 (Pll
Pli P1.2 0 rx 0 rx Fl,
This can only happen when t = O. Hence M 1 =F Mo. Q.E.D.
EXAMPLE 4. Ruled Surfaces (examples of surgery) Our ruled surfaces
will be IFDI bundles over IFD'. Let IFDI = {' I' E C U {oo}} (nonhomogeneous
coordinates).
M(m) =
VI x 1FD1 U Vl x.1FD
1 where VI u Vl = IFDI, VI = C,
V
l
= 1FD1 -{O}, and identification takes place as follows (recall Section 3):
Let (ZI' (I) E VI x 1FD1, (Zl' ~2) E Vl X IFDI. Then
REMARK. MC",) =F M(I) for m =F t' (not to be proved now).

26 DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS
THEOREM 4.2. M(t) is a deformation of M(m) if m -t = 0 (mod 2). Assume
that In> f. Then there is a complex analytic family {Mt I I E C} such that
Mo = M(m) and M, = M(f) for 1:1= o.
Proof Define M, as follows: M, = VI X pI U V2 X pI where (ZI' (I)
+-4(Z2' (2) if ZI = l/z2, (I = Z~(2 + tz~ where k = !(m -t). Then it is easy to
see
that {M, t E
q is a complex analytic family and that Mo = M(m).
Suppose 1:1= O. Introduce new coordinates on the first PI by
(' _ z~' I -t (linear fractional transformation).
1-1(1
On the second pi,
r' '2
'>2 = I '" kv + t2·
22 C,2
Then, using ZIZ2 = I, and (I = Z~'(2 + IzL we get
Hence, in the new coordinates, ZI Z2 = I, (~ = z£(~; so
for t:f= O. Q.E.D.
PROBLEM. Finda pair of complex analytic families {M,lltl < I}, {N,II/I < I}
such that
(a) Mo =1= No,
(b) M, = No for t =1= 0,
(c) N, = Mo for t =1= O.
(not complex analytically
homeomorphic)
There are no known examples
of this type.

[2]
Sheaves and Cohomology
I. Germs of Functions
Let M be a complex (or differentiable) manifold. A local holomorphic
(differentiable)function isaholomorphic (differentiable) function defined on an
open subset U £; M. We write D<f) for the domain of f Let p E M and suppose
given local functions
f, g such that D(f)
11 D(g) 3 p. We say that rand g are
equivalent at p if
I(z) = g(z) for z E W
£; D(!) 11 D(g), Wa neighborhood of
p. By a germ 01 a lunction at p we mean an equivalence class of local functions
at
p. Denote by Ip the germ of
1 at p, (!)p the set of germs of all hoi om orphic
functions
at p, and
£1)p the set of germs of all differentiable functions at p.
The definitions
rxlp + pgp = (af + pg)p
fp' gp = (fg)p,
rx, p E C,
are well defined, hence, (!)p, £1)p become linear spaces over IC. We also define,
We
put a topology on (!) and
El) as follows: Take any cp E (!) (or El); then
cp E (!)p (or El)p) for some p. Take any holomorphic (differentiable) 1 with
Ip = cp and define a neighborhood of cp as follows:
where
p E
U £; M, U is an open set in D(f). It is easy to see that the system of
neighborhoods (il1(cp;f, U) defines a topology on (!) (or £1).
EXAMPLE. (!) on the complex plane C. Let p E IC. Then if 1 and g are
holmorphic
at p we have expansions valid in some neighborhood of p,
co 00
fez) = L fk(z -pt, g(z) = L gk(Z -pt,
k=O k=O
so 1 and g are equivalent at p if and only if Ik = gk for all k. Hence, the germ at
p is represented by a convergent power series; (!)p = ring of convergent power
series. And an element cp E (!)p can be represented by cp = Ip = {p;/o ,fl' ... }
where Iimk .... oo I/kll/k < + 00 and the radius of convergence is r(cp) = 1/ lim.
27

28 SHEA YES AND COHOMOLOGY
We define
0fI{(J); E) = {t/I I t/I = Iq, Iq -pi < E where 0 < E < r{(J)}.
In terms of our representation we calculate
00 00
I(z) = L Ik(Z -p)k = I fm{Z _ q + q _ p)m
k;O 111;0
Hence
0fI«(J); E) = {t/I I t/I = (q; go,"', gk" .. ), I(q -p)1 < E
gk = m~k (;)fm(q -p)m-k}.
We note
that
t/I E d//{(J); E) means that t/I is a direct analytic continuation of (J).
The case of ~ on IR is not so simple. If (J) = Ip where I is a Coo function
atp,
III
j(x) = I fk(X -p)k + O(x _ p)m.
k=O
But I is not determined by the Ik'S since there exist COO functions I which are
not identically zero, but which have all derivatives zero
at some point.
Define
w: (!) (or ~) -. M by w «(!)p) = p.
PROPOSITION 1.1. (1) w is a local homeomorphism (that is, there exists
0fI such that w: 0fI«(J);f, U) -+ U is a homeomorphism).
(2) w-l(p) = (!)p (or ~p) (obvious).
(3) The module operations on w-l(p) are continuous (that is, IX(J) + IN
depends continuously on (J), t/I).
Proof (1) 0fI«(J);f, U) = {fq I q E U} and w: /q -+ q is certainly 1 - 1. It
is obvious that w is continuous. To show that w-
l is continuous, let OfI(w; g, V)
be a neighborhood of t/I = f . We want to find a neighborhood W of q so
4
that fw=W-I(W)EOfI(t/I; g,V) for wE W. We know that 9q=t/I =Iq, so /
and g are equivalent at q. Hence, 1= 9 in some neighborhood N of q. Let
W = N n V. Then/w =gw on W, so Iw E 0fI(t/I; g, V) for wE W. This proves
that the w-
l is continuous.
(3) Let (J)=/p,t/I=gp' Then 1X(J)+pt/l={~f+pg)p. Let OfI{IX(J)+pt/l;
h, U) be a neighborhood of IX(J) + Pt/l. Then IX(J) + pt/l = hp = (IX! + pg)p so

1. GERMS OF FUNCTIONS 29
h =rxf + flg in some neighborhood V£ U of p. Then if U E OJI(lp;J, V),
• E all(1/1; g, V), we have
rxu + fl. = rxfq + flgq
= (rxf + flg)q
= hq E OJI(rxlp + fll/l; h, V).
Since OJI(rxlp + fll/l; h, V) £ O//(rxlp + fll/l; h, U) we are done. Q.E.D.
We now give a formal definition. Let X be a paracompact Hausdorff
space.
DEFINITION 1.1. A sheaf
Y over X is a topological space with a map w:
Y --. X onto X such that
(1) iii is a local homeomorphism [that is, each point s E 9' has a neigh.
borhood all such that w: OJI -+ w(OJI) c X is a homeomorphism onto an open
neighborhood
of
w(s)].
(2) iii-1(x), x E X is an R-module where R = 71., IR, C, or principal
ideal ring.
(3) The module operation (s, t)
-+ rxs + flt is continuous on w-1(x)
where rx, fl E R.
(The reader can easily generalize this definition,
but for our purposes it
suffices.) The set
Y" = w-1(x) is called the stalk of Y over x.
EXAMPLES. (of sheaves)
(l) (!J on a complex manifold.
(2) ~ on a differentiable manifold.
(3) The sheaf over X of germs of continuous (Ill or C valued) functions.
(4) The sheaf over X of germs of constant functions.
In Example
(4)
9' = X x C with the following topology: Let s = (x, z);
then OJI(s) = {(y, z) lyE U, z fixed}. If r -+ f(r) is a continuous map into Y of
1= {r I a < r < b}, then f(l) = {(y, z) I z fixed and y = w(f(r»r E l}. In other
words
we give
X x C the product topology where X has its given topology
and C has the discrete topology.
DEFINITION 1.2. Let U be a subset (usually open) of X. By a section u
of 9' over U we mean a continuous map x --. u(x) such that iii u(x) = x.
Suppose X = M, a complex (or differentiable) manifold; and suppose Y =
(!J (or ~). If f(z) is a holomorphic (or differentiable) function on U, then
u: p -+ fp, p E U is a section.

30 SHEAVES AND COHOMOLOGY
PROPOSITION 1.2. Let (1: V ~ f/ be a section (f/ as above). Then (1 deter
mines a holomorphic (or differentiable function) 1 = I(z) on V such that
a(p) = Ip.
Proof a(p) E (!)p (or ~p). Hence there is a holomorphic (or differen
tiable)
g(z) defined on some neighborhood of p so that
a(p) = gp. Since g
depends on p we write, g(z) = gCpl(z). Define 1 as follows:
I(p) = g(P)(p).
Then 1 is obviously well defined. Then
(1) I(p) is a holomorphic (differentiable) function on V.
Proof Take Wa neighborhood of p, W ~ V. Let dIJ = dlt[a(p); g(Pl, W]
= {(g(P I q E W}. Since a is continuous, for any small neighborhood N
of p, N ~ W, we have a(N) ~ dIJ. Hence a(q) = (g(P . But we also
know a(q) = (g(q»q. Thus, (g(q = (g(P\, and g(q)(z) = g(Pl(z) for z in a
small neighborhood
V of q, V
c N. But I(q) = g(q)(q) = g(p)(q) for q E V. So
I(z) = g(P)(z) for z E V and g(P) holomorphic (or differentiable) in V implies
that 1 is also.
(2) By definition a(p) = (g(Pl)p = Ip for each p E V. Q.E.D.
Hence
we have the maps:
local holomorphic (differentiable) functions
t
germs
t
sections = holomorphic (or differentiable) functions.
re V, f/) will denote the R-module consisting of all sections of f/ over
V. We remark that re V, (!) are all holomorphic functions over V and
re V,~) are all differentiable functions over V. Let {V ;.11 ~ A. ~ n} be a
finite family
of open sets in
X such that n V). ::f: ljJ. Let a). E rc V)., f/) and
IX;. E R. Then L IX). a). E reV, f/) where V = n V).. Let W be an open set and
a E rc V, f/) for some open set V. Then x -+ a(x), x E W n V defines a
section
of
rc W n V, f/). We denote this section by r wa and call it the
restriction
of
a to W n V.
2. Cohomology Groups
Let X be a Hausdorff paracompact space and let f/ be a sheaf over X.
Fix a locally finite covering Ii/i = {Vj} of X. A O-cochain CO on X is a set
CO ={aj} of sections a
J E reVj')' A 1-cochain C· = {ajd is a set of sections

2. COHOMOLOGY GROUPS 31
Ujk E qUj n Uk, f/) such that Ujk = -Ukj (skew-symmetric). A q-cochain
c
q = {ujo '" A} is a set of sections ujo "'ik E qujo n ... n Uj.' f/) which
are skew-symmetric in the indices
jo ...
A. Let Cq(Olt) be the R-module of all
q-cochains. We define a
map
Cq(0lt)...!..Cq+1(0lt), the coboundary map as
follows:
For O-cochains,
t5Co = {Tjk} = {Uk -uJ where CO = {ud; for
l-cochains C
1 =
{ujd, t5C1 = {Tjk/} where Tjkl = Uu -ujl + ujk = Ujk + Ukl
+ Utj. In general, t5C
q = {Tjo ... j.+.} if C
q =
{Ujo ... j.}, where
+ (_l)q+lujo"'j'
= L (-l)kujo··· j~ ... i.+.' (1)
where 1 means "omit."
We denote the q-cocycles by
zq(Olt) = {Cq I t5Cq =O}.
The q-cohomology group (with respect to Olt) is
(2)
We should remark that t5C is always skew-symmetric and t5t5 = 0 so that
t5Cq-l(Olt) £; zq(Olt) and Equation (2) makes sense. The qth cohomology group
of X with coefficients in the sheaf f/ is defined to be
Hq(X, f/) = lim Hq(Olt, f/).
'II
This limiting process will now be explained. We say that the open covering
"Y = {V;J AeA of X is arefinementofOlt = {U JieJ if there isamap s: A -+ J such
that VA c U.(A) = Uj(A) , where we setj(A) = SeA). We define a homomorphism
where
It
is easy to check that
n~ : (q(OU) -+ U("Y),
nt: {uio·"j.} -+ {TAO'" A.}'
t5n~ = n~ t5,
(3)
(4)
so that
n~ maps zq(°lt) into zq("Y) and t5Cq-1(fI) into t5Cq-l('f"). Hence n~
induces a homomorphian n~: Hq(:5II) --+ Hq('f").
LEMMA 2.1. n~: Hq(:Jlt) -+ W('f") is independent of the choice of map
s : A ...... J in the definition of refinement.

32 SHEAVES AND COHOMOLOGY
Proof First some notation: fix indices IXo, ••• , IXq E A. Let
A.
V = V; (')... (') V; VI = V; (')... (') V; (')... (') V;
Clo CEq , (10 fl.1 IZq ,
./"-... ./"-...
Uil = Uf(ao) (') ... (') Uf(aj) (') Ug(a
j
)
(') ••• U9(II/) (') ... (') Ug(aq),
and
Ui = Uf(II,) (') ... (') Uf(IIj) (') Ug(II
J
)
(') ••• (') Ug(II.)'
where!, g: A ~ J are two refining maps. Define a function (kU)A, ... A. by
q
(kuh, ... A. = p"f;o ( -1)P-1 rVou f(A,)· .. f(Ap)g(Ap)·· ·g(A.) (5)
Let us call the maps n~, defined by f and g,f*, and g*. We claim that the
following
equation holds:
[(ok + kO)]IIo ... a. = (g*u -1*u)ao ... a •. (6)
The function ku is not necessarily skew-symmetric in its indices; so we
skew-symmetrize
-r~I .. · A = (k'-r)A' ... A =~, L sgn(Al
• • q. P.I
Next we use (6) to see that
[(15k' + k'b)u]ao ... aq = (g*u -1*u)IIo ... II •.
Hence, if bu = 0, bk'u = g*u -f*u E bC
q
-1(1').
Hence,f* and g* induce the
same map, Hq(CJlt) ~ Hq(1'). Therefore we prove (6). The reader can easily
check
the following calculations:
q (bku)ao ... II = L ( -1)' ry(ku)w··;, ... a.
• 1=0
= t ( -1 try [tt -l)i ryt u f(IIo)··· f(IIi9(IIi) ... g{;;) ... g(a.)
{=o i=O
+ t (-l)i -1 rv{ u f(IIo) ... ;(a';)f(IIJ)g(IIJ) ... g(a.)]
i=t+ 1
(bku)aO···II. = ~ (-1)(+iryu/(IIo)···f(IIj)g(aJ) ... ~) ... g(a.)
)<t
'( 1){+i+1
+ L... - rv u /(.0) ... f(II,) ... f(IIj)g(II ) ... g(II ).
j>t J q
Similarly,
(kbu)IIo···II = I,(-1)j+tryuJ(IIo)···;(a';)···f(II·)9(IIJ) ··g(II)
q tSj J q
(7)
(8)

2.
Equations (7) and (8) give
COHOMOLOGY GROUPS
q
-L ry 0j(IIo) "'/(IIj)g(IIj) ... g(IIq)
j=O
33
= ry O"g(IIO) ... g(IIq) -rV 0"/(110) "'/(IIq)' (9)
proving Equation (6). Q.E.D.
Knowing that the map n~ depends only on 1111 and "Y, we proceed to the
definition of the limit. We write 1111 < -H' if -H' is a locally-finite refinement of 1111.
Then < is a partial order and given 1111, "Y there is -H' so that 1111 < -H' and
"Y < -H'. Hence the set of all locally finite coverings of X forms a directed set
with respect to <, and the following equations can be verified (using Lemma
2.1):
n: = id,
n:,. = n~ 0 n~,
DEFINITION 2.1. Hq(X,!/) = lim Hq(l1I1, !/).
'"
REMARK. We recall the definition of the limit lim. We say that g, hE Hq
'" (1111, !/) are equivalent if there exists -H' > 1111 such that n:,. 9 = n:, h. Denote
the equivalence class of 9 by g. Let
Hq(I1I1,!/) = {g \g E HQ(I1I1, !/)}.
The map 9 ~ 9 defines a homomorphism II"',
n~ : HQ(I1I1, !/) ~ HQ("Y, !/),
and n~ induces a homomorphism n~,
_.'" - -
11.,. : HQ(I1I1, !/) ~ HQ("Y, !/).
LEMMA 2.2. n~ is injective.
Proof n~g = 0 if and only if n:,. 0 n~ 9 = 0 for some W. So n:,. 9 =
o and 9 = O. Q.E.D.
Hence, identifying H4(11I1,!/) with n~H4(11I1, !/), we may consider
HQ(I1I1, !/) c H4("Y, !/) provided that 1111 < "Y. Then by definition,
H9(X, !/) = U HQ(I1I1, !/),
'"

34 SHEAVES AND COHOMOLOGY
and n'1': Hq(lJIt, f/) -t Hq(lJIt, f/) £;; Hq(X, f/) is a homomorphism of
Hq(lJIt, f/) into Hq(X, f/).
PROPOSITION 2.1. HO(X, f/) = rex, f/).
Proof By definition C-
l = 0 so HO(IJIt, f/) = ZO(IJIt, f/).
ZO(IJIt, f/) = [0'10' = {O'j},O'j E r(Vj' f/), DO' = OJ.
But (j0' = 0 means O'j(z) -O'k(Z) = 0 on Vj n Vk. Hence O'(z) E r(X, f/), de
fined
by O'(z) = O'j(z) when Z E
Vj, is meaningful. This proves HO(IJIt, f/) =
r(X, f/) and implies HO(X, f/) = r(X, f/). Q.E.D.
PROPOSITION 2.2. H"": HI(IJIt, f/) -t HI(X, f/) is injective.
COROLLARY. HI(X, f/) = U HI(IJIt, f/).
""
Proof (of the proposition). Suppose hE HI(IJIt, f/) = ZI(IJIt)/DCO(d/t).
Then h = {O')k}' O'jk E r( Vj n Vk, f/) where 0' ij + O'jk + O'ki = O. We want to
show that n""h = 0 implies h = O. n""h = 0 means Ii = 0 and this is true if and
only if n~h = 0 for some 1', l' > 1JIt. Let 1(/ = {WjA I WjA = Vi n V).}. Then
"If/" is a locally finite refinement of l' and n~h = n~ 0 n~h = O. Also 1(/ > d/t
since 1(/ i). C Vi and we can use the maps(iA,) = ; in the definition of refinement.
Then we have
where
't(j).)(jll) = 'tjAjll = fW,.l.'" Wj,. O'ij'
Then n~v h = 0 implies {'t i).jll} = D{ 't i).}' that is, 't i).jll = 't jll -'t i).' Since
'tWIl = rW • .l.",W.,.O'ii = 0, we obtain 'till = 'ti). on Wi). n Will' Vi = U).WjA, and
't i = 't il' on Will defines an element 't i E r( Vi' f/). Then the equation 0' ij = 't j -
't i implies h = O. Q.E.D.
Consequently, in order to describe an element of HI(X, f/), it is sufficient
to give an element of HI(IJIt, f/) for some 1JIt.
EXAMPLE. Let M = {(Zl' z2)llzll < 1, IZzl < 1, (Zl' Z2) =F (0, O)}. Then
dime Hl(M, l!J) = + 00.
Proof. Set
VI = {(ZI' zz) I (ZI' zz) E M, ZI =I:-OJ,
VZ = {(Zl' zz) I (Zl' Z2) E M, Z2 =I:-OJ.

3. INFINITESIMAL DEFORMA nONS 35
In this case M = UI U Uz so chose as covering 0/1 = {VI' Uz}. Then
HI(o/1, (9) = ZI(OlI, (9)/bCO(o/1, (9) where ZI(o/1, (9) = {0'121 0'12 E r(VI () Uz,
(9)},Co(o/1, (9) = {t I t = (tl' tz), tit E r( V"' (9)}, and bCo(o/1, (9) = {tz -tl I tit E
r(V", (O)}.
We note that VI () Uz = {(ZI, zz) 10< IZII < 1,0 < IZzl < I}, so we have
a Laurent expansion for 0'12
m=-CX)n=-co
tl IS holomorphic on VI = {(ZI' zz) 10 < IZII < I, IZzl < I} so tl(Z) =
L~~ -ooL:'=obm"z/~z~. Similarly for tz, tz(z) = L~=oL:=OO_oocm"z~z~, and
tz -tl = Lm~oor"2:0 am"z~z~. Then HI (0/1) ~ {0'121 0'12 = L;;;! -00 L;:! -00
am"z'~z~}. Hence dim HI (0/1, f/) = +00 and since HI(o/1, f/) £;; HI(X, f/),
dim HI(X, f/) = + 00. Q.E.D.
PROPOSITION 2.3. If HI( V j, f/) = 0 for all Vj E 0/1, then HI(o/1, f/) ~
HI(X, f/) where d/I = {VJ.
Proof. We already know that HI(o/1, f/) £;; Hl(X, f/). Hence we only
need to show the following. Let "Y = {VA} be any locally finite covering.
Let if" = {WjAI W jA = Vj () VA}' Then it suffices to show that n:" : HI (0/1) -+
Hl("/Y) is surjective. Take a I-cocycle {O'jAb} of HI("/Y) where O'jAjlt + O'j"kv +
O'k,jA = O. Then {O'WIt} for each fixed i is a I-cocycle on the covering {Wj)J of
Uj' Since HI(Vj, f/) = O,HI({WU},f/) £;; HI(Vj, f/)givesHl({Wu}, f/) = 0
for each i. This implies the existence oft iA E r( Wu, f/) such that aWIt = tjlt -
tiA' Let t be the O-cochain {tiA} on "/Y. Then {aIAh} = {aUk,} -bt defines a
I-cocycle
on
"/Y which defines the same cohomology class in Hl("/Y) as 0'.
From the definition of t we see that 0'1J.i1t = O. So O'iAj" + O'iltkv + O'~,jA = 0
yields O'lltkv = alAh' Similarly, O'jAh = O';ltkw' Hence, O'ik = aiAkv = ai,kv' and
O'ik E r( V j () Uk, f/). Now we have found aik so that n:.( O'tk) = O'tU" and
{aIAkv} is cohomologous to {ajAkv}' Hence n::,. is surjective. Q.E.D.
3. Infinitesimal Deformations
Using cohomology groups we will give an answer to the following
problem: Let .;II = {M 1ft E B} be a complex analytic family of compact com
plex manifolds
M
I and let t = (tl, ... ,t") be a local coordinate on B. The
problem
is to define
(aMI/at').
For this we define the sheaf of germs of holomorphic vector fields. Let M
be a complex manifold and let W be an open subset of M. Let 0/1 = {V j, Z j}

36 SHEAVES AND COHOMOLOGY
be a covering of M with coordinates patches with coordinates p --+ Zj(p) =
[zl(p), .. " z7(p)]. A holomorphic vector field () on W is given by a family of
holomorphic functions {OJ} on W (' V j where
n a
0= L OJ(p)-IX
IX=I aZj
on W (' Uj• These functions should behave as follows: On W (' U",
n a
0= L Of(p)p'
(1= 1 az"
We want
so the transition equation
(1)
should be satisfied on W
(' Uj (' U". Thus we have a definition of local
holomorphic vector fields and
we can define germs of local holomorphic
vector fields.
As notation we denote by
0 the sheaf over M of germs of holo
morphic vector fields. (Later we shall give a formal definition of the holo
morphic tangent bundle of a complex manifold.)
Next
we want to define the infinitesimal deformation (aM,/at.). First we
consider the case B =
{tlltl < r} £; C. .I{ is a complex manifold and iij:
.I{ --+ B is a holomorphic map satisfying the usual conditions
(1) M, = i.ij-I(t);
(2) the rank of the Jacobian of iij = 1 = dim B.
We can find an e > 0 small enough so that iij-I(A), A = {tlltl < e} looks as
follows:
J iij-I(A) = U OU j
j= 1
(a union of a finite number
of open sets).
On each OU j there should be a coordinate system
p --+ [z}(p), .. " zj(p), t(p)],
where t(p) = iij(p) and such that OUj = {pi Izj(p)1 < ej. It(p)1 < e}. We write
p = (Zj' t) = (z}, ... , zj, t). This construction is possible because rank iij = 1
These charts are holomorphically related so
zj(p) =
fj,,[z~(p), . ", z~(p), t(p)] = fj"(z,,, t)
on Uj (' Uk' Let U'j = M, (' OUj, It I < e. Then set
{(z} "', zj, t)llzjl < ej} = V'j'

3. INFINITESIMAL DEFORMATIONS 37
so we can use {(Z], ... , zj)llzjl < eJ as coordinates on VIj. The transforma
tion zj = IMzk' t) depends on t. Consider p E dIIj n dIIj n dII". Then p =
(Zj' t) = (Zj' t) = (Zk, t). So z~ = IMzk' t) = Jfj(Zj, t) = lil/jiz), t], where
!jk = (fA .... .Jj,,). We set
(). ( ) = ~ ajj,,(Zk, t) ~
Jk P. t ~ :It :l 01·
01=1 u UZj
Obviously (}j,,(t) E r( V'j n V,,,, 0,) where 0, is the sheaf of germs of holo
morphic vector fields on M, .
Proof Before beginning the proof we remark that () ij + () jk + (}kj = 0
and (}ij = -(}jj is equivalent to (}j" = (}jj + OJ,,. To prove the lemma we
differentiate the transition equation to get
an" = ar~ + t arj af~k
at at P=1 az~ at
Then
(}." = L an" ~ = L af~j ~ + L af~k ~
I 01 at azQI QI at azi (J at az~
Q.E.D.
*DEFINITION 3.1. (dM,ldt) = {OJ,..(P' t)} E HI(M" 0,). We have made several
choices in this definition and
we must justify them.
PROPOSITION 3.1. (dM ,Idt) is independent of the choice of local coordinate
covering
{zj}.
Proof Let {r.J be a locally finite refinement of {dllj} such that (C~. t)
are coordinates on r;. where
r;. = {(C;., t)II'~1 < e;.. It I < e}.
Since {r;.} is a refinement of {dII j} we have a map s: A ..... +-J such that
"f"). £; dIIs().). We also have holomorphic transition functions C{J).v where
(~= C{J;..«(., t) on r). n r •.
Then the cocycle defined by this covering is
"aC{J~. a
fT)..(t) = ~ at a,~·

38 SHEAVES AND COHOMOLOGY
As before s induces a map s* : {Ojd ~ {OA.}, where
0A.(t) = r1'";.n1'"vnM.[Ojk(t)],j = S(A), k = s(v).
We must show that {Ih.} is cohomologous to {O;..}; that is, there exists a
cochain {Oit)} such that
Ihllt) -0dt) = 0.(1) -Oit).
Since "Y;. c;;;; o/ij, j = S(A), there is a holomorphic gj such that zj = gj«(A' t) on
"Y;.. The following equalities are clear:
gj[IP;..(C., t), t] = gj(C;., t) = zj
= fMzk' t)
= fjk[gk«(;" t), t] on "Y;.n"Y •.
Differentiating we obtain
L ogj 01P~. + ogj = L Ofjk ogf + Ofjk.
ac~ at ot azf ot at
(2)
Then (2) implies [multiplying by (iJlozj)]
~ azj ( a) og~ ~ a fjk a ~ ozj ( a ) olP~. ~ ogj a 3)
L" ozf ozj . at + L" at ozj = L" iX~ ozj at + L" at ozj' (
Hence,
~ Dg~().) [_0_] _ ~ ag~(\.) [_0_] 0
'1;.. + L" IX -L" II + A"
ot oz.(;.) ot oz'(')
(4)
on "Y, n "Y •. Therefore if we let
we get '1,\. -0;.. = 0. -0,\. Q.E.D.
So
we see that the infinitesimal deformation, dM ,Idt E
H
1(M" 0,} is de
termined uniquely by the family Jt = {M, It E B} and is thus well defined.
If we introduce new coordinates on B, I = t(s) so that t'(s) =1= 0 then the
relation
(5)
is obvious.
Now to return to the more general case, let
{M,
I t E B} be a family where
B is now a general connected complex manifold. Let A be a coordinate
neighborhood around
bE B and let (t
I , ... ,1m) be local coordinates. Then

3. INFINITESIMAL DEFORMATIONS 39
we may assume II so chosen that w-I(ll) = U} '¥Ij , a union of finitely
many coordinate neighborhoods
on each of which there are coordinates (zj, "', zj, tl, "', t
m
), where '¥Ij = {(Zj' t)llzjl < Bj' tEll}. Again we have
transition functions
fjk
zj = fjk(Zk' tl,
"', t
m
) on '¥Ij n '¥Ik'
DEFINITION 3.1. (oM,/ot
Y
) E HI(M" 01) is the cohomology class of
{ 0 jk I y(t)} where
If (a/at) denotes the tangent vector
a m a
at = .f:1 c. otY ,
then we define
We make the following definition:
DEFINITION 3.2. .It = {MIl t E B} is locally trivial (complex analytically) if
each point bE B has a neighborhood II such that w-I(ll) = Mb x II (complex
analytically). This means
that we can choose coordinates (zj, t) such that,
zj =
fMzk' b) (independent of t).
If .It is locally trivial, then each M I is complex analytically homeo
morphic to
M
0; hence M I is independent of t.
PROPOSITION 3.2. If .It is locally trivial then (oM,/ot") = O.
Proo}: Trivial.
We mention here a theorem
ofW. Fischer and H. Grauert (1965).
THEOREM. If each M I is complex analytically homeomorphic to Mb, then
.It is locally trivial.
We now study some examples:
EXAMPLE I. Let R be a compact Riemann surface. Fix a point a E R.
Let w be a coordinate in a neighborhood of a point bE R such that w(b) = O.
We define a family {M,} as follows: M, will be the branched two-sheeted

40 SHEAVES AND COHOMOLOGY
covering Rp of R with branch points at a and p; t = w(p). We have the ques
tion "is d:, = 07" Define the following neighborhoods on R:
Wb = {wllwl < r},
Wo = {wllwl < rI2},
WI = {wi rl4 < Iwl < r}.
We can write M, = Uo U UI U U2 ••• Uj••• where Uo = 1I:-
I(WO)' UI =
11: -I( WI), 11:( U j) n Wo = 4>for j ~ 0, 1 and 11: is the map 11: : M, -+ R defined by
the covering map 11: : Rp -+ R. We introduce local coordinates as follows on
M" (I E two):
Zo = Jw -ton Uo,
ZI = Fw on UI,
and Zj on Uj can be an arbitrary coordinate which should be fixed and inde
pendent
of t. Then we have Zj =
fjiZk' t) for holomorphic fjk. In fact,
Zo = fOI(zl, t) = Jw -t = Jz~ -t,
and
Zj = fjiZk) (independent of I)
for (j, k) ¢ {CO, 1), (1,0)}. Then OCt) = {Ojk(t)} has only one nonzero
component,
o fo I (0) 1 ( 0 ) 1 (0)
°Ol(t) = Tt· OZo = -2Jz~ _ t OZo = -2zo OZo .
Let Vo = Uo, VI = U J ~ I Uj• Then OCt) is a l-cocycIe on the covering "Y =
{Vo, Vd; OCt) E HI("Y, e,) s; HI(M" e,).
Suppose dM ,Id! = O. Then there are holomorphic vector fields Oy(t) on
Vy such that
so
(6)
We make the definition

3. INFINITESIMAL DEFORMA nONS 41
Then ,,(t) is a vector field on M, which is holomorphic on M, - {p} and
has a simple pole at p.
LEMMA 3.2. If the genus 9 of R is ~ 1, then such vector fields '1 do not exist.
COROLLARY. If 9 ~ 1, then dM,/dt"# 0, that is, the conformal structure of
the branched covering M, depends on t.
Proof (of lemma) By the Riemann-Hurwitz formula, we have
X(M,) = [2 -2g(M,)] = 2X(R) -2
where X(M) is the Euler characteristic of M. Then the genus geM,) equals 2g.
By the Riemann-Roch formula [see Hirzebruch (1962)], there is a holomor
phicdifferentialqJ(z) =h(z)dzonM,with2(2g) -2zerossince2g - 1 ~ 1 > O.
Since" = y(z)(d/dz) has one (simple) pole, fez) = h{z)y(z) is a meromorphic
function
on M, with more zeros than poles [2(2g) - 2
~ 2]. This is impossible
(the number
of zeros equals the number of poles). Q.E.D.
EXAMPLE 2. Ruled Surfaces (See Chapter 1, Sections 3 and 4.) Recall
that M, = U'I V U,z where each U'y = C X pi and
(ZI' (I) -(Z2, (2)
if and only if
'1 = ZiC2 + tz~, and ZI = Ijz2'
We are assuming m ~ 2k, k ~ 1. Then M, is independent of t "# 0 for t :p 0 so
dMr/dt = 0 for t :p O.
(For this, one could use the theorem of Fischer and Grauert.) What is
dM ,Idt 1,= 0 ? Consider the covering of M 0, dlJ = {U 01> U 02}; then
dM,/dtl,=o = 0(0) E Hl(dlJ, 00) £ Hl(Mo, 00),
Then
so that
(af~2) (0) k(a)
adO) = ---at ,=0 a'l = Z2 a'i E qUol (', V02, 0).
Suppose dM,/dt = 0 at t = O. Then
OdO) = ()2 -01
where each 0. is a holomorphic vector field on UOv = C X pl.

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punished as a sign of idleness;—and finally, a title of nobility
conferred upon a man for some signal service rendered to the state,
does not descend to his posterity, but goes backwards and ennobles
his ancestors.”
The Japanese Family.
Japan, consisting of a large island, that of Nipon, and seven other
smaller islands, of which the principal are Yesso, Sitkokf, and
Kiousiou, is inhabited by an industrious and intelligent people. The
Japanese, whilst resembling the Chinese in many points, differ from
them in many others, and are far superior in a moral point of view to
the inhabitants of the Celestial Empire.
The written character of Japan is the same as that of China, and its
literature is not a distinctive one, but entirely Chinese. The two
creeds of Buddha and of Confucius prevail in Japan as they do in
China. The worship of these creeds is carried on in both countries in
similar pagodas, and their ministers are the same bonzes with
shaven heads and long gray robes. The buildings and the junks of
both nations are identical. Their food is the same, a diet of
vegetables, principally rice, and fish, washed down by plenty of tea
and spirit. The coolies carry their loads in exactly the same manner
in Japan and in China, at Nangasaki and at Peking, and make the
streets resound with the same shrill measured cries. The Japanese
women wear their hair as the Chinese women used to do before
they adopted the fashion of pig-tails, and the townspeople in Yeddo,
as in Nankin, seclude themselves in their houses, which are
impervious both to heat and cold.
But the resemblance stops there. The Japanese, a warlike and feudal
nation, would be indignant at being confounded with the servile and
crafty inhabitants of the Celestial Empire, who despise war, and
whose sole aim is commerce. A Chinaman begins to laugh when he
is reproached with running away from the enemy, or when he is
convicted of having told a lie; such matters give him little concern. A

Japanese sets a different value on his life and on his honour; he is
warlike and haughty. A Japanese soldier always confronts his enemy.
To deprive him of his sword is to dishonour him, and he will only
consent to take it back stained with the life-blood of his conqueror.
The duello, unknown in China, is carried out in a terrible fashion
among the Japanese. The islander of Nipon disembowels himself
with a thrust of his own sword, and dares his adversary to follow his
example. The Chinese race live in a state of disgusting and perpetual
filth; every Japanese, on the contrary, without distinction of rank or
fortune, takes a warm bath every other day. Of a jovial and frank
disposition, and of great intelligence, they are always desirous of
knowing what is going on in the world, and ever anxious to learn;
whilst the Chinese, on the other hand, shut themselves up behind
their classic wall, and recoil from everything that is strange to them.
These characteristics show that the Japanese are a far superior race
to the Chinese.
A few peculiarities, more especially found in the inhabitants of the
sea coasts, the fishermen and the sailors, separate the Japanese
physical type from that of the Chinese. The former are small,
vigorous, active men with heavy jaws, thick lips, and a small nose,
flat at the bridge, but yet with an aquiline profile. Their hair is
somewhat inclined to be curly.
The Japanese are generally of middle height. They have a large
head, rather high shoulders, a broad chest, a long waist, fleshy hips,
slender short legs, and small hands and feet. The full face of those
who have a very retreating forehead and particularly prominent
cheek-bones is rather square than oval in shape. Their eyes are
more projecting than those of Europeans, and are rather more veiled
by the eyelid. The general effect is not that of the Chinese or
Mongolian type. The Japanese have a larger head than is customary
with individuals of these races, their face is longer, their features are
more regular, and their nose is more prominent and better shaped.
They have all thick, sleek, dark black hair, and a considerable
quantity of it on their faces. The colour of their skin varies according

to the class they belong to, from the sallow sunburnt complexion of
the inhabitants of southern Europe to the deep tawny hue of that of
the native of Java. The most general tint is a sallow brown, but none
remind you of the yellow skin of the Chinese. The women are fairer
than the men. Amongst the upper and even the middle classes,
some are to be met with with a perfectly white complexion.
Two indelible features distinguish the Japanese from the European
type. Their half-veiled eyes, and a disfiguring hollow in the breast,
which is noticeable in them in the flower of their youth, even in the
handsomest figures.
138.—JAPANESE.

Both men and women have black eyes, and white sound teeth. Their
countenance is mobile and possesses great variety of expression. It
is the custom for their married women to blacken their teeth. The
national Japanese costume is a kind of open dressing gown (fig.
138), which is made a little wider and a little more flowing for the
women than for the men. It is fastened round the waist by a belt.
That, worn by the men, is a narrow silk sash, that, by the women, a
broad piece of cloth tied in a peculiar knot at the back.
The Japanese wear no linen, but they bathe, as we have said, every
other day. The women wear an under-garment of red silk crape.
139.—A JAPANESE FATHER.
In summer, the peasants, the fishermen, the mechanics and the
Indian coolies follow their calling in a state of almost complete
nudity, and the women only wear a skirt from the waist downwards.
When it rains they cover themselves with capes made of straw, or
oiled paper, and with hats made, shield shape, of cane bark. In
winter the men of the lower classes wear, beneath their kirimon or
dressing-gown, a tight fitting vest and pair of trousers of blue cotton

stuff, and the women one or more wadded cloaks. The middle
classes always wear a vest and trousers out of doors.
Figs. 138, 139, 140, and 141 represent different Japanese types.
Their costume generally differs only in the material of which it is
made. The nobility alone have the right to wear silk. They only wear
their costlier dresses on the occasions of their going to court or
when they pay ceremonial visits. All classes wear linen socks and
sandals of plaited straw, or wooden shoes fastened by a string
looped round the big toe. They all, on their return to their own
house, or when entering that of a stranger, take off their shoes, and
leave them at the threshold.
140.—JAPANESE SOLDIER.
The floors of Japanese dwellings are covered with mattings, which
take the place of every other kind of furniture.

A Japanese has but one wife.
The Japanese have a taste for science and art, and are fond of
music and pageants. Their manufactures are largely developed. They
make all sorts of fine stuffs, work skilfully in iron and copper, make
capital sword-blades, and their wood carvings, their lacquer-work,
and their china, enjoy a wide reputation.
Political power is divided between an hereditary and despotic
governor, the Taïcoon, and a spiritual chief, the Mikado.
The creed of Buddhism, that of the Kamis, and the doctrines of
Confucius equally divide the religious tendencies of the Japanese.
141.—JAPANESE NOBLE.

We will give a few details on the interesting inhabitants of Japan,
from the account of a visit to that country written by M. Humbert,
the Swiss plenipotentiary there, which was published in 1870 under
the title of “Japan.”
M. Humbert was present at the ceremonies which took place on the
occasion of an official visit paid by the Taïcoon to the Mikado, and he
gives the following account of it:—
“While I was in Japan, it happened that the Taïcoon paid a visit of
courtesy to the Mikado.
“This was an extraordinary event. It made a great sensation,
inspired the brush of several native artists, and gave resident
foreigners a chance of seeing a little more clearly into the reciprocal
relation of the two powers of the empire. Their respective position is
really one of considerable interest.
“In the first place, the Mikado has over his temporal rival the
advantage of birth and the prestige of his sacred character.
Grandson of the Sun, he continues the traditions of the gods, the
demi-gods, the heroes, and the hereditary sovereigns who have
reigned over Japan in an uninterrupted succession since the creation
of the empire of the eight great islands. Supreme head of their
religion, under whatever form it may present itself to the people, he
officiates as the sovereign pontiff of the ancient national creed of the
Kamis. At the summer solstice, he offers sacrifices to the earth; at
the winter solstice, to heaven. A god is specially deputed to watch
over his precious destiny; from the shrine of the temple he inhabits
at the top of Mount Kamo, in the neighbourhood of the Mikado’s
residence, this deity watches night and day over the Daïri. And
finally at the death of a Mikado, his name, which it has been
ordained shall be inscribed in the temples of his ancestors, is
engraved at Kioto, in the temple of Hatchiman; and at Isyé, in the
temple of the Sun.

“It is indubitably from heaven that the Mikado, both theocratic
emperor and hereditary sovereign, derives the authority which he
exercises over his people. Though now-a-days, it must be
acknowledged, he scarcely knows how to employ it. However, from
time to time it seems proper to him to confer pompous titles, which
are entirely honorary, on a few old feudal nobles who have deserved
well of the altar. Sometimes also he allows himself the luxury of
openly protesting against those acts of the temporal power, which
seem to infringe on his prerogatives. This is the course he took with
special reference to the treaties made by the Taïcoon with several
western nations; it is true that he finally sanctioned them, but that
was because he could not help himself.
“Now the Taïcoon, as everybody knows, is the fortunate successor of
a common usurper. In fact, the founders of his dynasty, subjects of
the then Mikado, robbed their lord and master of his army, his navy,
his lands, and his treasure, as if they were desirous of depriving him
of any subject of earthly anxiety.
“Possibly the Mikado was too ready to fall in with their plans. The
offer of a two-wheeled chariot drawn by an ox, for his daily drive in
the parks of his residence, doubtless a considerable privilege in a
country where nobody uses a conveyance, should not have
persuaded him to sacrifice the manly exercises of archery, hawking,
and hunting the stag or wild boar. He might likewise, without making
himself absolutely invisible, have spared himself the fatigue of the
ceremonious receptions where, motionless on a raised platform, he
accepts the silent adoration of his courtiers prostrated at his feet.
The Mikado, now, they say, only communicates with the exterior
world through the medium of the female attendants intrusted with
the care of his person. It is they who dress and feed him, clothing
him daily in a fresh costume, and serving his meals on table utensils
fresh every morning from the manufactory which for centuries has
monopolized their supply. His sacred feet never touch the ground;
his countenance is never exposed in broad daylight to the common

gaze; in a word, the Mikado must be kept pure from all contact with
the elements, the sun, the moon, the earth, mankind, and himself.
“It was necessary that the interview should take place at Kioto, the
holy town which the Mikado is never allowed to leave. His palace,
and the ancient temples of his family are his sole personal
possessions there, the town itself being under the rule of the
temporal emperor; but the latter dedicates its revenues to the
expenses of the spiritual sovereign, and condescends to keep up a
permanent garrison within its walls for the protection of the
pontifical throne.
“The preliminaries on both sides having been carried out, a
proclamation announced the day when the Taïcoon intended to issue
forth from his capital, the immense and populous modern town of
Yeddo, the head-quarters of the political and civil government of the
empire, the seat of the Naval and Military Schools, of the
Interpreters’ College, and of the Academy of Medicine and
Philosophy.
“He was preceded by a division of his army equipped in the
European manner, and, while these picked troops, infantry, cavalry,
and artillery, were marching on Kioto by land along the great
Imperial highway of the Tokaïdo, the fleet received orders to set sail
for the inland sea. The temporal sovereign himself, embarked in the
splendid steamer, the Lycemoon, which he had purchased of the firm
of Dent and Co. for five hundred thousand dollars. Six other
steamers escorted him; the Kandimarrah, notorious for its voyage
from Yeddo to San-Francisco to convey the Japanese embassy sent
to the United States; the sloop of war, the Soembing, a gift from the
King of the Netherlands; the yacht Emperor, a present from Queen
Victoria; and some frigates built in America and in Holland to orders
given by the embassies of 1859 and 1862. Manned entirely by
Japanese crews, this squadron left the bay of Yeddo, doubled Cape
Sagami and the promontory of Idsou, crossed the Linschoten straits,
and coasting along the eastern shores of the island of Awadsi,

dropped its anchors in the Hiogo roadstead, where the Taïcoon
disembarked amid larboard and starboard salutes.
“His state entry into Kioto took place a few days later, with no
military parade but that of his own troops, as the Mikado possesses
neither soldiers nor artillery, with the exception of a body-guard of
archers, recruited from the families of his kinsmen or of the feudal
nobility. Indeed, he can hardly afford even on this moderate scale,
the expenses of his court; and his own revenue being insufficient, he
is obliged to accept with one hand an income the Taïcoon consents
to pay him out of his own private purse, and with the other, the
amounts that the brethren of a few monastic orders yearly collect for
him, from village to village, in even the furthest provinces of the
empire. Another circumstance that assists him to support his rank, is
the disinterested abnegation of many of his high officials. Some of
them serve him with no other remuneration but the free use of the
costly regulation dresses of the old imperial wardrobe. On their
return home, after doffing their court costume, these haughty
gentlemen are not ashamed to seat themselves at a weavers’ loom
or an embroidery frame. More than one piece of the rich silk
productions of Kioto, the handiwork of which is so much admired,
has issued from some of the princely houses, whose names are
inscribed in the register of the Kamis.

142.—JAPANESE PALANQUIN.
“These drawbacks did not prevent the Mikado from inaugurating the
day of the interview, by exhibiting to his royal visitor the spectacle of
the grand procession of the Daïri. Accompanied by his archers, by
his household, by his courtiers, and by the whole of his pontifical
staff, he left his palace by the southern gateway, which, towards the
close of the ninth century, was decorated by the historical
compositions of the celebrated painter-poet, Kosé Kanaoka. He
descended along the boulevards to the suburb washed by the
Yodogawa, and returned to the castle through the principal streets
of the town.
“The ancient insignia of his supreme power were carried in state at
the head of the procession; the mirror of his ancestress Izanami, the
beautiful goddess who gave birth to the sun in the island of Awadsi;
the glorious standard, the long paper streamers of which had waved
above the heads of the soldiery of Zinmou the conqueror; the

flaming sword of the hero of Yamato, who overcame the eight-
headed hydra to which virgins of princely blood used to be
sacrificed; the seal that stamped the first laws of the empire; and
the cedar wood fan, shaped like a lath and used as a sceptre, which
for more than two thousand years has descended from the hands of
the dead Mikado to those of his successor.
“I will not stop to describe another part of the pageant, intended
doubtless to complete and enhance the effect of the rest, namely
the banners embroidered with the armorial bearings of all the
ancient noble families of the empire. Perhaps they were intended to
remind the Taïcoon, that, in the eyes of the old territorial nobility, he
was nothing but a parvenu; if so, the parvenu could smile
complacently at the thought, that the whole of the Japanese
grandees, the great as well as the lesser daïmios, are, nevertheless,
obliged to pass six months of the year, at his Court in Yeddo, and
offer him their homage in the midst of the nobles of his own
creation.
“The most numerous and the most picturesque ranks of the
procession were those of the representatives of all the sects who
recognise the spiritual supremacy of the Mikado. The dignitaries of
the ancient creed of the Kamis are scarcely distinguishable, as to
dress, from the high officials of the palace. I have already described
their costume, it reminds the spectators that the Japanese
possessed originally a religion without a priesthood. Buddhism, on
the contrary, which came from China, and rapidly spread throughout
the empire, has an immense variety of sects, rites, orders, and
brotherhoods. The bonzes and the monks belonging to this faith
composed in the procession endless ranks of devout-looking
individuals, with the tonsure or with entirely shaven heads, some of
them uncovered, and some wearing curiously shaped caps, mitres,
and hats with wide brims. Some of them carried a crozier in their
right hand, others a rosary, others again, a fly-brush, a sea-shell, or
a holy water sprinkler made of paper. They were dressed in
cassocks, surplices, and cloaks of every shape and hue.

“Behind them came the household of the Mikado. The pontifical
body-guard in their full dress, aim beyond everything at elegance.
Leaving breast-plates and coats of mail to the men-at-arms of the
Taïcoon, they wear a little lacquer-work cap, ornamented on both
sides with rosettes, and a rich silk tunic trimmed with lace edgings.
The width of their trousers conceals their feet. They are equipped
with a large curved sabre, a bow, and a quiver full of arrows.
“Some of the mounted ones had a long riding-whip fastened to their
wrist by a coarse silken cord.
“A great deal of brutality is too often hidden beneath this imposing
exterior. The wildness and the dissipation of the young nobles of the
Japanese pontifical court have supplied history with pages recalling
the worst period of papal Rome, the days of Cæsar Borgia. Conrad
Kramer, the envoy of the Dutch West Indian islands to the court of
Kioto, was allowed to be present in 1626 at a festival held in honour
of a visit of the temporal emperor to his spiritual sovereign. He
relates that the following day, corpses of women, young girls, and
children, who had fallen victims to nocturnal outrages, were found in
the streets of the capital. A still larger number of married women
and maidens, whom curiosity had attracted to Kioto, were lost by
their husbands and parents in the turmoil of the crowded streets,
and were only found a week or a fortnight later, their families being
utterly unable to bring their abducers to justice.
“Polygamy being a legal institution for the Mikado only, it was
perhaps natural for him to make some display of his prerogative. It
costs him sufficiently dear. It is the abyss hidden with flowers that
the first usurpers of the imperial power dug for the feet of the
successors of Zinmou. It is easy to imagine the cynical smile on the
lips of the Taïcoon as he saw the long row of the equipages of the
Daïri making its appearance.
“A pair of black buffaloes, driven by pages in white smocks, were
harnessed to each of these cumbrous vehicles which were made of
precious woods and glistened with coats of varnish of different tints.

They contained the empress and the twelve other legitimate wives of
the Mikado seated behind doors of open latticework. His favourite
concubines, and the fifty ladies of honour of the empress followed
close behind, in covered palanquins.
“When the Mikado himself leaves his residence, it is always in his
pontifical litter. This litter, fastened on long shafts, and borne by fifty
porters in white liveries, can be seen from a long distance off
towering above the crowd. It is constructed in the shape of a
mikosis, the kind of shrine in which the holy relics of the Kamis are
exposed. It may be compared to a garden summer-house, with a
cupola roof with bells hanging all round its base. On the top of the
cupola there is a ball, and on top of the ball there is a kind of cock
couchant on its spurs, with its wings extended and its tail spread:
this is meant as a representation of the mythological bird known in
China and Japan under the name of Foô.
“This portable summer-house, glistening all over with gold, is so very
hermetically closed that it is difficult to believe that any body could
be put inside it. A proof, however, that it is really used for the high
purpose attributed to it, is that on each side of it are seen walking
the women who are the domestic attendants of the Mikado. They
alone have the privilege of surrounding his person. To the rest of his
court as well as to his people, the Mikado remains an invisible,
dumb, and inapproachable divinity. He kept up this character even in
the interview with the Taïcoon.
“Amongst the group of buildings that constitute the right of Kioto to
be styled the pontifical residence, there is one that might be called
the Temple of Audience, for it is constructed in the sacred style of
architecture peculiar to the religious edifices of the faith of the
Kamis, and it bears like them the name of Mia. Adjoining the
apartments inhabited by the Mikado, it stands at the bottom of a
large court paved and planted with trees, in which are marshalled
the escorts of honour on high and solemn festivals.

“A detachment of officers of the artillery and of the body-guards of
the Taïcoon (fig. 143), and several groups of dignitaries of the
Mikado’s suite drew up successively in this open space.
“The women had retired to their own apartments.
143.—THE TAÏCOON’S GUARDS.
“Deputations of bonzes and different monastic orders occupied the
corridors along the surrounding walls. Soldiers of the Taïcoonal
garrison of Kioto, posted at intervals, kept the line of the avenue
which led to the broad steps reaching up to the front of the building.
Up this avenue the courtiers of the Mikado, clad in mantles with long
trains, passed with measured tread, majestically ascended the steps,

and placed themselves right and left on the verandah with their
faces turned towards the still closed doors of the great throne room.
Before taking up their position they took care to lift the trains of
their mantles and throw them over the balustrade of the verandah,
so as to display to the crowd the coats of arms which were
embroidered on these portions of their garments. The whole
verandah was soon curtained with this brilliant kind of tapestry.
“Presently the sound of flutes, of sea-shells and of the gongs of the
pontifical chapel, proceeding from the left wing of the building,
announced that the Mikado was entering the sanctuary. A deep
silence fell upon the crowd. An hour passed away in solemn
expectation, whilst the preliminaries of the reception were being
performed. Suddenly a flourish of trumpets announced the arrival of
the Taïcoon. He advanced up the avenue on foot and without any
escort; his prime minister, the commanders in chief of the army and
navy, and a few members of the council of the Court of Yeddo,
walked at a respectful distance behind him. He stopped for a
moment at the foot of the great staircase, and immediately the
doors of the temple slowly opened, gliding from right to left in their
grooves. He then ascended the steps, and the spectacle which had
held in suspense the expectation of the multitude at last unveiled
itself to their eyes.
“A large green awning of cane-bark fastened to the ceiling of the
hall, hung within two or three feet of the floor. Through this narrow
space, could be perceived a couch of mats and carpets, on which the
broad folds of an ample white robe spread themselves out. This was
all that could be seen of the spectacle of the Mikado on his throne.
“The chinks in the plaits of the cane awning allowed him to see
everything without being seen. Wherever he directed his gaze, he
perceived nothing but heads bent before his invisible majesty. One
alone remained erect on the summit of the stairs of the temple, but
it was one crowned with the lofty golden coronet, the royal symbol
of the temporal head of the empire. And even he too, the powerful
sovereign whose might is boundless, when he had reached the last

step, bent his head, and sinking slowly, fell on his knees, stretched
his arms forward towards the threshold of the throne-room, and
bowed his forehead to the very ground.
“From that moment, the ceremony of the interview was
accomplished, the aim of the solemnity was gained. The Taïcoon had
openly prostrated himself at the feet of the Mikado.
144.—A LADY OF THE COURT.
“The interview at Kioto, had for its result two facts. By the first, the
bending of the knee, the temporal sovereign showed that he
continued to be the traditional obedient son of the high pontiff of the
national religion; but, by the second, that is to say by accepting this
act of homage, the theocratic emperor formally recognised the
representative of a dynasty sprung from a source alien to the only
legitimate one.”

As the art of war is of some importance in Japan, we quote a few
details from M. Humbert, on the equipments and the uniforms of the
Taïcoon’s soldiers.
“The common soldiers are,” M. Humbert tells us, “inhabitants of the
mountains of Akoui. They return to their homes after a short service
of two or three years. Their uniform is made of blue cotton stuff,
striped with white across the shoulders, and consists of a tight-fitting
pair of trousers, and a shirt like that worn by the followers of
Garibaldi. They wear cotton socks, leather sandals, and a waist-belt
supporting a large sword in a japanned scabbard. Their cartridge-
pouch and their bayonet are slung to their right side by a baldric.
Their get-up is completed by a pointed hat, sloping at the sides, and
made of lacquered cardboard; but they only wear it when on guard
or at drill.
“As for the muskets of the Japanese troops, they have all, it is true,
percussion-locks, but they vary both in calibre and in make,
according to where they happen to come from. I saw four different
kinds in the racks of some barracks at Benten, which a Yakounine
did me the favour to show me. He showed me first a Dutch sample
musket, and then one of an inferior quality manufactured in some
workshops that had been started in Yeddo to turn out arms copied
from this sample; he then pointed out an American gun; and finally,
a Minié rifle, the use of which a young officer was teaching a squad
of soldiers in the barrack-yard.”
The dress of the Japanese soldiery is curious in this respect, that it
reproduces and preserves the whole military paraphernalia of
European feudal times. A helmet, a coat-of-mail, a halberd, and a
two-handed sword, such are the equipment of the better class of
soldiery.
Fencing is held in high esteem in the Japanese army. The men are
very clever at this exercise, which keeps up their vigour and their
skill. Even the women practise it. Their weapon is a lance with a
bent piece of iron at the end of it. The ladies learn how to use it in a

series of regular positions and attitudes. The Japanese Amazons can
also skilfully make use of a kind of knife, fastened to the wrist with a
long silken string. When they have hurled this weapon at the head
of their enemy, they draw it back again by means of the cord. The
men also hurl the knife, but without fastening it to their wrist, and in
the same way as they practise throwing the knife in Spain.
The Japanese nobles carry very costly weapons. The temper of their
sword-blades is matchless, and their sword-hilts and scabbards are
enriched with finely chased and engraved metal ornaments. But the
chief value of their swords lies in their great age and reputation. In
old families, every sword has a history and tradition of its own,
whose brilliancy corresponds with the blood it has shed. A maiden
sword must not remain so in the hand of its purchaser. Till an
opportunity turns up of dyeing it with human blood, its possessor
tries its prowess on living animals, or better still, on the corpses of
executed criminals. The executioner, having obtained permission,
hands him over two or three dead bodies. Our Japanese then
proceeds to fasten them to crosses, or on trestles, in a courtyard of
his house, and practises cutting, slashing, and thrusting, till he has
acquired enough strength and skill to cut a couple of bodies in two
at one stroke.
The sword, in Japan, is the classical, the national weapon.
Nevertheless, in process of time, it will have to give way to the new
improved firearms. In spite of the traditional prestige with which the
Japanese nobility still endeavour to surround the former old-
fashioned weapon; in spite of the contempt they affect for military
innovations; the rifle, the democratic arm of arms, is becoming more
and more used in Japan. This weapon will inaugurate a social
revolution that will put an end to the feudal system. The rifle will
cause an Eastern ’89 in Japan.
We have said that two creeds are followed in Japan, the Buddhist
faith and the religion of the Kamis. The latter, with its ancient rites,
has been replaced, however, nearly throughout the empire by the
former.

We quote some of M. Humbert’s remarks on Buddhism.
“Our imagination can hardly conceive,” says this traveller, “that
nearly a third of the human race has no religious belief but that of
Buddhism, a creed without a God, a faith of negation, an invention
of despair.
“One would wish to persuade oneself that the multitudes who follow
its doctrines, do not understand the faith they profess, or at least
refuse to admit its natural consequences. The idolatrous practices
engrafted on the book of its law seem in fact to bear witness that
Buddhism has neither been able to satisfy or destroy the religious
instinct innate in man, and germinating in the bosoms of all nations.
“On the other hand, it is impossible not to recognize the influence of
the philosophy of final annihilation in many of the habits and
customs of Japanese life. The Irowa teaches the school children that
life disappears like a dream, and leaves no trace behind. A Japanese,
arrived at man’s estate, sacrifices with the most disdainful
indifference his own life or that of his neighbour, to appease his
pride, or for some trifling cause of anger. Murders and suicides are
of such every-day occurrence in Japan, that there are few families of
gentle birth who do not make it a point of honour to boast at least
one sword that has been dyed in blood.
“Buddhism is, however, superior in some respects to the creeds it
has dethroned. It owes this relative superiority to the justice of its
fundamental axiom, which is an avowal of a need for a redeeming
principle, grounded on the double fact of the existence of evil in the
nature of man, and of an universal state of misery and suffering in
the world.
“The promises of the religion of the Kamis had all reference to this
life. A strict observance of the rules of purification would preserve
the faithful from the five great ills, which are the fire of heaven,
sickness, poverty, exile, and early death. The aim of their religious
festivals was the glorification of the heroes of the empire. But were

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Complex Manifolds 1 Reprint With Errata James Morrow Kunihiko Kodaira

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