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 CataloginginPublication Data Morrow, James A., 1941Complex manifolds / James Morrow, Kunihiko Kodaira. p. cm. Originally published: New York: Holt, Rinehart and Winston, 1971. Includes bibliographical references and index. ISBN 082184055X (alk. paper) 1. Complex manifolds. I. Kodaira, Kunihiko, 1915 II. Title.
QA331.M82 2005 515'.946dc22
20051
© 1971 held by the American Mathematical Society. Reprinted with errata by the American Mathematical Society, 2006 Printed in the United States of America. §
The paper used in this book is acidfree 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 mathematicians. 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 19651966. 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 mid1950s 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 semicontinuity 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 selfcontained. We wish to thank the publisher for patience shown to the authors and Nancy Monroe for her excellent typing. James A. Morrow Kunihiko Kodaira
Seattle, Washington January 1971 v
Contents Preface
v
Chapter 1. Definitions and Examples of Complex Manifolds 1. Holomorphic Functions 2. Complex Manifolds and Pseudogroup Structures 3. Some Examples of Construction (or Description) of Compact Complex Manifolds 4. Analytic Families; Deformations
1 1 7 11 18
Chapter 2. Sheaves and Cohomology 1. Germs of Functions 2. Cohomology Groups 3. Infinitesimal Deformations 4. Exact Sequences 5. Vector Bundles 6. A Theorem of Dolbeault (A fine resolution of (I))
27 27 30 35 56 62 73
Chapter 3. Geometry of Complex Maoifolds 1. Hermitian Metrics; Kahler Structures 2. Norms and Dual Forms 3. Norms for Holomorphic Vector Bundles 4. Applications of Results on Elliptic Operators 5. Covariant Differentiation on Kahler Manifolds 6. Curvatures on Kahler Manifolds 7. Vanishing Theorems 8. Hodge Manifolds Chapter 4. Applications of Elliptic Partial Differential Equations to Deformations 1. Infinitesimal Deformations 2. An Existence Theorem for Deformations I. (No Obstructions) 3. An Existence Theorem for Deformations II. (Kuranishi's Theorem) 4. Stability Theorem
83 83 92 100 102 106 116 125 134
147 147 155 165 173 186 189 193
Bibliography Index Errata
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 complexvalued 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, ... kn (Z1
a 1)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 '
Iek, .. ·kn W"l1
.••
wknl , ,< _C .
Hence
I
Iek, ... k zk,1 ... zknl ,,<  C 2Z Ik' '" n
W1
I
2Z
Ik" •
(1)
W"
If Izdwil < 1 for 1 :S i:S n, (1) gives
L Ie", "'knZ~'
'"
zktl
n( 1
:S C.1=1
Zi
1 
Wi
1
I)
< + 00.
Q.E.D.
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 complexvalued function/(z) = /(x1 + iYI, ... , Xn + iYn), where i = 1 can be considered as a function of 2n real variables. Then:
J
DEFINITION 1.2. A complexvalued function of n complex variables is continuous 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( ZI, . . . , f( WI>
and so on.
Z2'
...
)  _1 f Z" 
J,
• 2Xl Iw,lId=r,
) 1 2x!
,Z"   .
f.
IwzlIzl=rz
f(w l ,
z2, ... ,
WI 
Zn) d wI>
Zl
f(w l ,W2 ,Z3,···,z")d
W2,
W2 
Z2
1.
HOLOMORPHIC FUNCTIONS
3
Substituting we get
We are assuming
< 1. Iz.w.  a'l a. Hence the series
1 w. 
Z.
1
= (w. 
=(
+ (a.  z.) =
a.) 1
)
w.  a.
[ 1
(Zy 
1 ] 1 ay/w.  aJ w.  a.
L (Z  a )k 00
v
k=O
v
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 CauchyRiemann equations. Let/(z) be a differentiable function on domain n f; en. DEFINITION
a/az a/oz., 1 ~ v ~ n are defined by af 1 (aJ . Of) o~. = 2 ax.  I oY. ' af 1 (a f . OJ) oz. = 2 OX + I Oy. '
1.3. The operators
y ,
y
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 CauchyRiemann 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)] 2 ax ax ay ay
OZ
=
~ [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 calculation. 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 = oz.
f
(Of ow;. + !L ow;.) oz. ow). oz. '
..= lOW).
(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.
5
HOLOMORPHIC FUNCTIONS
Making the substitutions, 1
UA
= 2(g A + 9 A),
we get
f
oj[g(Z)] = {OJ! (09A + 09A) OZ, A= I oU A2 oz. oz. oj
+ OVA =
(1 )(09A 09A)} OZ.  OZ. 2i
f {2I (OJOUA 
A= I
. I
Of) og A OVA oz.
. of) 09 +I (oj +1 A} 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 OZI
ojm OZI
ojl OZn
ojm OZn
=
(iz:);.=I . . .
m
v= 1, ...• n
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 REMARK.
DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS
IfI 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 CauchyRiemann equations and set a. A= aUA/aX. = aVA/ay., bVA = aVA/aX. = au}../ay•. Then
au, ax,
av, ax,
oU z av z ax, ax,
aUI ay,
av,
aU 2 av z = ay, aYI
aYI
all
b ll
al2
bl2
b'l
all
bJ2
al 2
a21
b21
a22
b 22
b 21
a2'
b 22
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 a(u, v) gZI = 0 o(x, y) 0
gl2 g22 0 0
* *
* *
gz,
gZ2
gil gl2
= Idet(g.A)1 2
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 holomorphic 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 II is differentiable on I(N). Set qJ(w) = /I(W); then z" = cp,,[J(z)]. Computing,
o = a~1l = az.
±
aCPIl a~A A=I aw}.az.
+ a~" a~A
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.
7
COMPLEX MANIFOLDS AND PSEUDOGROUP STRUCTURES
COROLLARY. (Implicit Mapping Theorem) Letf)., A. = I, ... , m be holomorphic on V ~ en. Let rank (fJf)./fJz.) = r at each point z of V and suppose in fact that det(iJf;./fJzvhsr :# o. Iff;.(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 complex 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 coordinates Zj: Vj + C" such that: (I)
X=UU J • JEI
(2) The maps fjk: Zk(P) + Zj(p) are biholomorphic [that is, Zj Zk 1 = fjk and r;,/ = Zk 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 V k :# ljJ. 0
0
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 manifold 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 {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 IP n = en + I {O}/'" is a compact Hausdorff space and one can construct a system of complex coordinates as follows: We let Vj = {p E IPnlpj ¥ O}. Then {Vj}jsn is an , f 10. rrM 0 n V j themapzj= (ZO n) h opencovenngo j , " 'J ,Z j I ,Zjj+ I ,,,·,zj,were z/ = pA/pj gives a local coordinate on V j ; in fact, Zj(V) = en. Then fjk: Zk + Zj is given by zj = z:/zt for A =F k, z~ = I/z{. (One simply multiplies by pk/p j ,) Thus we see that {V j , zJ is a complex analytic system defining a complex structure on IPn.
en+ 1 _
Generalizing this procedure we introduce the idea of a pseudogroup structure. All spaces will be Hausdorff in what follows. 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 (diffeomorphism) of X is such a map with X = Y. Let 9 be a domain of IR n 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 onfI[W nf(V)], DEFINITION
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 I Er. (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
2.7. Let r (a pseudogroup on 9) and X (a paracompact Hausdorff space) be given. By a system of local rcoordinates 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 (fequivalent) if W;. a zj' E r when defined. A rstructure on X is an equivalence class of systems of local rcoordinates on X. A rmanifold is a paracompact Hausdorff space X together with a rstructure on X. DEFINITION
EXAMPLES 1. 9 = en, re = (all local biholomorphic maps of e").Thenarcstructure is a complex structure, and a remanifold is a complex manifold. 2. 9 = ~", fd = (all local diffeomorphisms of ~n). Then a fdstructure is a differentiable structure and a fdmanifold 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
1
0
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 fstructure is called a canonical structure. 4. Let r = (local affine transformations of ~"). These transformations have the form n
/A(X) =
L a! x + b\ Y
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
f 2 structure. By assumption f c fd for all f. So every fstructure on X determines a differentiable structure on X and every fmanifold is a differentiable structure on X and every fmanifold is a differentiable manifold. The fstructure M is defined on the differentiable manifold X. The problem of determining the fstructures 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 differentiable 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 2n + I
[s2n
=
{(Xl' ••• , X2n+l) 1
i~2
xf, (Xl'···' X 2n + 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 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. 0
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 iJf;.(P)] ran k [ = r. IS .10 d epen d ent 0 f 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 V j • 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: 2.9. A meromorphic function f on M is a complexvalued function 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. DEFINITION
EXAMPLES l. Any holomorphic mapf:M ..... [pI1 = C U {co}, [S =fI(oo)]. 2. In C 2,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 hypersurface 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 number X(M d) = d(d 2  4d + 6). [The formulas in Examples 1 and 2 can be obtained from Hirzebruch (1962), p. 91, Equation (5). They are wellknown classical formulas. The simple connectivity is also well known and it follows from the Lefschetz theorems on hyperplane sectionssee 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, submanifolds give many examples; but for general istructures one does not usually get sub istructures. 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 automorphisms of M if for any pair of compact subsets K" K2 £; M, the set {g E G I gK, n K2 =t= 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 replace 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
Hirzebruch (1951) Let M = 1Jl>' X 1Jl>'. In homogeneous coordinates, 1Jl>' = {C/ ( = «(0' ~,)}; = {C u {(Xl in inhomogeneous coordinates, (= (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: EXAMPLE I.
j(z, (*) .... (z,
0
= [z,«(*/z'")]
where 0 < Izl
' 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 2cycIe C is homologous ("') to as, + bS 2 , a, b e Z. The intersection multiplicity I(C, C) = J(aS, + bS 2 , aS I + bS 2 ) = 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, Proof
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
./"" ~
V
I
.
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 2cycle and Ao '" Ac. Hence /(Ao, Ao) = /(Ao, Ac) = 1. Since for any 2cycle Z on M, /(Z, Z) == 0 (mod 2) we see M::f: Mr. REMARKS
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).
1.
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 CE 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}. T
w
s Figure 7
Then set W* = D x T = {z, [(*] I zED, [(*] T. Define/: W*  S*~ W  S as follows: /: (z, [(*]) ~ {z, [(*
where 0
O} [some sort of a generalization of 1m W > 0 in Example (1)]. Let C§ = the set of all transformations 0+ (An
where
+ B)(CO + D)I = n',
(~ ~) 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 KodairaSpencer 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 M t = M for t t= to and M to 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 Gt =
{gm I mEl} and g: (ZI' Z2) + (azl where 0
0, and
1 1. few, 2m Iwl=r w 
F(Zl' Z2) =  . j
Z2)
dw,
ZI
for Izd < rand Z2 arbitrary. Then F(zl' Z2) is an analytic function in its cylinder 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 holomorphic if Z2 #= O. So Cauchy's theorem gives
Fix Zl' 0 < Izd < r. Then F(ZI' in Z2; therefore,
Z2)
= f(zl' Z2) for
Z2
#= O. Both are analytic
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
25
a map I: W t W which is biholomorphic, such that
W~W
G'l
I
Go
f
Mt+M o commutes. It follows that Gt = 1' Go! Hence for generator 9t of Gt ,
9, =1'9"5' f.
(9)
Write the map I in coordinates as
I(z" zz) = [J,(z" zz)'/z(z" Z2)]. 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 11 to F which satisfies F[F(z)] = z on Wand by continuity, F[F(O)] = O. But if F(O) =F 0, £[F(O)] = II [F(O)] =F O. This contradiction gives the result. Now expand F)"
F;,(ZI' zz) = F)"z, We know thatf[9,(z)]
+ F).,zz + F;'3ZT + F).• ZIZ2 + ....
= 9"5 I [f(z)] so F[g,(z)J =
(~ O)±I rx F(z).
Rewriting this gives
F,(rxz, + lZ 2 , rxZ2) = rx±IF,(ZI' zz), Fz«(1.z1 + lz 2 , cxzz) = rx±IP1(ZI, zz)· Expanding these and taking the linear terms yields
(P P Il
li
t)
0) (PF
P'z) (rx = (rx P1.2 0 rx 0 rx
±1
ll l ,
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 V l x.1FD 1 where VI u V l = 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 V l 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 {M t I I E C} such that Mo = M(m) and M, = M(f) for 1:1= o. Define M, as follows: M, = VI X pI U V 2 X pI where (ZI' (I) if ZI = l/z 2 , (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 Proof
+4(Z2' (2)
(' _z~' 1
I 
1(1
t (linear fractional transformation).
On the second pi, r'
'2 +
'>2 = I '" kv 22
Then, using
ZIZ2 =
I, and
(I =
Hence, in the new coordinates,
Z~'(2
+ IzL
ZI Z2 =
for PROBLEM. such that
C,2
t2·
we get
I, (~ = z£(~; so
t:f= O.
Q.E.D.
Finda pair ofcomplex analytic families
(a)
Mo
(b) (c)
M, = No N, = Mo
=1=
No, for t for t
=1= =1=
0, O.
{M,lltl
t L...( 
1){+i+ 1 rv u /(.0) ... f(II,) ... f(IIj)g(II ) ... g(II ). J
(7)
q
Similarly, (kbu)IIo···II = I,(1)j+tryuJ(IIo)···;(a';)···f(II·)9(IIJ) ··g(II) q tSj J q
(8)
COHOMOLOGY GROUPS
2.
33
Equations (7) and (8) give
q
 L ry
0j(IIo) "'/(IIj)g(IIj) ... g(IIq)
j=O
= ry O"g(IIO) ... g(IIq)

rV 0"/(110) "'/(II q) '
(9)
Q.E.D.
proving Equation (6).
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 locallyfinite 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 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. dime Hl(M, l!J)
Proof.
Let M = {(Zl' z2)llzll < 1, IZzl < 1, (Zl' Z2) =F (0, O)}. Then
= + 00.
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 DEFORMAnONS
35
In this case M = UI U U z so chose as covering 0/1 = {VI' U z }. Then HI(o/1, (9) = ZI(OlI, (9)/bCO(o/1, (9) where ZI(o/1, (9) = {0'121 0'12 E r(VI () U z , (9)},Co(o/1, (9) = {t It = (tl' tz), tit E r( V"' (9)}, and bCo(o/1, (9) = {tz  tl Itit E r(V", (O)}. We note that VI () U z = {(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.
HI(X, f/) where d/I
If HI( V j, f/) = 0 for all Vj E 0/1, then HI(o/1, f/) ~ = {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 WjA = Vj () VA}' Then it suffices to show that n:" : HI (0/1) + Hl("/Y) is surjective. Take a Icocycle {O'jAb} of HI("/Y) where O'jAjlt + O'j"kv + O'k,jA = O. Then {O'WIt} for each fixed i is a Icocycle on the covering {Wj)J of Uj' Since HI(Vj, f/) = O,HI({WU},f/) £;; HI(V j , 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 Ocochain {tiA} on "/Y. Then {aIAh} = {aUk,}  bt defines a Icocycle 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 is surjective. Q.E.D.
n::,.
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 complex 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
0=
a
L OJ(p)IX aZ
IX=I
j
on W ('\ Uj • These functions should behave as follows: On W ('\ U",
a L Of(p)p' 1 az" n
0=
(1=
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 holomorphic vector fields. (Later we shall give a formal definition of the holomorphic 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
M, = i.ijI(t); (2) the rank of the Jacobian of iij = 1 = dim B. We can find an e > 0 small enough so that iijI(A), A = {tlltl < e} looks as follows: (1)
J
iijI(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 OU j = {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, ('\ OU j , It I < e. Then set
{(z} "', zj, t)llzjl < ej } =
V'j'
3.
37
INFINITESIMAL DEFORMATIONS
so we can use {(Z], ... , zj)llzjl < eJ as coordinates on VIj. The transformation zj = IMzk' t) depends on t. Consider p E dII j n dII j n dII". Then p = (Zj' t) = (Zj' t) = (Zk, t). So z~ = IMzk' t) = Jfj(Zj, t) = lil/ji z), t], where !jk = (fA .... .Jj,,). We set (). ( Jk
P. t
)= ~ ~
01=1
ajj,,(Zk, t) ~ :It :l 01· u UZj
Obviously (}j,,(t) E r( V'j n V,,,, 0,) where 0, is the sheaf of germs of holomorphic 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
t
an" = ar~ + 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 {dll j } such that are coordinates on r;. where
(C~.
t)
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"). £; dII s ().). We also have holomorphic transition functions C{J).v where (~=
C{J;..«(., t)
on
r). n
Then the cocycle defined by this covering is
at a
"aC{J~. fT)..(t) = ~ a,~·
r •.
38
SHEAVES AND COHOMOLOGY
As before s induces a map s* : {Ojd
0A.(t) =
r1'";.n1'"v n
~
{OA.}, where
M.[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 "Y;.. The following equalities are clear: gj[IP;..(C., t), t]
= gj«(A' t) on
= 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
(2)
at
Then (2) implies [multiplying by (iJlozj)]
~ azj ( a) og~ ~ afjk a ~ ozj ( a ) olP~. ~ ogj a ozf ozj . at + L" at ozj = L" iX~ ozj at + L" at ozj'
L"
3)
(
Hence, '1;..
~ Dg~().)
+ L" ot
[_0_] _~ ag~(\.) [_0_] + 0 IX oz.(;.)

L"
ot
II oz'(')
A"
(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 determined 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
U}
we may assume II so chosen that wI(ll) = '¥I j , 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' DEFINITION 3.1. { 0jk I y(t)} where
(oM,/ot Y )
E
tl, " ' ,
HI(M" 0
tm ) on '¥I j n '¥Ik' is the
1)
cohomology
class
of
If (a/at) denotes the tangent vector
a at =
m
a
.f:1 c. ot
Y ,
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 wI(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 homeomorphic 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 M b , 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 twosheeted
40
SHEAVES AND COHOMOLOGY
covering Rp of R with branch points at a and p; t = w(p). We have the question "is
d:,
= 07" Define the following neighborhoods on R:
Wb = Wo =
WI =
{wllwl < r}, {wllwl < rI2}, {wi rl4 < Iwl < r}.
We can write M, = Uo U UI U U 2 ••• Uj ••• where U o = 1I: I (WO)' UI = 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): 11: I( WI), 11:( U j)
Jw  ton Uo , = Fw on U
Zo = ZI
I,
and Zj on Uj can be an arbitrary coordinate which should be fixed and independent 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) component,
= {Ojk(t)}
has only one nonzero
(0) =  2Jz~1 _ t ( OZo0) =  2zo1 (0) OZo .
ofo I °Ol(t) = Tt· OZo
Let Vo = Uo , VI = UJ ~ I Uj • Then OCt) is a lcocycIe 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 RiemannHurwitz 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 RiemannRoch formula [see Hirzebruch (1962)], there is a holomorphicdifferentialqJ(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 Q.E.D. (the number of zeros equals the number of poles). 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
We are assuming m
~
= ZiC2 + tz~,
2k, k
~
and
ZI
= Ijz2'
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> U02}; then
dM,/dtl,=o = 0(0)
E
Hl(dlJ, 0 0 ) £ Hl(Mo, 0 0 ),
Then
so that adO)
af~2) ,=0 = ( at
(0) a'l = k(a) a'i Z2
E
qU ol (', V
02 , 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.
42
SHEAVES AND COHOMOLOGY
LEMMA 3.3.
Any holomorphic vector field on C x pi is of the form 0= 9(Z)(:z)
+ [a(zK2 + b(zK + C(Z)](:c),
where g, a, b, care holomorphic functions on C. Assume Lemma 3.3. We have the following relations:
(o~J = Z~(a~J. (o~J = mZICI(o~J  zf(o~J,
(7)
where (Zy, Cv) are coordinates on VOy = C X pl. Let us compare the coefficients of (0/oc 1) in O2  01 and 012 (0). From Equations (7), we get Z~
I
= zic 2(Z2)  C (Zl) = zic 2 (Z2) 
CI
CIJ,
where the cy(z.,) are entire functions. Expanding,
and 0 < k < m. This is impossible. Hence,
dMtl /1=0 =F O. dt For the lemma we have: Proof. Let (z, 0 E C X pi, where Cis a nonhomogeneous coordinate on At C= 00., the local coordinate on pi is 1'/ = llC. Restrict the vector field to C x pi  C X {oo} = C 2 • Here pl.
0= 9(Z,C)(:z) + h(Z,C)(:C)' where 9 and hare holomorphic on C 2 • At O=y(z, I'/)(:z)
where C=
00
we have
+ P(Z,I'/)(:I'/)'
1/'1 and y, pare holomorphic. Then oC/ol'/ = 1/1'/2 so (0/01'/)
 C2 (0/00. Hence at
00,
0= y(z,
I'/)(:J 
,2{3(Z,
'1)(:,),
=
3.
INFINITESIMAL DEFORMATIONS
43
since g(z, ') = y(z, "'), g(z, ~) is holomorphic on C x pl. So g(z, ') is constant as a function of ,
g(z,
0
= g(z).
Finally, h(z, 0 =  ,2 P(z, "') implies that h(z, ') has a pole of order::;; 2 at So h(z, ') = a(zK2 + b(zK + c(z). Q.E.D.
00.
REMARK 1. The dime HO[M(m), 0] is the number of (complex) linearly independent holomorphic vector fields on M(m). We want to compute it. As usual, M(m) = UI U U2 , U. = C X pi, and
if and only if
We must count the number of parameters involved in representing a HO[M(m), 0]. By the lemma,
() E
() = (). = g.(Z')(a~v) + a.(z.K~ + bv(z.K. + c.(Zv)(a~J on each U., and
(}I
=
(}2
on UI n U2. Changing coordinates,
Hence
(}2 = Zig2(a~J + mZlg2'I(a~J + (a2 zim'i + b2 z~' I + C2)Z2(a~
J
=
Zig2(a~J + [a2 z~'i + (b 2 + mz l g 2)'1 + C2 zim](a~J
=
(}I = gl (a~J + (alCi + b l ,! + Ct)(a~J·
Equating coefficients,
gl(Zt) = zIgiz2), a1(zt) = zTaiz2)' b1(z.) = b2(Z2)
+ mz.g 2(z2), C!(Zl) = Zi'"C2(Z2).
44
SHEAVES AND COHOMOLOGY
These functions are all entire functions of Zl' Let us investigate their behavior at ZI = 00. Since Z2 = lIz!> we see that 91 has a pole of order ~2 at 00, al has a pole of order ~m at 00 and CI has a zero at 00. Assume that m ~ 1. Then
z~
gl
=
al
= alOz~
CI
= 0
glO
+ gl1 Z 1 + g12' + ... + aim,
(by Liouville's theorem).
Consider the b terms:
So bl(zl) = mg lo zl blO . 0 depends linearly on (910,911,912' alO' "', aim, blo), Hence,
(8) We therefore have: THEOREM 3.1.
M(m)
=1=
=1=
M(n)
REMARK 2.
M(2n)
REMARK 3.
Let {M tit E
(complex analytically) if n =1= m.
M(2nl)
topol09 ically.
q, M t given y
':01
=
my Z2':o2
as before. Then Mo = Ml m), M, =
dM,ldt={
by
+ tz k2 , ZI M(m2k)
0, =1=0,
1
=Z2
for t =1= O. And we have shown
fort=l=O
for t = O.
Suppose we "reparametrize" and consider {M.21 sEq.
Then Mo
= M(m),
Ms2
= M(m2k),
S
M.2
is defined by
=1= 0 as above. But
dMs2 = dM t • ds 2 = 2s dM, = 0 ds dt ds dt for all SEC. We know that M t independent of t implies dMt/dt = O. We have just seen that dM,/dt = 0 does not imply that M, is independent of t. However, we have the following theorems:
3.
INFINITESIMAL DEFORMAnONS
4S
THEOREM Ol. If dim HI(M I> 0,) is independent of I and if iJM ,liJI' = 0 for all v and I, then {M, I E B} is locally trivial and hence M, is independent of I. THEOREM p. The function t function of I. That is
+
H1(M" 0,) is an upper semicontinuous
dim HI(M" 0,) ~ HI(Ms' 0 s), if I is in a sufficiently small neighborhood of s; that is, lim dim H1(M" 0,) ~ Hl(Ms. 0 s). , .... s
THEOREM y. of s.
If Hl(Ms' Os) = 0, then M, = Ms for t in a small neighborhood
Theorem Ol is proved in Kodaira and Spencer (l958a), Theorem p in Kodaira and Spencer (1960), and Theorem y is due to Frolicher and Nijenhuis (1951). Theorem p follows from some results which we will prove in a later chapter. Theorem ex will not be proved here. DEFINITION 3.3. We say that a compact complex manifold M is rigid if, for any complex analytic family {M, It E B} such that M,o = M, we can find a neighborhood N of to such that M, = M,o for lEN. (More precisely, if w: J( + B is the family {M,}, then wI(N) = N x M,o complex analytically.) The following theorem follows from Theorem "/. THEOREM 3.2. If Hl(M, 0) = 0, then M is rigid. We will give a proof of this using elementary methods. We have the following: PROBLEM. (Not easy?)
Find an example of an M which is rigid, but Hl(M, 0) '=F O.
REMARK. IfDn is rigid. For n ~ 2 the only known proof is to show H 1(lfD n, 0) = 0 [Bott (1957)]. Let us proceed to the proof. Proof (of Theorem 3.2) The proof will be elementary in that it consists of two elementary ideas:
(I) (2)
Construction of a formal power series, and proof of convergence.
The proof is actually long and computational, so please stay with us. It makes no difference for the proof and it makes the writing much easier if we assume
46
SHEAVES AND COHOMOLOGY
dim B = 1. The result is local so we may assume B = {tlltl < r} and to = O. We can cover (jjl(d£), d a = {tlltl < e} with coordinates o//j = {(Zj' t)llzjl < ej' It I < e}. Then
Zj=fjk(Zk,t)
on
o//jno//k'
M is covered by u UJ = M where j
UJ Then M x B
= u(UJ
= {z j Ilzjl < l)X{O}
x B) where for (wv, t)e
!:
U~
o//j.
x B,
J
if and only if
that is, (9)
wj = gMwk), We can rephrase our result:
If b is sufficiently small there is a biholomorphic map qJ of (jjl(d,) onto M x d, such that qJ: maps (jjl(t) onto M x t and qJ: M = (jjl(O) + M x 0 is the identity map. Suppose we choose b so that qJ maps
THEOREM.
into U~ x B, qJ(o//i)!: U~ x B. Let (zj,t)e0//1. Then, qJ(Zj,t) =(wj,t) = [qJj(Zj, t), t] so on each qJ is represented by holomorphic functions qJ~(Zj, t) where qJj(Zj' 0) = zj. On o//~ n
0//1,
all:,
zj = fMzk' t); so
implies (10)
Therefore we see that we can prove the theorem if we can construct holomorphic functions qJj(zJ' t) on o//~ satisfying (10) and
qJj(Zj' 0)
= zj.
(11)
3.
INFINITESIMAL DEFORMATIONS
47
For simplicity we may as well assume that dIl j is of the form
= {(Zj' t)llzjl < 1, It I < e}, M = U UJ, {zjllzjl < 1 + v for some v > O} and UJ:::> M n dIl j • If expand CPj(Zj, t) dIl j
UJ = into a power series, we get CPj(Zj, t)
= Zj + CPjI1(Zj)t + CPjI2(Zj)t 2 + '" + CPjlm(Zj)t m + ... ,
(12)
where each CPj/m(z) is a holomorphic vector valued function. If we expand both sides of (10) we get co
co
L Fm(CPjll' ... , CPjlm) = m=O L Gm(CPt/1' ... , CPltlm)tm, m=O
(13)
where Fm and Gm are polynomials. We introduce some notation: If P(t) = Pn t" and Q(t) = Qn t n are two power series, P(t) == Q(t) means Qn = P n
L
L
m
un to n = m [that is, P(t) == Q(t) mod (tm+l)]. Therefore, to solve (formally) Equation (13) we need only solve CPj[fjk(Zk, t), t] == gjk[CPk'(Zk' 0], m
for each m, where cPj(Zj' t) =
Zj
+ ... + CPj/m(z)t m.
(14)
First consider m = 1. We have co
Zj
= fjk(Zk, t) = Yj/.(zk)
+ L fjklm(Zk)t m • m=1
Using (13)10 gjk(Zk)
+ fjkll(Zk)t + CPjJ1[gjk(Zk)]t == gjk[Zk + CPkl1(Zk)t] 1
So
Now
ejk = ~ (a~:k),=oC~~j) = L fjkll(a~j) belongs to Hl(M, 0). By assumption HI(M, 0) = 0 so {Ojk} is cohomologous to zero. But Equation (13)1 says we must find {CPk)!} so that Ojk = CPk/10 so that 00
gjk(Z"
+ y) ~
L "m(YI + ... + Yn)m. m=O
(17)
For the moment, let 1/I"(z,,, t) = qJ~(Zk' t)  Zk' We want to estimate [gj,,(qJ~)]m+l = [gjk(Z" + 1/IJJ+ml where m + 1 ~ 2. But,
since I/I"(z,,, t) is a polynomial of degree we get
~
m in t. From Equation (17),
52
SHEAVES AND COHOMOLOGY
hence, 00
[gjk(Zk
+ !/Ik)]m+ 1 ~
L KTnA(t)J ,=2
~ A(t)
L K'n' (b)'1 C 00
,=2 K2
1
n2 b
= A{t)c1 _ (Knh/c) ,
because by induction!/lk
~
A(t).lfwe choose c so large that (Knb/c)
bo , c> Co and then A(t)
~
Ao(t), so
K2 K2b 7i Ao(t)A(t) ~ Ii ~ A(t). Finally,
If  2K 2n 2) ~b A(t),
K2 ~ ( for
ZE
U; n Uk. Hence
r.
Jk
K b 2 n2 ) A(t) t m + 1 ~K ( 2+2K 1
(3
(20)
c'
~ ozj p () r ajk = L.. :;p CPklm+ 1 Zk vZk
a
(_ )
qJjlm+ 1 "" j
and we want to estimate cpjlm+l(Z). As before, consider r jk = L rjizj)(%zj), CPjlm+l = L cpjlm+l(O/Ozj). With these notations, r jk = CPklm+l  CPjlm+l. At this point we need a lemma which plays a crucial role in arguments of this type. Let r = {r jk } be a lcocyc1e where r jk is a holomorphic vector field on Uj n Uk. Let '" = {'"j} be an Ocochain where '" j is a holomorphic vector field on U j • We define
lin = max j,k
sup
max
ZeVj"Vk
a
WMz)l.
""'II = max sup max l"'j(z)l. j
ZE
VJ
a
LEMMA 3.7. There exists a constant K such that if r is cohomologous to 0, then we can find", with fJ", = r satisfying" "''' ~ K where M is a compact complex manifold, Hl(M, 0) need not be zero, and K does not depend on r.
"n
Proof Remember Vj = {zjllzjl < I} and U;{zjllzjl < I the lemma is not true. Let t(r) = {""''' I fJ", = r}
.)} such that I/I~V,l)(Zj) converges uniformly on Ur If Z E Uk then Z is in some Ur Hence for each k, 1/11·,l)(z) = I/I}·,l)(z) + r~~,l)(z) converges uniformly on Uk so cp~·,l)(z) + I/Ik(Z) uniformly on Uk. Since f~~,l) + 0, I/Iiz) = I/Ik(Z) on Uj n Uk. Let tfr)V,l) = I/Ii·,l)  I/Ij. Then q:,l) = tfr~·,l)  tfr}V,l). But IItfrc· . . )1I < 1 for large v, and thus r[r c.... )] < t. This contradiction proves the lemma. Q.E.D. By Lemma 3.7 we can find K and {CPjlm+l} so that r jk = CPklm+l  CPjlm+l and IICPm+ll1 ~ Kllfll· Hence
(2 2+73 K2) ~A(t). b
CPjlm+l(Z)t m+l ~KKI 2K n
We can choose c so large that KKl(2K 2 n 2 + K2/P)(b/c) < l. Then CPjlm+1 t m+ 1 ~ A(t) so cpj+l(Zj' t)  Zj ~ A(t). This completes the induction and proves that cP j(Zj, t)  Zj converges for small t, thus finishing the proof of the theorem. Q.E.D. There is the following: THEOREM. [Kodaira, Nirenberg, and Spencer, (1958)] Assume H2(M, 8) = 0 (M is compact complex as always). Then for any element e E Hl(M, 8), there is a complex analytic family {M,lltl < r, r > O}, such that Mo = M and (dM,/dt),;o = e. PROBLEM. Find an elementary proof of this. (In the analogous idea of proof the convergence gives trouble.) We also have the completeness theorems. DEFINITION 3.4. We say that the family (.it, B, w) is complete at be B if, for any family (.AI", A, n) such that n 1 (a) = w 1 (b) = Mb there is a neighborhood U 3 a and holomorphic maps : n 1(U) +.it, h: U + B such that h(a) = b, 7[l(V) _ _ .it
·1 . 1
commutes,
0
UB
maps 7[l(S) biholomorphically onto w1[h(s)] for each s E U, and : nI(a) = Mb + Mb is the identity map.
SHEAVES AND COHOMOLOGY
56
Roughly, (...II, B, w) is complete if it contains all sufficiently small deformations of M b. The (holomorphic) tangent space Tb at b is the set
{:t \:t vt/v(o~v)}. =
map Pb: (a/at) + (OMrlOt)r=b Tb + Hl(Mb' 0 b).
The
THEOREM.
[of completeness; see
E
Hl(Mb' 0 b) defines a linear map
Kodaira and Spencer (1958b)]
If
Pb(Tb) = Hl(Mb' 0 b), then (...II, B, w) is complete at b. REMARKS. (1) Theorem 3.2 is a corollary of this theorem. (2) This theorem of completeness can be proved by the same elementary method used to prove Theorem 3.2.
4.
Exact Sequences As usual X is a paracompact Hausdorff space, [/ is a sheaf over X and map.
w : [/ + X is its projection DEFINITION 4.1. (1) (2)
(3)
[/'!; [/ is a subsheaf if
[/' is open, w([/') = X, wl(x) n [/' = [/~ is an Rsubmodule of [/".
Let [/" be a sheaf over X with projection
w".
DEFINITION 4.2. A homomorphism h of [/ into [/" is a continuous map of [/ into [/" such that (1) (2)
w"
0
h = W,
h: [/" + [/; is an Rhomomorphism. h is a local homeomorphism. We define the kernel of h to be
REMARK.
ker h = {s I h (s) = O} where" t = 0" means t" = Ox E [/x. LEMMA 4.1.
ker h is a subsheaf of [/.
LEMMA 4.2. h([/) is a subsheaf of [/". The proofs are left to the reader.
4.
57
EXACT SEQUENCES
Let f/' E f/ be a subsheaf. Let Qx = f/x/f/~ which is an Rmodule. Define hx: f/ x + Qx to be the natural homomorphism. Let Q = UXEX Qx' Define 11: on Q by 1I:(Qx) = x; h is defined h : f/ + Q by h(s) = hx(s) for s E Y X' We give Q a topology by saying i1IJ is open if and only if h 1(011) is open in f/ (the quotient topology).
LEMMA 4.3. Proof
Q is a sheaf and h : f/
+
Q is a surjective homomorphism.
Left to the reader.
DEFINITION 4.3.
Q is the quotient shea/of Y by Y' and we write Q =Y/f/'.
A sequence ho
III
f/ 0   + f/ 1   + f/ 2   +
...   +
IJ ,.+
hn
J
Y n   + Y n + 1    + ...
of sheaves is exact if h.(f/.) = ker(h.+ 1) for all v. Suppose given sheaves Y and Y" over X and a homomorphism h : Y + Y". Let V be an open subset of X and Y) be the set of all sections of Y over V, Then h 0 U E Y") if U E Y) so h induces a map h: Y) + V, f/"). Let i1IJ = {Vi} be a locally finite covering and Cq(OlI, Y) be the space of qcochains cq = {u io ... i.} where U io ... j. E V jo n ... n V j . ' Y). Then h induces a map h : CQ(i1IJ, f/) + Cq(i1IJ, f/") defined by h cq = {hUio'" i.}' Then we have:
re
LEMMA 4.4. Proof
reV, reV,
reV,
reV,
re
h0
()
= () h. 0
Obvious.
Hence h maps zq(OU, Y) into zq(OU, f/") and thus h induces a homomorphism h: Hq( i1IJ, f/) + Hq( i1IJ, f/"). Let 1f/ = {1f/./.} be a refinement of i1IJ, 1f/ > i1IJ. Then HCJ(u71, f/)~Hq(i1IJ, f/")
ln~
In~r
Hq( "If", Y)..!:. Hq( 1f/ f/")
commutes. Hence h induces a homomorphism h: Hq(X, f/)
+
Hq(X, Y").
SHEA YES AND COHOMOLOGY
58 THEOREM 4.1.
Assume that
is exact. Then there is a homomorphism c5* such that
is exact.
Proof i is injective so f/' ~ i(f/') c f/, and i(f/') = ker h. Thus we consider f/' c f/ where f/' = ker hand i is the inclusion map. Recall that HO(X, f/) = ZO(X, f/) = f/). Since nX, f/') c nX, f/), we see that 0+ HO(X, f/') ~ HO(X, f/) is exact. If a E nX, f/), then ha = 0 if and only if a E nX, f/'); so HO(X, f/') ~ HO(X, f/)..!!. HO(X, f/") is exact.
rex,
LEMMA 4.5. HO(X, f/)..!!. HO(X, f/")~ Hl(X, f/') is exact (where we must define c5*).
Proof Let a" E nX, f/"). Since h is a local homomorphism there is a section Ty E Uy , f/) over a small neighborhood Uy of y such that h Ty(X) = a"(x) for x E Uy . Now {Uy lYE X} covers X and we have a locally finite refinement tft = {Uj } of {Uy}, that is, there is a map j + y(j) such that Uj S;; Uy(j)' Set Tj = r Uj ry(j) E nUj' f/). Then h Tj = a" where defined. Let CO = {Tj } E CO (tft, f/). Then:
n
DEFINITION (4.4)1' c5*a" = [c5eO] E Hl(X, f/') where for any eq E zq(tft, f/), [e q] denotes the cohomology class in Hq(X, f/) of eq. (One should check that c5* is well defined.) Since c5eo = {Tk  Tj} and hTk  hTj = a"  a" = 0, we see that c5eo E ZI(tft, f/') so Definition (4.4)1 makes sense. Exactness means c5*a" = 0 ifand only if a" = ha for some a E nX, f/). So suppose (j*a" = [c5eO] = O. Then c5eo = (jeo' where co' = {Tj} E eO(tft, f/'). So CO  co' = a E ZO(X, f/) = nX f/), and ha = heo = {hT j } = a". Now suppose a" = ha. Then h(Tj  a) = 0, so TjaEnUj,f/'). Set C~={Tja}=coaECO(tft,f/'). Then c5e~ = c5eo since a E ZO(X, f/) and hence c5*a" = [c5eo] = [c5e~] = O. Q.E.D. We now turn to check that
4.
59
EXACT SEQUENCES
is exact. Take e l ' E ZI(Olt, f/'). If [e l '] = {)*q" = [{)eO], then i[e l '] = 0, and if ;[e l '] = 0, then e l ' = {)eO; so 0 = he l ' = Meo = ()heo. Thus heo defines an element q" E reX, f/"). By definition, [e l '] = {)*q". We want to prove exactness
LEMMA 4.6. Given ell" E Cq(qj,f/"), then we can find a locally finite refinement "if'" and ell E CIl("if'", 9') so that n~eq" = heq.
Proof We give proof for q = 2. Let Olt = {Vi}' eq" = {qijd, where q/jk E r( Vi (1 Vi (1 V k , f/"). Choose a covering "Y = {Vi} such that Vi C V J • Since Olt is locally finite, a given Y E X belongs to only finitely many Vi' We choose a neighborhood Ny of Y sufficiently small so that x
E
(I) if Y E Uk (1 Ui (1 V k there is r E r(Ny , f/) with q/jk(X) = hr(x) for Ny (remember h is a local homeomorphism), (2) for each y there is Vi such that Ny C Vi' (3) if Ny (1 tj :F cP then Ny C Ui .
Then {Ny lYE X} covers X and we can choose "if'" = {WJ a locally finite refinement of {Ny}. Hence there is a map A.  YA such that WA C Ny). • By (2) Ny). C Vi;>.' so 1(/ > "Y > Olt. Define r = {r A".} E C 2 (1(/, f/) as follows: we have WAC VI' W"c Vi' WycCk where i=jA' j=jA, k=jy. By (3) if Ny). n Vi :F cP, Ny). C Ui , and Ny). n Vk:F cP gives NYJ.. C Ub and so on. We are assuming WA n W" n W. :F cP, and WA C Ny). C Vj , and so on. Hence it follows that YA E Ny). C V k (1 Ui n Uk' By (1) q/jk(X) = hr(x) for x E Ny). where L E r(Ny". f/). Let LA". = rWA" W .... " Wy(L). Then
Q.E.D. Let us prove that h
j
Hq(X, f/)Hq(X, f/)Hq(f/")
is exact. hi = 0 is clear. Suppose,., E BIl(X, f/) and h,., = O. Then,., = [eq], eq E ZIl(Olt, f/) for some Olt and n!!hell = ()e ll  I " for some "Y and eq I " E Cql("Y, f/"). By the lemma n~eqI" = h IqI for some 1(/ and E cqI(1(/, f/). Thus
,,1
hn~ ell  h {)t,I =
0,
so n~ eq  ()t q I = eq'
where eql E zq(1(/, f/') ~ cq(1(/, f/'). Finally we get ,., " = i,,' where ,.,' = [eq,] E Bq(X, f/').
= [eq] = [e"]
so
60
SHEAVES AND COHOMOLOGY Next we prove that
is exact. We must define 15*. Take '1" E Hq(X, f/"). Then '1" = [eq"], eq" E zq(O/t, f/"). By the lemma there is t q E Hq(1I', f/) such that h t q = IT:' e q". DEFINITION 4.4. 15*'1" = [t5tq] E Hq+l(X, 9"). Again we should check that 15* is well defined. For the moment denote n:. by n. Suppose 15*'1 = O. Then TIt5t q = bbq' for some bq' E C q(1Ji, 9"). Thus t5(IT t q  bq,) = 0 and n t q  bq = eq E zq(1I', 9'). So let ,,= Ceq] E Hq(X ,9'). Then h'1 = [he q] = [TI ht q] = [IT eq,,] = ,,". Suppose conversely that '1" = h". Let" = Ceq]. Then nil = [he q] and ht' = IT heq by definition of t q. So t q  neq = aq, E Cq(11', f/') and (j*'1" = [bt q ] = [t5a q ,]. Thus (j*'1" = O. Finally we prove that 6'
i
Hq(x, f/")..Hq+l(X, f/')..Hq+l(X, f/)
is exact. Certainly i (j*,," = O. Since by definition (j*'1 = [(jt q] where t q E Cq (11', f/) so i(j*" = O. Suppose i'1' = 0, ,,' = [e q+1 '] E Hq+l(X, 9"). Then TIe'+!' = (jt q for t q E Cq(1I', f/). Then 0 = (j ht q E 2'(11',9''') and ,," = [htq] E Hq(X, f/"). Since (j*,," = [(jt q] = '1' we are finished. Q.E.D. [If the reader wishes more details for these elementary properties of sheaves he may consult Hirzebruch (1962).] Next we prove functoriality. THEOREM 4.2.
If i
h
0  . . f/'  . . f /  . . f/"..O
j~
j~' t
j~. k
O..ff'..ff..ff"..O
is exact and commutative, then
is exact and commutative.
4. Proof.
61
EXACT SEQUENCES
We need only prove commutativity. We check that Hq(X, !I''') 
6'
Hq+ I(X, !I")
\..
\. ."
Hq(X, ffll) 
Hq+ I(X, ff')
commutes. The rest is easy. Let,," E Hq(X, !I'''). Then there is cq" E zq(1I/, !I''') and t 4 E C4(1r,!I') such that ,," = [c q"] and cq" = ht q. Thus ~*17" = [c5tq]. However, cp'~*,," = [cp'~tq] = [~cptq], and cp",," = [cp"htq] = [cp"ht q] = [h cpt q ]. Thus ~*cp",," = [c5 cpt q] = cp'c5*,,". Q.E.D. We give a brief discussion of fine sheaves. DEFINITION 4.5. !I' is afine sheaf if for any locally finite covering {Uj} of X there exists a set {hj} of homomorphisms hj: 9..9 such that (1) (2)
hj!l'" = 0 for x ¢ Wj' where Wj j = id.
Lh
£;
Uj is a closed subset of Uj,
j
EXAMPLE. Let!l} be the sheaf of germs of differentiable functons on a differentiable manifold X. We have a partition of unity subordinate to Uj ; that is, a set {Pj} of differentiable functions Pj = pix) on X such that (1) (2)
Pj(x) = 0 for x ¢ Wj'
L Pj = 1.
For any local differentiable function f Then h j induces a homomorphism hj : P} is fine. THEOREM 4.3.
= f(x)
on X, define h jf = pix) f(x). !l} +!l}. Using these {hJ we see that
If 9 is a fine, then Hq(X, 9) = 0 for q
~
1.
Proof. We give the proof for the case q = 2. Let c 2 be a cocycle, = {O'ijk} E Z2(atJ, !I'), with O'ijk E n Uj n Uk' 9). By fineness we have the {hj} in Definition 4.5. Since ~C2 = 0, if Ui n Uj n Uk n Ut =F tP then O'jkl  O'iM + O'ijl  O'ijk = o. Since hiO'ij/,(x) = 0 for x ¢ Wi' hiO'ijk can be extended to T ijk E U j n Uk,!I') by setting T ijk(X) = 0 for x E Uj n Uk  Wi. For fixed (j, k) we have only a finite number of Ui with Ui n Uj n Uk =F cp and for each i we have Tijk from hiO'ijk. We set Tjk = Li Tijk. Then
reUi
c2
re
hi O'jkl = hi O'ikl  hi O'jjt + hi O'jjk· Thus O'jkt = Tkl  Tjt proof for any q ~ 1.
+ Tjk so c 2 = &1 Q.E.D.
where c l
= {Tjd. It is easy to give this
62
SHEAVES AND COHOMOLOGY
5.
Vector Bundles
We give a brief review of vector bundles. Again, a good reference for this section is Hirzebruch (1962). Let M be a complex (differentiable) manifold.
5.1. By a complex analytic (differentiable) vector bundle (e" or bundle) we mean a complex (differentiable) manifold F together with a holomorphic (differentiable) map rc: F + M onto M such that, for a sufficiently fine locally finite covering dIJ = {V j} of M: DEFINITION ~"
(1) There is an analytic (differentiable) equivalence Ij between rc 1(V j ) and Vj x en (or U j x ~II) such that
rc 1(V j ).!!. Vj x
en
·1 " 1·, Vj
commutes, where rciZj' () (2) (or Vj
If(z, X ~"),
=
+
Vj
Zj'
(J. ... , (j) E Vj x e" (or Vj x ~") and (z, (L"', me V
k
x
e"
then ~I f j Jk 1( Z, ~k' 0
YII)" = l.J"
••• ,C,k
fIXjkP (Z )~k' ~P
P:l
are holomorphic (differentiable) functions on Vj n V k • In vector notation.ljk is the matrix (.Ij~p) and (i = (j), and then
where.lj~p(z)
('J ' .. "
We call rc 1 (z) the fibre of F over. By (1) and (2) we can give it a vector space structure rc 1(z) = z X e" [or rc 1(z) = Z X ~"]. DEFINITION 5.2. We say that Fand F' are holomorphically (or differentiably) equivalent if there is a biholomorphic (bidifferentiable) map cp : F + F' such that
(I)
F~F'
\/ M
commutes.
5.
63
VECTOR BUNDLES
(2) On each fibre cp is a linear transformation, that is, if Uj is chosen as in Definition 5.1, then there is a holomorphic (differentiable) matrixvalued function h j on U j such that fi 0 cp OfjI = hj, that is, C,/ = hjp(z) C~. In particular, if Fis a C l bundle and Fis trivial over Uj (that is, nI(U j ) = U j x C), then z E Uj n Uk implies that (z, C) is identified with (z, (k) if and only if (j = fjk(zKk wherefjiz) is a nonvanishing holomorphic (differentiable function) of Z E Uj n Uk. Let (J* (or .@*) be the sheaf over M of non vanishing holomorphic (or differentiable) functions in which the module operation on each stalk is multiplication and the ring R = 7L so here instead of a.f(z) + f3 g(z) we have [f(Z)JIZ[g(Z)]P, a., f3 E 7L. Consider fjk as an element of
Lp
Hence {fjk} E Zl(au, (J* (or .@*» and two bundles F and F' are equivalent if there are nonvanishing functions h j(z) on U j such that fi  I 0 hj 0 fj = R  1 0 hk 0 fk on Uj n Uk. This is equivalent to
or Uik} = {fjd . b{h;I}, that is, Uik} is cohomologous to {.Ijk}. Thus an equivalence class of bundles defines an element [{fjdJ E HI(au, l!l*(or .@*») s;;; HI(M, l!l*(or .@*». Conversely it is easy to construct a bundle from an element of HI(M, l!l*(or .@*». Thus we have a 11 correspondence between equivalence classes of C l bundles and classes in HI(M, (!)*(or .@*». We shall always identify equivalent C l bundles and we call Ht(M, l!l*(or .@*» the group of c i bundles over M. It has a natural group structure if we define F· G = {fjkgjk} where F = Ujk} and G = {gjk}. We construct an important invariant of C I bundles. For any germ of a holomorphic function!, e 2l1i / E l!l*. Thus we get an exact sequence
o+ 7L + l!l + l!l* + 0, where 7l.. is the sheaf of germs of locally constant integer valued functions on M. We also have the following commuting, exact diagram:
O~r~[~u~O 0+ 7L    + l!l    + l!l *    + 0,
64
SHEA YES AND COHOMOLOGY
since lJ7 c
~
and lJ7* c
~*.
This yields the exact commutative sequences
... _HI(M, l)HI(M, lJ7);;+HI(M, lJ7*) lJ'
_H2(M, l).H 2(M, lJ7) ...
···_HI(M,l)HI(M,~)HI(M,~*) IJ'
_H2(M.l)_H2(M,~)_···.
Now ~ is a fine sheaf, so Hq(M,
c5* : Hl(M,
~*) + H2(M, l).
DEFINITION
5.3.
~)
= 0 for q ~ 1. Thus
c5* is an isomorphism
c(F) = t5*(F) is the (first) Chern class of F.
We remark that c(F) = c(G) if and only if qJ(F) and qJ(G) are equal in Hl(M, .@*) where qJ : Hl(M. lJ7*) + Hl(M, .@*) is induced by lJ7* c .@*, where F, G E Hl(M, lJ7*). Hence F and G are difJerentiably equivalent; that is, there are nonvanishing differentiable functions hj such that {ht} = {hjgjt h;;l} where F = {hk}' G = {gik}. Thus: 5.1. The Chern class c(F) of a complex analytic represents the differentiable equivalence class of F.
PROPOSITION
el
Let us give an explicit description of c(F). If F = {ht}, hj . h,.
bundle
.hi =
1
and logJij
+ loghk + IOgJki =
2ni Ciik,
where i = )=1. Then c(F) = {Cijk} E H2(M, l). Next consider C" bundles. Let n ~ z and F be a en bundle defined by {fjk}. Let (D be the sheaf over M of germs of matrixvalued holomorphic functionsJ(z) = J;(z) with detJ;(z) "# 0 where the module operation is matrix multiplication. Note that the operation is not commutative. We cannot define the higher cohomology groups of (D but we can define the following objects: Let dlt = {V j} be a locally finite open covering of X. A Ocochain CO = {h}, Jj E nV j , (D) is a set of sections of (D over V j ; a lcochain c l = {hd where hk E nVjk' (D); a lcocycle c l is a lcochain such thathiz) = hj(z) . Jjk(Z) for z E Vi n Vj n V k . Let Zl(dlt, (D) be the set of all lcocycles. We note that Zl(dlt, (D) is not a group. DEFINITION 5.4. We say that {fik} and {gid E Zl(dlt, (D) are equivalent if there exists {hj} such that gik = hihkh;; I. Let HI(dlt, (D) be the set of equivalence classes of lcocycles. We define HI(M, (D) = lim HI(dlt, (D). cfI
5.
65
VECTOR BUNDLES
PROPOSITION 5.2. Each element of Hl(M, 6)) represents an equivalence class of complex analytic en bundles.
Proof
Left to the reader.
We now list some methods of forming new bundles from old bundles. Let F be a en bundle, G be a em bundle, q[ = {U i} a trivializing covering and F = {fik}, G = {gik}' We define the following new objects: (1) Whitney Sum Fffi G. the cocycle {hik} where
This is a
hik(z) =
Tensor Product F®G. This is a hik(z) =Jjk(z)®gik(Z). Recall that hjk 
(
hll 21
h21 11
•••
h~~
...
by
(~k g~J.
(2)
hll 11
en + m bundle which is defined
...
enm bundle defined by {hik} where
h
11 nl
•••
hll) nm
,
It::
and here hjtp", =fjkpgjk",. A point in F®G has coordinates (z, e}l, "', qm, .. " ejm), where (z, e1) and (z, ek) are identified for z E Vi n Vk if and only if
ejA =
L fjkP(Z)gjkizKf"'.
(3) Dual bundle F* of F. This is the bundle defined by {Jjt} where fit = (fjk 1)' = Uie)' which is the transposed inverse of Jjk' Then (z, n") is identified with (z, et P) if and only if
ej« = L fjkP(zKt P = Sometimes we write
L ffj ..(zKt p.
et.. for ej«. Then we have n
(t.. = P=1 L Hk. (Z)(:p. (4)
Complex Conjugate F of F. This bundle is defined by the cocycle
{fik}' Let us now define subbundles and quotient bundles. Suppose that, by a suitable choice of q[ = {U i} and of fibre coordinates G, the matrices Jjk in the lcocycle {fik} defining F can be written as follows:
66
SHEAVES AND COHOMOLOGY
Hencej!tp(z) = 0 for I ~ f3 ~ m, m + I ~ ex ~ n. Thus ej = :Lil=m+tf/t/i cf for ex> m, and if ef = 0 for f3 > m, then ej = 0 for ex > m. Let F' = u UJ X em where em = {(ej, "', ej, 0, . ", On £; and we identify (z, e) and (z, ek) if j = Ajk(zKk' Then F' is a subbundle of F. The quotient bundle F" = FIF' m bundle defined by the lcocyc1e {C jk }. is a
e
en,
en
DEFINITION 5.5. A holomorphic (or differentiable) section of F over V£; M is a holomorphic (differentiable) map cp: z + cp(z) of V + F such that 1lCP(z) = z where F is a holomorphic (or differentiable) en bundle. We see that locally cp is a set of nfunctions. Since local sections and germs of sections are defined, we get a sheaf of germs of sections of F. We denote by t9(F) (or !7)(F» for sheaf over M of germs of hoi omorphic (or differentiable) sections of F. Then locally, t9(F) = t9IUj E9'" E9 t91 Uj (sum ntimes), where t91 U j means t9 restricted to Vj and t9 z(F) = t9 z E9'" E9 t9 z (ntimes). We now review tangent bundles and tensor bundles. Let M be a complex manifold and {U j} an open covering of M with coordinate patches with coordinates (z), .. " zj) on V j • A (holomorphic) tangent vector at z is an element of the form v = :L:=1 ej(olozj). It is easy to see that the set Tz(M) of all complex tangent vectors at z is a complex vector space Tz(M) ~ If z E Uk another chart at z, then we identify :Lil=l Cf(818zC) with :L ej(818zj) if
en.
I jk/i z II
()
8zj
=::;p. OZk
This is a linear identification so the vector space structure of Tz(M) is well defined. The set T(M) = UzeM Tz(M) is a complex analytic vector bundle defined by the lcocyc1e {fj~/i(z)}. T(M) is the holomorphic tangent bundle of M. T(M) is the conjugate (holomorphic) tangent bundle of M. And !/(M) = T(M)E9 T(M) is the (complexified) tangent bundle of M. Then e, the sheaf of germs of holomorphic vector fields, is t9(T(M». If M is a differentiable manifold with local coordinates (xj, . ", xj), then the (complex) tangent bundle !/(M) = UxeM!/iM), where
The real tangent bundle !/JI,CM)
= UxeM!/ xlR(M), where
The relation is !/(M) = IT IR(M) ®IR e, where e is the trivial (complex) line bundle over M considered as a real bundle.
5.
67
VECTOR BUNDLES
Let T*(M) be the dual bundle of T(M). If Tz*(M) the transition relations are
= {(ejt> "', ejll)}, then
We use the following notation: An element u E ~*(M) shall be written u = L" ej" dzj, where dzj(a;i}Z~) = b p as an element of T.*(M). Briefly let T = T(M), T* = T*(M). A tensor bundle is a bundle of the form T ® ... ® T® T* ® . " ® T ® ... ® T*.
We denote T®'" ® T = (® T)P for the pfold tensor product of T. We remark that T is not a hoi om orphic bundle so (® T)P ®(® T*)q is a holomorphic bundle but (® T)P ®( ® T*)q ®( ® T)' ®( ® T*Y is only differentiable. A holomorphic (differentiable) tensor field is a holomorphic (differentiable) section of a tensor bundle. We now give a brief treatment of differential forms. 5.6. A differential form of type (p, q) (or a (p, q)form) over an open set W~ M is a differentiable section cP: z + [z, q> jill ... "pp, ... P4(Z)] of (® T*)P ®(T*)q over W such that the fibre coordinates CPj""""pPI"'P q are skewsymmetric with respect to OCI ••• OCpPI ••• pq • If P = 1, q = 0, cp(z) = 1 cP j..(Z) dzj. In general, we represent the (p, q) form as follows:
DEFINITION
L:=
1 cp(z) =  ,,
L
p. q. """, IIp 111 ..... Pq
CPj"I'''''PP''''Pq(z) dzj'
1\ ... 1\
dzjP
1\ ... 1\
dz~q,
where dz P, = dz P, and " 1\" is the wedge product of skewsymmetric forms and satisfies for example, dz" 1\ dz P = dzfl 1\ dz". [Note: We write z" = z" = Zii.] If M is only differentiable we still have !/(M) and !/*(M). Then if W is
open in M, we make the following definition: DEFINITION
5.6'. A differential form of degree p over W 1 cP = p!
L CPj"I"'''p (x) dxjl 1\ .. , 1\ dxjp
is a section over W of !/P which is skewsymmetric in the indices OC1 •• , oc p . If cP is a pform and x is a qform, we define the wedge product of forms
68
SHEAVES AND COHOMOLOGY
For example, if cp
= t L CPap dx" CP" 1/1
" dxP and 1/1 = L I/Iy dxY, then
1
= 2" L CP,.pl/ly dx"
A
dx P " dx Y
1 = 3! "L.. X,.py dx"" dx P " dx Y ,
where
Hence
DEFINITION
5.7. If
then the exterior derivative dcp is
1 din . .=, . , p.
L
acpJ'''' ... ,. P d x~ !\x~
CI.Cl:I,° .. ,CZ:p
1
V
J
J
A
d x":'"···,, dx,,:p N
N
J
J
(1)
L
= ( p+ 1)',,,o,,,,,,.p I/Ij,.o"·"p dx? " '" " dxj",
where
1/1"0"'"'' = (a:"o)CP"""""  (a:,..)CP,.O,.l"'"'' + ...
+(
l)p(a:"" )cp,.o ... ,.p •.
5.3. dcp is well defined (that is, the definition is independent of the choice of local coordinates Xj)'
PROPOSITION
Proof We write out the proof for p = 2 and leave the general case to the reader. So consider cp on Uj
(i
= t L ({) jl1P dxj "
dx~ =
t
L ({)U,. dX; "
dx:
Uk' We want to see that
acpj,.p ZL..a Y
1 "
Xj
dx~J " dx~J A dx~J =
acpu,. 2L..a.
.1 "
Xk
dx·k " dx kA " dx"k'
(2)
5.
69
VECTOR BUNDLES
The following rules of transformation are given to us. ox~ ox~
CPj«P =
L o" 0 and R  e > O. Proof
(of Theorem 6.1) We may as well assume U = UR
= {zllzl < R, where lal = maxlzlll}.
Then we want to prove that if oqJ = 0,
1
qJ
= q! L qJIII"'II, dzlll
1\ "'1\
is a Coo(O, q)form on UR , then for any Il > 0, R form 1/1 on UR  t such that
01/1
= qJ on URt
dill,
8
> 0, there is a Coo(q  1)
(q ~ 1).
Suppose
By this we mean that form qJ does not involve differentials of coordinates for i > m. The proof will be by induction on m with fixed q,n. First we consider m = q. Then Zi
and
=
~ L"
o
!III""I"'q(z) dill {fI
l12:q+ 1 uZ
Hence, (O/Oi'")qJl '"
q
= 0 for ex ~ q + 1. Define
1 g(Zl, ... , z") = 7r
II
1\
di 1
1\ '"
1\
di q •
78
SHEA VES AND COHOMOLOGY
Theng is Coo on UR 0 .. ,A=lOZk
11 = 1
if C=/::. O. Thus ds 2 is positive definite and the rest is easy to check. Suppose given an Hermitian metric ds 2 = ). Jl.E{12 ... n I ... jj} by , "'"
( )_(0 gjA/J 
gji/l
L 9 M dzj dz~.
Q.E.D.
Define 9 jAjJ for
9M) 0 .
Then 2'L9M dzj dZ~='LA'/Je(l,2, ... ,jjI9jA"dz1dz~, where, as has been our custom, z~ = zj. We associate to ds 2 a differential form of type (1, 1),w = iL 9j..11 dzj 1\ dz~, where i = ~. REMARK. prefers
Or if one w(C, ,,) = ligM«(j,,~  "j(~) = ligj/lii«(~"~  ,,~C~)
= w«(, ,,). So w is a real form. DEFINITION 1.2. 2 'L 9..11 dz" dz ll is called a Kahler metric if dw = O. Then we call w a Kahler form. M is called a Kahler manifold if we can define a Kihler metric on M. REMARKS (1)
zero.
An Hermitian metric can always be defined, but generally dw is not
1.
HERMITIAN METRICS; KAHLER STRUCTURES
85
(2) Compact Kahler manifolds have many properties similar to (projective) algebraic manifolds. (3) Every algebraic manifold is a Kahler manifold. (Recall that algebraic manifolds are by definition compact submanifolds of IPN for some N.) . (4) There are many nonalgebraic Kahler manifolds. CONJECTURE. A compact complex manifold of complex dimension 2 is Kahler if and only if the first Betti number is ~ven. We know that M Kahler implies that the first Betti number is even (Theorem 5.4, Corollary 2). We have the following facts: (1) If hI (M2) is even, then M2 is a deformation of an algebraic manifold where by M2 we mean a compact complex manifold of complex dimension 2. Thus there is a complex family {Mtlltl < 2} such that M~ is algebraic and Mf = M2 [Kodaira (1964)]. (2) Any small deformation of a compact Kahler manifold Mii is Kahler; that is, if M(j is Kahler, then M7 is Kahler for small enough t. (3) It was conjectured that any deformation of a (compact) Kahler manifold is Kahler. This turned out to be false [H. Hironaka (1962)] for dimension ~ 3. So we make the conjecture for M2. Now we collect some facts about Kahler manifolds. Let U = {zllzl < I}
c
en.
PROPOSITION 1.1. A form qJ = L qJ,,/J dz" only if there is a Coo function! such that
A
dz/J on U satisfies dqJ = 0 if and
02j ( OJ) = "'.LfJ  dz" A dzfJ . oz." ozfJ '
qJ = ooj= 0 Ldz/J ozfJ that is, if and only if qJ,,/J
= (fP!lozQ ozfJ).
Proof (I) If qJ = aGf, then d(oo!) = (0 + o)(oof) =ooo! = ooo! =0. (2) Suppose dqJ = oqJ + oqJ = O. Since oqJ is of type (2, 1) and oqJ is of type (1, 2), oqJ = 0 and oqJ = O. Dolbeault's lemma implies that there is ",(1.0) such that 0",(1.0) = qJ on U, tIP· 0) = dz". We have
L "'"
0= oqJ = 00",(1·0) = _00", 0 on Uj n Uk' Any local coordinate (Xv, ev) is positively oriented if det(oe~/i.lx~) > 0 for allj such that Vj n Xv i: . Let {Pj} be a partition of unity subordinate to {U j}.
J ep
f Pj(x)epj, ...m(x) dx) ... dxj. Using Equation (4) and the orientability it is easy to check that f cp is well defined indepenDEFINITION
1.3.
M
= Lj
Uj
M
1.
89
HERMITIAN METRICS; KAHLER STRUCTURES
dent of the choice of covering by local (positively oriented) coordinate patches and partition of unity subordinate to it. We leave this to the reader.
PROPOSITION
If cp = dt/l, then
1.3.
f
cp = O.
M
Proof
t/I =
Ij Pj(x) t/I /x), so
where
Uj = {xjllxjl < rj} and Pj(x) = 0 for x ¢ Wj We will show that each
f
£;
Uj .
d(pjt/lj) = O. To simplify notation let us drop the
Uj
subscripts. Then m
I
Pj t/I/x) =
11=
hix) dx 1
/\ •.•
dx~ t /\ dx~+ 1
/\ ••• /\
dxm
1
and hix) = 0 for x ¢ Wj . We calculate
fu. d(p t/I): j
J
f
f =f
d(pjt/l) =
Uj
Uj
d(Ih ll dx 1 a.
/\ ••• /\
dXa.l /\ dXa.+l /\ ... /\ dx m )
I (l)~+ 8h~ dx ' ... dx m 1
Uj
8x~
a.
= I(l)'z+1 "
f
dX'''' dX~l dX~+I ... dxm
Ix'i O} e Vj if Vj n M = cpo supp Pj ~ {xjlrj < xj ~ r j , Ixjl < r i , oc ~2}. (3) Pj is Coo, Pj ~ 0 and L Pi = I.
(I) (2)
Then Iwdl/l = LjIwd(Pil/l) = LjIujd(pjl/l). From Proposition 1.3 Iujd(Pk 1/1) = 0 if V k n M = cpo Suppose Vj n M =F cp and 1/1 = L::: I/Ij dxj, /\ ... dxjl /\ dX~+l /\ ... /\ dxj+1 on V j . Then
and
Foroc=I,
Hence,
f
Uj
d(p J'I' .. 1,)
=
f Uj
p.J'I'J ·"~(r.J' x~J' ... • x'!+ 1) dx~J ... dx'!+ I J J
Thus,
where M=oW. For the proof of the theorem, suppose M n is compact Kahler and N m is a complex submanifold N m eM". Suppose N =oW eM for Wan embedded submanifold. Then if w is the Kahler form on M, 0 < IN w m = Iw dw m = O.
92
GEOMETRY OF COMPLEX MANIFOLDS
This contradiction proves the theorem in this case. Generally if N is homologous to zero, we do not have such a convenient situation. One must change the proof, and we supply no details here.
2.
Norms and Dual Forms
Let OP be the sheaf over M (a compact complex manifold) of hoI om orphic pforms. Let AM be the sheaf of Coo (p, q)forms on M. We want to introduce an Hermitian scalar product (cp, 1/1) for cp, 1/1 E r(AM), which makes r(AM) into an (incomplete) inner product space. We introduce an Hermitian metric 2 L gj,.p dzj dz~ = 2 L gap dz" dz P on M. Associated to this metric we have the form w = i L gap dz" 1\ dz P and wn = 2n n! 9 dx 1 1\ ••• 1\ dx 2" as before where X 2a  1 + i x2a = z,. and 9 = det(gaP)' We denote the inverse (g,.p)l of (g,.p) by (gi P) = (g,.p)l, that is,
L giPgp'Y = ;
and L gapgPY = ~.
P
P
The length 1'1 of a tangent vector C is given by inner product (c,,,) = La,pg,.pC""p. Let cp(z)
= _1_ L cp p! q!
p dz'"
1(21 = L,.,p gap cafi and
A ••• 1\
dz Pq
1\ ... A
dz Pq
the
q
al'"
and 1/I(z) = _1_ L 1/1 p dZ"1 p! q! al'" q Then at each point
ZE
M we define
= _1_ " glIal (cp , .I.)(Z) 'I' I I i..J p.q. a.II .... JJ
DEFINITION
2.1.
••• gPIJJI •••
gPqJJqcpal "',.q ~ '1'''1 ~ ... JJq'
(1)
The inner product of two forms cp, 1/1 is
(cp, 1/1) =
f (cp, 1/I)(z) w:n. f (cp, 1/I)(z)2"g dx =
M
M
(,) satisfies the following properties: (1) (cp, 1/1) = (1/1, cp), (2) (a1/l + b X, cp) = a(1/I, cp) + b(X, cp), (3) (cp, cp) ~ 0, (4) (cp, cp) = 0 if and only if cp = O. We define
•
IIcpli =J(CP, cp) as usual.
1 •••
dx2n.
(2)
2.
93
NORMS AND DUAL FORMS
THEOREM 2.1. There is a linear map *: r(AM) + r(A n  p,n q) such that: (I)
ron (cp, 1jJ)(z)  = cp(z) " *i/i(z), n!
(2)
*1jJ = *i/i (that is, * is a real operator), **IjJ(p,q) = (I)P+q ljJ(p,q).
(3)
Proof Before giving the proof let us fix some notation. Let n = dim M. We denote as follows:
and (0(1' .•. , a p ' O(p+1' ••• , O(n) is a permutation of (I, ... , n). For example, = 5 and A2 = (2, 4), A n  2 = (I, 3, 5). Similarly for Bq = (Ph···' Pq), Bn q = (P q +1> ••• , Pn). Then with this notation we write a (p, q)form
n
(3)
where dzAp =dza,
" ... " dZ«p, and so on. We denote .1,ApS, 'I'
= "giilAI ••• gil.P •• I, L'I'AI",A
__ p !ll"'!l.'
A,!l
Then
L L (IYQIjJ Apli, dz B , " dz/l p = L L (i/ih,::t, dz B, " dz::t,.
=
Thus, (i/i)S,A p = ( 1)PQIjJ ApBq·
We can now write Equation (1) as
(cp, 1jJ)(z) =
L
Ap,B,
CPA
II
P'
IjJApB, = (lyQ
L
Ap,B,
CPA B i/iS,A P • p,
Remember
Then we define *,1, = (i)n( _1)+ = 0 if and only if oq> = 8q>
= O.
(Oq>, q» = (oq>, oq» + (9q>, 9q» = lIoq>1I 2 + 119q>f.
(t!, oq»
103
Q.E.D.
= (91'/, q» = 0,
= (0". 1/1) = 0, (oq>, 91/1) = (ooq>, 1/1) = o. (", 91/1)
Thus, yt'q, o.!l" 1, and 9.!l'q+ 1 are orthogonal and .!l'q = Jf(q Ef) D.!l'q = Jf(q Ef) o8.!l'q EEl 80.!l'q £ Jf(q £ .!l'q.
ffi o.!l'q ffi :}.!l'q Q.E.D.
We next have the important theorem relating cohomology groups and harmonic forms. THEOREM 4.1. [Kodaira (1953)] Let F be a holomorphic vector bundle on a compact complex manifold. If QP(F) is the sheaf of germs of holomorphic pforms with values in F, then
Proof
Recall HP(M, QP(F)
~
f(oA (P.,I)(F))/of(A(P.,I)(F)) and
f(oA(P.ql(F)) = {q> I q>
Let Z;; (.!l'q) = {q> I q>
E
E
f(A(P·q)(F)), oq> = O}.
.!l'q, oq> = OJ. Then
We claim (1)
The inclusion
104
GEOMETRY OF COMPLEX MANIFOLDS
is obvious. Let cP E ZlJ(!£q), cP
= '1 + 01/1 + 90'. Then ocp = 0 so 090'
=0
(090', 0') = 0 (90', 0') = 0 and 90' = O. Thus Equation (1) is true and implies the theorem. COROLLARY.
dim Hq(M, OP(F»
O. Q.E.D. From Theorem 1.4 we already know h2p
REMARK.
COROLLARY 2.
~
I.
On a compact Kahler manifold b2 k+ 1 == 0 (mod 2)
Q.E.D.
Proof
PROPOSITION 5.5.
On a compact Kahler manifold, every holomorphic = O.
pform
0 satisfying aj \.fj1e\2 = ale such that
i
y= 21l a0 log a
Proof. and
a l.fjkl j
2
j'
Choose any metric il = {aj} on F. That is, il j = k • Then define
a
1 . L X Ail dz A 1\ dz il , ~ =2111 where that is,
i ~ = 2n:
a0 log a
j .
E
Coo(U j ), ti j > 0
7. Then as in Theorem 7.2 dcp is a (1, I)form and
VANISHING THEOREMS
[~]
129
= c(F)c so ~ 
y = dcp, where cp is a Iform. Thus,
dcp = ~  y = 11
+ Otjl,
where 11 and tjI are (I, I)forms and 011 = O. But then
A11 = 2 011 = 0
d11
so
= t511 = o.
Also ddcp = 0 so dcp = 11
+ !(dt5 + t5d) tjI
implies 0= dt5dtjl
(1)
and hence (t5dtjl, t5dtjl) = (dtjl, dt5dtjl) = O.
Thus, so Then (11, dcp) = (t511, cp) = 0
= (11, 11) + ! (11, dt5tj1) = (11, t7). Hence t7 = o. Using Equation (J) 0= (t5dtjl, tjI) = (dtjl, dtjl)
so dtjl=O.
(2)
From (2)
o=
dtjl = atjl
+ atjl.
Thus, atjl = atjl = 0 since tjI is of type (I, 1). So dcp =
e y = a8tj1 = i(aAatjl a aAtjI) = i a oAtjI.
Thus, i 00/, 211:
~ y=
130
GEOMETRY OF COMPLEX MANIFOLDS
where I is a Coo function on M. But
e y = e  y = 
; ~ 2n u oj
i
= 2n 0 oj
e
e
since y is a real form. Hence, y = (ij2n)0 vU)(1 + j) and thus we may assume that y = (ij2n)o 0I where I is real valued. Finally
e
Y=
i_ e 2n ; 0 of_ = 2n  0 o(log a.  f). J
i
(1) Y =  0 0 log a . 2n J
Q.E.D.
REMARK.
Perhaps we should explain this proof a little more clearly. We
claim: PROPOSITION
7.1.
If 1
1/1 = 2. I l/Ia11 dz" nl
1\
dz 11
and jf [1/1] = 0 (that is, 1/1 = dcp), then there is a Coo function 1/1 = 0 01 when M is a Kahler manifold.
!')
I such that
Proof Let 'P={I/III/I=dcp, 1/1 of type (I,I)}. Then 00!,)~'P, where is the space of differentiable functions, and 'P = YEt> 0 0 f!}, where
Y
= {'71 '7 e 'P, ('7, 0(/) = 0,
for all/e f!}}.
We note that ('7, 0 0f) = 0 if and only if 88 '7 = O. We claim that if M is Kahler, then Y = {O}. For Kahler implies t~ = 0 = 0 and 08 + 80 = 0 = 08 + 90. Thus t~2'7 = 00'7 = (v8 + 80)(09 + 80)'7 for '7 e Y. Since '7 is of type (I, I) and '7 = dcp, 0 = d'7 = 0'7 = 0'7. Thus, t~2'7 = (0809
+ 8009)'7
=  0089 '7 + 8080 '7 =0. Thus t1 2 '7 = 0 and (~2'7, '7) = (t1'7, ~'7) = O. So t1'7 = 0 and hence 15'7 Finally. ('7, '7) = (dcp, '7) = (cp, 15'7) = 0 and '7 = O. Q.E.D.
=0
7.
131
VANISHING THEOREMS
DEFINITION 7.1. A complex line bundle F over any compact complex manifold is said to be positive if there is a y = (l/2ni) X).il dz). /\ dz il , dy = 0, y = y, and [y] = c(F)c such that X).il(z) is positive definite at every point z of M.
L
L
REMARK. If F over M is positive, then w = i X).jj dz). /\ dz jj is a Kahler form. Hence M is a Kahler manifold. Rewording Theorem 7.1 gives: THEOREM 7.S.
If F  K is positive, then Hq(M, l!J(F» = 0 for q ~ I.
THEOREM 7.6.
If F is positive, then Hq(M, l!J(F» = 0 for q :::;; n  1.
Proof Serre duality gives Hq(M, OP(F» ~ Hnq(M, onpc F» where dim M = n. Notice that on ~ l!J(K) and let p = O. Then
Hq(M, OO(F»
~
Hnq(M, on( F»
~
Hnq(M, l!J(K  F»
= 0 fO[ n  q ~ if K  F  K
=
F is positive.
1
Q.E.D.
We also have: THEOREM 7.7. If F is "sufficiently" positive, then Hq(M, OP(F» = 0 (where F is a line bundle) for q ~ I.
Proof
Again we use
Hq(M, OP(F»
~
.yt>P·q(F)
= {cp I DaCP
For cP
I
0>
M
E
= 0, cP of type (p, q)}.
ytJ(P, q)(F) we let the reader check the following inequality:
,q { L L (X ij n. t.1' «,.P"
Wn
t
_
t
_
Rij )CPl1.i ... l1. p tP2 ••. /I. cP
ii, ...
iiptllh·· P.
Thus if X'ij is sufficiently positive definite, then the integrand is positive for cP =1= 0 we see ytJp.q = O. Q.E.D. We now proceed to a generalization of Theorem 7.6 due to Nakano (1955). As usual M is a compact Kahler manifold and F = {.fjk} is a complex
GEOMETRY OF COMPLEX MANIFOLDS
132
line bundle with metric {aj}. Remember that (8"cp)j = {(l/a)8(ajcpj)}, and so forth.
(00" + o"o)cp = X /\ cP, where X = 00 log aj and
LEMMA 7.2.
r
cP
E
(AP·q(F)).
Proof
We have (o"cp)j
=
{~j o(aj' CP)}
= {ocpj
+ alog aj /\ CPj}.
Thus,
O(o"cp)j={Oocpj+oologaj /\ cpologaj /\ oCPj}. Add o"oCPj
= {oocpj + alogaj /\ oCPj}
to get (00"
+ o"o)cp = X
/\ cpo Q.E.D.
THEOREM 7.8. [Nakano (1955); Calabi and Vesentini (1960)] r (AP·q(F» be such that ocp = 8" cP = O. Then
Let cP E
0::::;; )=l(X /\ Acp  A(X /\ cp), cp). Proof since
0::::;;
FI (o"cp, o"cp) = (90"cp, cp)
(01/1,1/1) = (cp, 8" 1/1) and (9cp, 1/1) = (cp, Ao. Hence
FI9 = oA 
0::::;; (90"cp, cp) =
FI (oAo"cp 
a" 1/1).
By
Proposition
Aoo"cp, cp)
5.4
(3)
=)=1 (Ao"cp, 8"cp)  )=l(A(oo" + o/)cp, cp) = J=l (A (X /\ cp), cp). But we also have
0::::;; (9cp, 9cp)
since ocp = 8"cp
= O.
= (0,,9cp, cp) = )~I (olJAcp  o"Aocp, cp) = )=1 (o"oAcp, cp) = )=1 (a" 0Arp + oo"Acp, cp)
Thus,
)=1 (X /\
Acp, cp) ~
O.
Now as Equations (3) and (4) to get the theorem. THEOREM 7.9.
[Nakano (1955)]
Q.E.D.
If F is negative, then
Hq(M, QP(F») = 0 when n = dim M.
(4)
for p
+ q ::::;; n 
I
7. Proof
133
VANISHING THEOREMS
By the harmonic theory
Hq(M, QP(F»
~ {q> I q> E
P,q(F), aq> = 911. q> = O} = Jf('p.q(F),
where P,q(F) = r(Ap.q(F». By Theorem 7.8 if q>
o :s; ~l(X
E
Jlfp,q(F)
" Aq>  A(X " q», q».
We let the reader verify the following equations: (X " III) 'F 11.0 ... «q/lo ... /lq = (l)P '" L... X «0/10 III 'F«,
•..
«p/l, ... /lq
P
+
L(lix«./loq>«o ... a, ... /I, ...
,=1 q
+ k=l L(ltX«./lkq>«\···«p/l.··.Pk ..
/lq
P
~
~
L
L ( _l)kgjiAX A/lk q>«, ... ji/l • ... Pk"
i;iA + /.t,L...A i=L...(l)g X«,jiq>A« •... ri.···/I\/ll··· 1 q
+
/l,A k= 1
p,q
+
'L... " 'L... " ( /l,). i=l,k=l
1) iHX Cli/l. 9 jiA q> A··· ji ....
Thus,
Since F is negative,  X Aji is positive definite at each point. (We should use Theorem 7.4 here; that is, we choose aj so that  X Aji is positive definite.) Now
GEOMETRY OF COMPLEX MANIFOLDS
134 and
00
satisfies
doo = deL 
xA" dz A 1\ dz") = o.
Thus we may use  X A" as a Kahler metric on M. Hence, assume g",P Then
=  X",p .
and
Finally
O$f
M
0$ So 0 $ (n
1 ,,,{(n)L+PL+qL}, n. p.q.
00"
1 '" (n+p+q)L.,.CPA fMn!p!q! 00"
+ p + q)(cp, cp).
But p
+ q < n and
B
cP TB p q.
P4
we see that cP must be
o. Q.E.D.
8.
Hodge Manifolds
Recall that by de Rham's theorem a Kahler form on a manifold M determines an element of H2(M, C). We also have the image of the canonical map
H2(M, Z) + H2(M, C) which we denote c + Cc •
L
DEFINITION 8.1. gall dz a dz P is a Hodge metric on M if [00] = Cc for some c E H2(M, Z) where 00 = i gaP dz a 1\ dz ll . If M has a Hodge metric, then we say that M is a Hodge manifold.
L
THEOREM 8.1. M is a Hodge manifold if and only if there exists a positive line bundle FE HI(M, l!!*).
Proof Suppose FE Hl(M, l!!*) is positive. Then, by Theorem 7.4, c(F) E H2(M, Z) is cohomologous to (\/21[;) X where X = 00 log aj = X A" dz A 1\ dz ii and (X Aii ) is positive definite. Thus gAji = (1/21[)X A" defines a Hodge metric on M.
L
L
Next we assume M has a Hodge metric, that is, W = i gall dz a 1\ dz P with W cohomologous to Cc for C E H2(M, Z), and (gall) positive definite. It
8.
135
HODGE MANIFOLDS
suffices to show that there is a line bundle F such that c(F) = c; because then 1 c( F) ""'  . 21t1
L X«fJ dz« "
dz lJ ,
with X «fJ = 21t g«fJ' and hence  F is positive. Recall the exact sequence ... _
HI(M, (!)*)..:..... H2(M, Z)~H2(M, ( ! ) _ ... Fc(F).
Thus, it suffices to show JlC = 0. Let c be defined by the 2cocyc\e c = {c ijk}' The proof involves chasing through the de Rham and Dolbeault isomorphisms. Consider the following diagram: /H2(M, C) ~ J'(dAl)/dr(AI) CEH2(M,Z)~
cct/I
(I)
H2(M, (!) ~ r(oA O• 1 )/or(A o•1 )
/J
JlC _
fP(0.2).
As in the argument of Theorem 7.2, we can find differentiable functions Aij such that Cijk = t5(A};jk = Ajk + Aki + Aij' Then we can find differentiable Iforms t/lj such that dAjk = I/Ik  t/lj. Then t/I in Diagram (1) is obtained by 1/1 = dt/lk = dl/l j . For the Dolbeault isomorphism, OAjk = fPk  fPj' where the fPj are (0, I)forms. Then fP = OfPk' We can split upl/lj = I/IP' O) + I/IP' O) into forms of type (\,0) and of type (0, 1). We know that d = a+ 0 so we compute ;)1
_ .1.(1.0) 'l'k 
.1.(1.0) 'I' j ,
31
_ .1.(0.1)
.1.(0.1)
UlI.jk 
UlI.jk'I'k
'I'j
•
Thus we may assume that fPk = I/Ik(O.1). Then fP = ol/l}O.I) = 1/1(0.2) [the (0,2) part of t/I]. Thus, if Cc ++ 1/1, JlC ++ 1/1(0. 2). Now we have assumed Cc '" w which is of type (1,1). Thus 1/1 = W(I.I) + d'1, with '1 = '1 0 •0 ) + '1(0.1). Thus 1/1(0.2) = 0'1(0.1) which means JlC = O. Q.E.D. With the obvious definition of elements of type (\, I) in H2(M, Z) we have: COROLLARY. Let M be a compact complex manifold. Then the il1'age of the map Hl(M, (!)*)~H2(M, Z) is the set of elements of type (I, I).
We now give the proof of the main theorem of this chapter which can be considered as a generalization of the fact that every compact Riemann surface is algebraic.
136
GEOMETRY OF COMPLEX MANIFOLDS
THEOREM 8.2. [Kodaira (1954)] Every Hodge manifold is algebraic (that is it is a submanifold of some pH). We first outline the idea. We know there is a posItIve line bundle
E E H 1(M, £D*). Let F = mE where m is a large positive integer. Let dim HO(M, £D(F» = N
+1
choose a basis {Po, ... ,PN} for HO(M, £D(F», and let F be defined by the Icocycle {jjk} with respect to some covering {U j } of M (remembering that the jjk are never zero). By definition
fJv = {fJvlz)}, fJVj(z) = jjk(Z) . PVk(Z), where the Pv/z) are holomorphic on U j : M + pH given by
•
Consider the candidate for a map for
ZE
Uj
•
It is easy to see this is well defined as a point of pH if for every Z E M there is an index v such that Pv(z) oF O. We want to be an embedding. To prove this it suffices to prove: (1) Given Z E M, at least one fJv(z) oF 0, that is, there is a cp E HO(M, £D(F», cP = CvPv such that lp(z) oF O. [Then (I) implies that is well defined and holomorphic on M.J (2) is injective, that is, for any pair of points p, q E M, there is lp E HO(M, £D(F» such that lp(p) oF 0, cp(q) = o. [In fact, this also implies (I).J (3) is biholomorphic, that is, for each point P there exist n (= dim M) elements lpl' ... , lpn E HO(M, £D(F» such that
L
det where P e U j
(alp~~~Z)) oF 0
•
We first prove (2). Let 1/ = £D(F  P  q) be the subsheaf of (9(F) consisting of germs of holomorphic sections of F which are zero at p and q. Let us investigate the stalks of 1/. Clearly,
1/% = £D(F)% ,
if Z oF p, z oF q I/p={lpE£D(F)p!lpj(p)=O, ;fpeU j }
and similarly for q. We have the exact sequence
o. 1/ . £D( F) . 1/" . 0,
(2)
where 1/" = £D(F)/I/ is the quotient sheaf. Then 1/; = 0 except at p or q. Clearly 1/; ~ C, ~ C and the isomorphism depends on the choice of local
9';
8.
137
HODGE MANIFOLDS
coordinates around p and q. This shows that HO(M, f/") cohomology sequence of (2) is
= C Ei3 C. The exact
O+HO(M, f/)+HO(M, (!J(F»+C Ef> C+Hl(M,!J')+ ... (fJ«(fJj(P), (fJk(q»·
We sometimes use the suggestive notation f/ = (!)(F  P  q). To prove (2) it is sufficient to prove: PROPOSITION 8.1. If M is compact and Kahler and Fe Hl(M, (!J*) is "sufficiently positive," then
Hl(M, (!)(F  P  q» =
o.
Proof The proof makes use of the quadric transformations Qp, Qq. Let Nt = Qp Qq(M) and let P be the holomorphic map P : Nt + M of Nt onto M such that C = pl(p) and D = pl(q) are isomorphic to pnl with dim M = n, and P is a biholomorphic map on Nt  c  D, P : Nt  c  D+ M  P  q. Let f/ = (!J(F  P  q), IJ = (!J(p  C  D), where P is the holomorphic line bundle on Nt induced by P and IJ is the sheaf of germs of holomorphic sections of P which vanish on C and D. Let dlt = {U j } be a covering of M. Then dii = {OJ}, OJ = P1(U J) is a covering of Nt. We recall that Hq(M, !J')
= lim Hq(dlt, f/) u
and for q = 1, the map Hl(dlt, f/) + Hl(M, f/) is injective. We prove: LEMMA
8.1.
If
then Hl(M, (!J(F  P  q» = O. Proof It suffices to show Hl(dlt, f/) = 0 for all coverings dlt. Take a lcocycle (fJ = {(fJij} e Hl(dlt, f/), (fJij e rcU i n U j , f/), where (fJij is a holomorphic section of F over U i n Uj such that (fJiip) = 0, (fJij(q) = 0 if p, q e U i n U j • P induces cPij = P*(fJij = (fJij 0 P e rcOi n OJ, (!J(F»,
where cPij vanishes on C and D if C £ Ojn OJ, D £ Oi n OJ. Thus {cPij} represents an element of Hl(dii, IJ) £ Hl(M, fJ) = o. Hence cPij"" 0, that is, cPij = t/Jj t/Ji where each t/Ji e reUi> (!J(F» and vanishes on C and D. If Ui £ M  P  q, then P : OJ + U i is biholomorphic. In this case there is (fJjer(Uj,(!J(F» such that t/Jj=P*«(fJj). If, for instance, peU j, then
138
GEOMETRY OF COMPLEX MANIFOLDS
P: OJCVjp is biholomorphic. Hence there is CPjEr(Vjp,~(F» such that P*(CPi) = '" j on aC. We can always assume F is trivial over Vi' so nVi> ~(F»;;; nV j , ~). Thus we consider cPj as a holomorphic function on Vj  p. By Hartog's theorem cPj can be extended to all of U j ' Then P*cpj is defined on all of OJ and must equal (by continuity, or the identity theorem). Thus C{Jj(p) = O. Hence we have found cPj E r(V j , ~(F  p» such that'" j = P*C{Jj. We have proved that there is a ocochain {cp;}, cPj E i' //) such that'" j = P*CPj, and thus P*CPij = P*cpj  P*cp j ' But P is surjective, so C{Jij = CPi  CPj· Thus {CPij}"" 0, and Hl(O/t, //) = O. Q.E.D.
"'j
nU
REMARK.
Relations between Hq(M, //) and Hq(M, IJ) are not easy to see.
To prove Proposition 8.1 it now suffices to prove HI(M, ~(p  C  D) = O. Let [C] and [D] be the corresponding bundles of the divisors C and D. Then we must show that Hl(M, ~(p  [C]  [D]» = O. To prove this it suffices to show that F  [C]  [D]  K(A:!) is positive, and then quote the vanishing theorem. We want to show that F  [C]  [D]  K(M) > 0
if m is sufficiently large, where F = mE, and K(M) is the canonical bundle of A:!. Therefore we would like to compute c([C]) and c([D]). First we find a lcocycle on A:! representing [C]. Let z be a coordinate chart map centered atp E M, and let V = {zllzl < 2E}. Let P: A:! ~ M. Let us describe the normal bundle W of C in A:!. Let VA
where Ul' "',
Un
= {u E pn11 U = (Ul' "', Un),
UA
#: O},
are homogeneous coordinates for pnl. Then
and n
W=
U(VA X
C),
1=1
where we identify (u,
WA)
and (v, wI') if and only if U
= v,
(3)
We could define P: W  U by
(4)
8.
139
HODGE MANIfOLDS
Then pnl = U). (V). x {O}) s;;; Wand we can identify C = Qp(p) with pnl in W. Thus we consider a small neighborhood pl(U) = of pnl in Was a small neighborhood of C in M. On each VA x C, C is defined by IV). = O. Let OJ. = n (VJ. x C) for it = I, ''', n, and let 0 0 s M  C be such that
a
a
QpQq(M)
We set
Wo
=I
on
00 ,
=M = 0 0 u 0 1 u .. · u
Then the line bundle [C] is given by the Icocycle on
Then (5) implies
gJ.O
an.
OJ. nOv.
(5)
= w)..
Recall that, in general, if F is defined by {Fjd and if aj Ifj kl 2 = ak for positive COO functions {at}, then i _ c(F) ,..",  iJ log a .. 2lt J
a
(6)
We want to find such COO functions for C. We make use of a Coo function ex on M with the properties (1) (2)
ex(z) = Izl2 for z E U, Izl < e ex(z) = 1 for z E M  V.
We define Ao(w) = cx(P(w», w E cx(z) A;.(w) =
Notice, on
Iw).1 2 '
WE
00
_ VJ..
en A). local coordinates are (it :F 0)
( ~ , ••• , U).l , U).+ 1 " •• , Un) . U). U). U). U). Thus,
so the definition has meaning, and
A;.>OonU).
A. = 0, "', n.
140
GEOMETRY OF COMPLEX MANIFOLDS
The A;. satisfies so
i a0 log A;.. 2n
(7)
c([C]) '" 
We notice that
a0 log A;. = a0 log(1 + ...L.1. IU'U.I.1 2 ) and this is just the standard Kahler metric on pnl = C. We also remark that
a0 (X(z) is a C 2form on M of type (1, 1). Since a0 = d(o(X) (X
a 0 (X is cohomologous to zero on M, and the induced form
n = p*(ao(X) is
cohomologous to zero on M. We then define
Then 0" c '"
(8)
c(  [ C])
by (7), and in a neighborhood of C,
2niO"c
=
Uv 2 aa10g(1 + v";' L IU;. I)
+ ao(lzI 2 ).
Recall that Z
W;.
= (z l' ". ' n z) = u = (. .. , wA, ... ) . U;.
Then
aO(L ZvZv) =
L dz v
dz v = dw;. " dw;. + .. '. "
Hence O"c is positive definite in a neighborhood of C. We get similar results for D. Next we want to find a relation between K(M) = K and K(M). We prove: PROPOSITION 8.2. K(M) = K + (n  1) [C] + (n  1)[D], where bundle over M induced from the canonical bundle K of M.
Proof
Suppose
M=
K is the
Qp(M). It is sufficient to prove
K(M) =
K + (n  1) [C].
(9)
8.
HODGE MANIFOLDS
141
We choose U 3 P as in the previous proof. Then we choose {U j , Zj} coordinate systems on M such that {U} v {U j} covers M. Let (Zl, ... , z") be a coordinate system on M. Then the canonical bundle K of M is defined by the lcocycle {J jk} where
dz) /\ ... /\ dzj =
J;;.' dzl/\ ... /\ dz~
dz l
J;'/ dzl/\ ... /\ dZ k
/\ ... /\
dz"
=
On M we may use {OA' OJ} as a coordinate covering where OJ = Pl(V) = Vj and 0 = pl(U) = vi= lOA' using the notation of the previous theorem. On A we have the local coordinate system
a
and .. Z
WAU ..
=,
if ex =I A.
UA
ZA
= wA,
if ex = A..
Computing, we get
since
dz" =
d(W~~") = w
A
d(::) + (::) dw A •
Let {Ijk' [A., [Ak} be the Iacobians on M [which are used to define K(M)]. Then [ I
jk
1 [ Ak
=
JI jk
1
WA
so
This proves (9) since
WA =
1
= ;;=i J ok
0 defines c.
142
GEOMETRY OF COMPLEX MANIFOLDS
We now return to the proof of Proposition 8.1. Recall that E is positive, that is,
L '}',.p dz" "
c(E) '" '}' =  i
dz ll ,
where (y,.lI) is positive definite. For simplicity we write y > 0 or E> O. We want to show
p
[C]  [D]  K(M) =
for large m. Let c(K) c(p 
K
K.
F K
n[C]  n[D] > 0
(to)
Then
n[C]  n[D])
my 
K + nUe
+ nUD,
where y = P*y and K = P*K. We choose m so large that my  K is positive definite on M. Then my  K is positive semidefinite on M and is positive definite on M  C  D. But Ue > 0 near C and UD > 0 near D. Then Equation (to) follows. This proves Proposition 8.1, and thus part (1) and (2). REMARK. It is an easy compactness argument to see that one can find an integer m such that HO(M, mE) separates points for all p, q E M, P "# q.
The proof of C is almost the same as the proof of B. We want to show that lI> is biholomorphic at each p EM. Consider f/' = (f)(F  2p) which is the sheaf of germs of holomorphic sections of F which vanish at p up to order 2. Again we compute the stalks f/'z and write down the exact sequence 0!/Q(F)!/"0,
z"# P
!/z = (f)(F)z, !/p
= {cp I cP = CPj' CPj(z) = k,
L
ak • ••• kn z;' ... zJ", cP
E
(f)(F)p}.
+"'+k"~2
Then
!/; = 0, =
Thus HO(M, f/'II) ~
ifz"# p
{cp I cP = Qo +
c+ 1.
t a,. z,.},
,.=1
if z = p.
We write down the exact cohomology sequence
It is easily seen that to prove lI> is biholomorphic at p we need only show Hl(M, !/) = O. To prove this we once again use M = Qp(M), C = Qp(p).
8. LEMMA 8.2.
143
HODGE MANIFOLDS
If HICM, (!)(F  2[C]» = 0,
then HI(M, (!)(F  2p» = O. Proof The proof is the same as that of Lemma 8.1. One only has to notice that if (() has a zero of order 2 at p, P*({) has a zero of order 2 (at least) on C and vice versa.
LEMMA 8.3. Proof
HI(M, (!)(F  2[C]» = 0 if m is large enough where F = mE.
Using Proposition 8.1. we find
F  2[C]  K(M) = m£ =
K
(n  l)[C]  2[C]
mE  K  (n + 1)[C].
Hence c(F  2[C]  K(M»,..,
if m is large enough.
my 
K
+ (n + l)uc >
0
Q.E.D.
REMARK. We again use compactness to see that there is an m which will work for all P E M. This completes the proof of Theorem 8.2. We now derive some consequences: THEOREM. 8.3. [Kodaira (1960)] If M is compact Kahler and H2(M, then M is projective algebraic. Proof
(!) =
0,
The exact cohomology sequence of 01L(!)(!)*0
yields ···.Hi(M, (!)*)~H2(M, 1L)0.
Thus everything in H 2 (M,1L) is the Chern class of some bundle. Let {bl' ... , bm } be a basis for the free part of H2(M, 1L) so that H2(M, C) = Cb l
+ ... + Cbm •
144
GEOMETRY OF COMPLEX MANIFOLDS
Each b). = c(F).) and hence is cohomologous to a real 2form of type (1, I). Let
w=
iL gll.lI dzll. "
dzll
be a Kahler form on M. We wish to modify w to get a Hodge metric on M. Since w E H 2(M, C)
WLP).b). where PA E R (w is real and the b). are real). Given e, we can always find integers k)., r E 1L such that A. = 1, ... , m.
But then for a small enough e I
W = W 
~  kJ.) Y). t... (PJ. r
defines a Kahler form on M where yJ.  bJ. is a real (I, I) form. Hence w= rw' is also a Kahler form. But
Thus
w defines a Hodge metric on M, and M
is algebraic.
Q.E.D.
Theorem 8.4. [Kodaira (1954)] Let M be a compact complex manifold. If the universal covering manifold M is complex analytically homeomorphic to a bounded domain fJI £; Cn , then M is algebraic. Proof We make use of the Bergmann metric on fJI [see Helgason (1962)]. We have M = fJI/G where G, the set of covering transformations of fJI, is a collection of biholomorphic maps from fJI to fJI. Let ds 2 = L gll.lI dzll. dztJ be the Bergmann metric on (fl. We claim
(I) ds 2 is invariant under G and hence induces a metric L gll.lI dza. dz tJ on M = fJI/G. (2) If w = (i/2n) L ga.lI dza. " dztJ, w'" c( K); so we have a Hodge metric on M.
This gives the theorem, thus we need only prove (I) and (2). Let .Yf be the Hilbert space of aU holomorphic functions f on fJI which have bounded norm
IIfII2 =
f If(zW dX, 91
8.
145
HODGE MANIFOLDS
where
dX = dX 1
•••
dX2ft and
= X2a:l
Za:
+ iX2a:.
Let {I.} be any orthonormal base of JIf. Then the Bergmann kernel K(z, i) is given by co
K(z, i) =
_
L 1.(z)/.(z)
[= K(z)].
."'1
The kernel K(z) is actually independent of the choice of orthonormal basis {Iv} [see Helgason (1962)]. Then L ha:p dza: diP is a positive definite Hermitian metric where = 02 log K(z) hliP() z :l:l. uZez uZp
Let 'I : f!J LEMMA
+ f!J
be a biholomorphic map, y(z) = z'.
8.4. K(z)
= Idet a(z~, ... , z~) 12 K(z'). O(ZI' ••. , Zft)
Proof co
_
L 1.(z)/.(z)
K(z) =
."'1
and
f I.(z')j~(z') dX' f 1.(z)/iz) dX = Ov).· =
~
~
Let
O(Z'»)
F.(z) = I.(z') det ( o(z)
and notice
Thus,
and {F} gives a new base. Hence, co
K(z) = v~tv(z)F.(z)
=
I
o(z') \2
det o(z)
I
£1(z') 12
= det o(z)
"'iJ.(z')Jv(z')
,
K(z).
Q.E.D.
146
GEOMETRY OF COMPLEX MANIFOLDS
Since G is a group of biholomorphic maps this proves (I). Now let K be the canonical bundle of M. Let n : B + BIG = M. Let U j be an open set in M on which a local inverse of n is defined, and choose one J,lj = n 1 to use as a coordinate chart for U j (J,lj(p) E C" if P E Uj). Suppose P E U j n Uk. Then there is "Ijk E G such that J,lip) = "Ijk(pip». The canonical bundle K on M is defined by the lcocyc1e
and we have K(Zj)
= l./jkl 2
K(Zk)
by the lemma. Recall that if we have positive Coo functions a j on U j such that
aj 1./j1c1 2 = ak , then i ~ "I c(  K) =  2n 0 u log a j .
Therefore, let aj
= K 1(zJ Then
c( K) =
~ oJ log K(zj) = ~ L glJll dzj " dz~. 2n
2n
This proves the theorem. There is much interest in nonalgebraic Kahler manifolds. Kahler manifolds give examples of the minimal surfaces of differential geometry.
REMARK.
[4] Applications of Elliptic Partial Differential Equations to Deformations I.
Infinitesimal Deformations
We want to study analytic families of compact, complex manifolds. Informally, we are only interested in small deformations. We may as well assume our base space B, = {til tl < r, tEem} is an open disk around the origin of em. We want a manifold"'" and a holomorphic map w: "'" + B, with maximal rank so that w is proper and each fibre Mr = wI(t) has the structure of a complex manifold which varies analytically with t. We want a covering {au j} of "'" so that
au j
e= j
t)1"jl < 1, It I < r} (e),···, ej), w(e j , t) = t,
= {(e j
,
and
Cj = fMe j , t) on au j n auk'
e
wherefjk is holomorphic in j and t. We notice that under these circumstances Mr is diffeomorphic to M 0' and in fact, "'" is diffeomorphic to X x B, , where X is the underlying differentiable manifold of Mo. Thus au j = U j x B, where U j = {e j IC j < I}, and
M =
UU
j
x B,.
j
If x is a point of X, t E B, we notice that
ej = ej(x, t) is a differentiable function of (x, t) and we have
ej(x, t) = fjk(ek(x, t), t).
(1)
Let M = M 0 = X and use the complex coordinates z of M as differentiable coordinates so that
ej(x, t) = ej(z, t), where ej(z, t) is a differentiable function of z and t. Because t = 0, ej(z,O) is holomorphic in z (otherwise it is only differentiable). 147
148 DEFINITION
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
1.1.
Let
(a) (ata) =Lc .=1· at. m
belong to the tangent space to B, at the origin. We define (aM,/at),=o to be the cohomology class in Hl(M, 0) given by the Icocycle
(J;k=I afMek,t) .. = 1 at
I (';) . ,=0
aei
We want to represent {(J;k} with a Iform by using Dolbeault's theorem. Let T be the holomorphic tangent bundle on M and 0 = @(T) be the sheaf of sections. Then we have the resolution D
D
0 _ 0 _ A o ( T ) _ A o . 1(T)_···, where AO·q(T) is the sheaf of Cal vector (0, q)forms. Locally, such a thing has the representation
where 1 rn P = .." q!
~ rn~
_
L... .." .. , .....q
dz'" /\ ... /\ dz ..q
and
Let us trace through the Dolbeault isomorphism 1 r(oAO(T» H (M, 0) ~ or(AO(T» .
Let (aM,/at),=o 11. Then 11 is defined as follows: Pick such that Then
PROPOSITION
1.1. 11 = 
I
«= 1
o(amz, t) at
I )(';). ael ,=0
ei
E
r(U;, AO(T»
1.
Proof
INFINITESIMAL DEFORMATIONS
149
Let
Then Equation (I) yields
Ci = L Of~k er + (a fik )
,
at
a~
1=0
Thus,
If we set
ek =  L cf(ajoef),
 L oCi(ojaei).
we get
Bil:
= el: 
ei'
Q.E.D.
Therefore,
,,= oel
=
We want to define a vector (0, I)form q>(t), t E B which describes the complex structure of M,. With respect to the local complex coordinate z on a neighborhood W of M we have the differential operators
a, (a~')' 0, (a:')' =
=
and
Recall that where fjt is a holomorphic function of el:' Thus
oej(z, t) = and
t ~~k oef(z,
t),
150
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
At t = 0, (Zl, .. " z") are local complex coordinates on M and «(J(z, 0), .. " (j(z, 0» are also local coordinates. Hence
So, for small enough
Itl,
Let
(
!l~«)I u~·
A
Aj«= azi
.
Consider the local Iform,
L A1« a(j(z, t) =
qJ~(z, t).
«
We claim the form qJ~(a, t) is well defined independent of (j. For fjk«(k' t) holomorphic in (k implies
which yields A( z, t ) = ~ A oYII( L. ()YII Aj« .. ,. Z, t ) «.11 ':ok
oq
qJj
=
L Atll oCf(z, t). II
We therefore define cpJ. = cp~(z, t). Then qJJ.(z, t) is a (0, I)form independent of (j' but it still depends on the local coordinate z. Let z = Zw. Suppose V is another open set in M with local coordinate z.,. Let qJ~ = qJJ.(zw, t) and qJ~ = qJl(Z." t). If we set
then cp(t) =
=
~ qJ~(zw, t)(a:~)
~ cpecz." t)(a:~)
l.
INFINITESIMAL DEFORMATIONS
151
is a welldefined, global vector (0, I )form on M. Now by the definition of A?2'
~r«( t) = "~ cp ). az o'jA.
u'>j Z,
Therefore (2)
1.2. The complex structure on M, is determined by cp(t). More specifically a differentiable function f defined on any open subset of M is holomorphic with respect to the complex structure of M, if and only if PROPOSITION
(0 Proof
L cp'\t)op)f(z) = O.
(3)
We represent cpp(t) by
I
cpP(t) =
cp(t)fdz~
tl
If f satisfies (2), then
( 0« 
t cp( t)~ a
P)
f
=
for all cx.
0
(4)
We use
ol :l_P =
v.
I
Y
of o~}
of o'}
o~rY a(I + Ll v'>j :lj'Y azP '>j"
and Equation (4) to get
Equation (2) implies that the first term is zero so
(5) Since 'j(z, 0) is holomorphic in z, cp(O) = small for small t. Thus,
o.
Continuity tells us that cpU) is
t
(Va  CP(t)~D/l)'~ is invertible for small t. Hence Equation (5) yields
a! = o.
o'j
152
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
Thus f is holomorphic in Cj(z, t). We can read the argument backwards, so the proof is concluded. Q.E.D. We want to introduce a bracket operation in the algebra of vector 0, q)forms. Let us recall the Lie bracket of vector fields. If u = ulJ OIJ , W = L ~OIJ' then [u, wJ = L(UIJO.. wp  wIJOIJUP)op .
L
... 11
For the generalization, let
DEFINITION
1.2.
"
[cp,I/I]= L (cpIJ A a.. I/I II (I)pQl/lIJ Ao.. cpll)oll IJ. fJ= 1 where a.. .1.fJ 'I' 
PROPOSITION
1
_
"
;:0
p.,~u..
cpfJ
_
AI''')'
dz.l. 1
p
A
"
•••
A
"
dz.l. p •
1.3.
I/IJ is bilinear [1/1, CPJ = ( l)pq[CP, I/IJ (J[cp, I/IJ = [acp, I/IJ + (I)P[cp, al/lJ (4) (I)PTcp, [1/1, rJ] + (I)QP[I/I, [r, cp]] + (_l}rq[r, [cp, I/I]J
(I) (2) (3)
[cp,
=
0,
where cp is a (0, p)form, 1/1 is (0, q) and, r is (0, r). (This is the Jacobi identity.)
Proof Uninteresting; we leave it to the reader. We collect our facts into the following theorem: THEOREM 1.1. If iii : vH + Dr is a complex analytic family of compact complex manifolds, then the complex structure on M, = iiilet) is represented by a vector (0, I)form cp(t) on M o such that
=
°
(1) (2)
ocp(t)  t[cp(t), cp(t)] cp(O) =
(3)
(OM,) __ '1 = _ (oCP(t»)
°
ot
,=0
ot
E 1=0
quAo(T».
1. Proof
INFINITESIMAL DEFORMATIONS
153
(2) has already been done. As for (1), by Equation (2)
(~cpP(t)iJp'j 
= o.
O,j)
So
o = o(~ cpP(t)Op,j) =
L ocpP(t)opej  L cpp(t) A oOp'j. P
P
Thus,
L OcpP(t)Op 'j = L cpp(t) A Op o'j P
P
~ cpfl(t) A OP(~ cp(t)YOy Cj) = L L cpfl(t) A OflCP(t)Y· Oyej
=
P y
+ L L cpp(t) A
cp(tYOpOy'j.
P y
The last term in this expression is zero, since cpP(t) A cpY(t) is skewsymmetric in p, y and op Oy Cj is symmetric in p, y. So we have (6) Now det(op ej(z, t» is nonzero if t is small, so afl ej is invertible for small t. Then (6) yields
ocpp(t) =
L cpY(t) A OycpP(t) Y
= ![cp(l), cp(t)JI1,
since
[cp, cp]p = L cplZ A OIZCPP  (_l)lcplZ A OlZcpP = 2
L cplZ A OlZcpl1.
Finally, for (3), we already know '1 = 
~ O~~(a~~)
where
,~ = (oe~) at
. 1=0
154
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
But
and the last term is zero, so
= _
.
(a~(t)) of
Q.E.D.
1:0
We wish to find conditions on a cohomology class p for it to represent an infinitesimal deformation. We first check to see that the bracket [,] extends to cohomology. Let
cp E qaAO,pl(T»,
t/J E qDAo.ql(T», so that
Then it is obvious
acp = at/J = O. that D[cp, t/J] = O. Also if t/J = D(1, then [cp, t/J] = [cp, C(1] = ±o[cp, (1].
Hence, Eo] induces a map HP(M, 0) Hq(M, 0)~ Hp+q(M, 0)
by using Dolbeault, and the facts just noticed, qDAo. pI(T» HP(M, 0) ~ ur(Ao. PI(T»'
Now let (tl' .. " fm) be coordinates on Br . Then any infinitesimal deformation is a linear combination of ones of the form
We claim: THEOREM 1.2.
If P E HI(M. 0) is an infinitesimal deformation, then
[p,p] =0.
2. Proof
AN EXISTENCE THEOREM FOR DEfORMATIONS I
155
By using the previous remark we need only check that ['1,., '1)] = 0
for all A, v. We differentiate the equation
ccp(t)  [cp(t), cp(t)]
=0
twice at t = 0 to get
= 2
acp aCP] [at;.' at•.
Let
Then we get
Q.E.D. We remark that this is not a sufficient condition, and the higher derivatives give more information. It is quite difficult to compute [p, p] for a general p E Hl(M, 0), but in most specific examples we get [p, p] = O.
2. An Existence Theorem for Deformations I. (No Obstructions) We aim to prove the following theorem: THEOREM 2.1. [Kodaira, Nirenberg, and Spencer (1958)]. Let M be a compact complex manifold. Assume that H2(M, 0) = o. Then there exists a complex analytic family vH ~ B., where
B. = {tlltl < e} ~
em, In =
dim Ht(M, 0),
such that: Mo = wt(O) = M. (2) The map To(Bt) + Ht(M, 0) given by (D/Dt) + (iJM,/iJt),=o is surjective (in fact, an isomorphism). More specifically if {Pl' ... , Pm} is a base
(I)
156
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
for Hl(M, 0), then we can define ..A so that
(OMI) ot. Proof
(1)
=
P
1=0
for v = 1, "', m.
•
We will accomplish the proof in the following two steps:
Construction of a vector (0, I)form
m(t) = "L.. 'f'k""km m tk' 'f' 1
••• t km In
such that
cp(o) = 0, ocp(t)  t[cp(t), cp(t)] = 0, and = P. E Hl(M, 0). ( O~(t)) ot. 1=0
(2) Show that cp(t) determines a complex analytic family by using the NewlanderNirenberg theorem. First we survey the NewlanderNirenberg theorem, which is sometimes called a "complex" Frobenius theorem. Let U !;;;;; en be an open domain, and
cp =
L cp~ dZ"(o~1I )
a vector (0, I)form on U. Let
La =
(o~~) IItl CP~(Z)(a~1I ).
We want to consider solutions to the equations Laf(z)
=
°
(1)
on the domain U. The theorem [of Newlander and Nirenberg (1957)] is: THEOREM.
If Li and Ii are (complex) linearly independent, and if
ocp  t[cp, cp] = 0, then Equation (I) has n COO solutionsft(z), .. ',J,,(z) such that det(O(fl' .. "
O(Zl> "',
In' I}, .. " In)) ~
°
Zn' Zl' " ' , Zn)
(that is,ft, ... ,J" define a differentiable coordinate system on U).
2.
AN EXISTENCE THEOREM FOR DEFORMATIONS I
La will be linearly independent.
REMARK
1.
If t is small, cp(t) is small and L i
REMARK
2.
Linear independence is needed; for if
,
157
iJ 0 0 L==+cz
az"
oxcz
OZ"
then La! = 0 implies! is independent of x". REMARK 3. If M is a complex manifold and qJ is given satisfying the conditions of the theorem, then by using (the proof of) Proposition 1.2 we see M has another structure as a complex manifold which is described by the form qJ. We say the almost complex structure qJ is integrable, and hence associated to a complex structure.
In order to construct our form qJ(t) we need to do some more potential theory. We want to define the Green's operator on
,f£q = r(Ao·q(T))
= the space of vector (0, q)forms.
To do this we introduce an Hermitian metric g«ll on M, and define an inner product,
where the * operator has been defined before. We have the adjoint B of 0, (BqJ, 1/1) = (qJ, 01/1); and the Laplacian 0 = Bo + oB. Then the space of harmonic forms
IHI q
= {qJ I qJ E ,f£q, OqJ = O} ~
Hq(M, 0),
defines a Hodge decomposition,
,f£q
= IHI q liB O,f£q::2 W liBo,f£q1 + B,f£q+ 1
into an orthogonal direct sum of subspaces. Thus, for qJ E ,f£q, qJ = " " E
IHI q,
1/1 E ,f£q.
+ 01/1,
Since 1/1 E ,f£q,
1/1 = , + 1/11' ,
E
IHI q,
1/11 E O,f£q,
and
Thus (2)
158
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
LEMMA
2.1.
Proof
The decomposition in Equation (2) is unique. Surely '1 is unique. If 1/1',1/1 are both orthogonal to IHI q and
ep = '1 + 01/1', ep = '1 + 01/1, then
0(1/1'  1/1) = 0 and
I/I'I/IEW. But
1/1'  1/1
.L
IHI q so 1/1' = 1/1.
Q.E.D.
DEFINITION 2.1. Given ep, the unique 1/11 making Equation (2) true is denoted Gep, and the mapping ep + Glp defines G : ,Pq + o,Pq. G is called the Green's operator, and is a linear map. We write '1 = Hep and call H the harmonic projection operator. Then
+ OGep.
ep = Hep PROPOSITION
2.1.
oH
=
Ho
= 0,
9H
=
H9
(3)
GH
= 0,
=
HG
= 0,
oG
=
Go,
9G = G9.
Proof
oHep = 0 since Hep E W
=
{I/I I 01/1 =
91/1 = OJ.
Hoep = 0 since oep E o,Pq .L W. The proof that 0 HGep = 0 since Gep .L W. For GH = 0 notice Hep = HHep
=
Hep
=
9H
=
H9 is analogous.
+ OGH IP
and uniqueness yields GHep = O. The proofs of the last two are similar to each other so we only prove the first of them. Recall 00 = 0(09
+ 90) = 090
and
Do = 090. Thus 00
= Do and oep = 00 Gep = OoGep = Hoep = o Goep.
2.
AN EXISTENCE THEOREM FOR DEFORMATIONS I
159
Since oGcp ..L Wand Hocp = 0, we use uniqueness of decomposition Equation (3) to see oGcp = Gocp. Q.E.D. To proceed further. we need to introduce the Holder norms in the spaces fi7 q • To do this we fix a finite covering {Vj} of M such that (Zj) are coordinates on U j • Let cp E fi7 q ,
~ CP~(Z)(a:~)
cp =
1
cp j).  q! "L.. cpAjii,..
ii.
dza, j
1\ .•. 1\
dz"' j .
Let k E 7L, k ~ O;!X E~, 0 < !X < 1. Let h = (hi' "', h 2n ), hi ~ 0, 2}~1 hi where n = dim M. Then denote
z} = Then the Holder norm
Ilcpllk+a. =
X;a.l
= Ihl
+ ixj.
Ilcpllk+a. is defined as follows:
m~x{ Lh (sup ID~cpfci, .. ii.(z)l) :eV) J
Ihl ~k
+ J1,,,,Vj sup
ID~ CP;a, ... iiq(Y)  DJ cpfa, ... !i.(Z)I)
Iyzl
Ihl =k
where the sup is over all A, !XI' ••• , of DougHs and Nirenberg (I 955).
!X q •
a . '
(4)
We have the following a priori estimate
(5) where k
~
2, C is a constant which is independent of cp and
IIcpllo =
max A,
sup ze Vj
j (lIt"
I
Icp;il, ".il (z)l. •
12q
REMARK. One can see that two norms defined as in Equation (4) for two different coverings {U j }, {Uj} induce equivalent topologies on gq.
PROPOSITION 2.2. dent of cp and "'.
Proof
II [cp, "'] 1Ik+. :::;; C Ilcpllk+ 1 +II IIcpllk+ 1 +/%' where C is indepen
We leave the simple check to the reader.
We need to know the following strong kind of continuity for the Green's operator G:
160
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
2.3. not on n/2
ID"/(x)1 ~ c Ilfll~+ 1"1'
(6)
where c is a constant depending only on k, I(XI, V, (6) we have: LEMMA 3.1.
There is a constant c such that if k
Ilfglli Proof
Let
It I ~ k.
~
~
u.
n
As a consequence of
+ 2,
c Ilfll~ . IIglif.
(7)
Then D(jg)
=
L
D'j' D'g.
r+.=(
Then either Itl ~ k nd k that
Irl + nil + I ~ k or lsi + nil + I ~ k when r + s = t, since + 1 implies It I + n + 1 ~ 2k. Thus, there is a constant K so
~ n
IDflg(x)1 ~
L
K r+.=f
(lIfIIf ID'g(x)1 + ID'j(x)llIglI~)
3. for x
E
AN EXISTENCE THEOREM FOR DEFORMATIONS
167
II
V, by Equation (6). Squaring,
IDtfg(xW
~ ,,' L+~=}llfIIf)2IDsg(xW + (1Igllf)2IDrf(XW].
Now (7) follows easily.
Q.E.D.
By using a partition of unity, one defines IIcpllk for any cp E A P = reAD. P(T». The estimates of the previous section are essentially the same and we list the ones we need.
II[cp,
t/I] Ilk ~ ckllcpllk+ Ilit/lllk+ 1 [for alllarge k. (k
~
2n
+ 2where dime M = n).] (8)
IIHcpllk ~ Ck Ilcpllk
(9)
lI.9gcpllk ~ ck Ilcpllkl·
(to)
From now on k will be chosen so that the conditions of Lemma 3 hold.
PROPOSITION solution. Proof
3.2. Let
T
For fixed '1(t), Equation (I) has only one small
(1lcpllk < e)
= cp  cp(t). Then
• = t.9G([cp, cp]  [cp(t), cp(t)]) = t.9G([., cp(t)] = t.9G(2[.,
Estimating
+ [cp(t),.] + [T, T])
cp(t)] + [" .]).
11.llk gives 11.llk ~ D(II'llk Ilcp(t)lIk + 1I'lIi) ~ D IITMllcp(t)llk + 1I.llk).
If
IIcp(t)IIJ. is
small enough, the only way
o~ x ~
Dx(lIcp(t)lIk
can happen with x close to 0 is for x =
L
o.
+ x) Q.E.D.
The set N = {'1(t) = '1v tv Iitl < e} describes a small neighborhood of For small enough B there is a onetoone correspondence between '1 E N and solutions cp of (1). This is proved as follows: If cp = '1(t) + t.9G[cp, cp], then H.9 = 0 implies '1 = Hcp. Thus '1 is uniquely determined by cpo Given '1. call the small solution of Equation (1) (given in Proposition 3.2) cpo Then a correspondence F is defined by F'1 = cpo
o E IHJI •
168
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
LEMMA 3.2. Suppose M", is given, If 8", = 0, then'" = qJ(t) for some t if 1I"'lIk < tJ for small tJ.
Proof
0'"  t["" "'J = 0, hence
0'" = =
80'" + 08", t8[""
"'J
since 8", = O. Then
'"  H", = GO'" = tG.9[""
"'J.
Let 11 = H "'. Then '" = 11 + t8G[ "', '" J; and by assumption II'" I k is small so that 1II111k is small by Equation (9). Hence 11 = 11(t) for some It I < e, and the Q.E.D. remark .iust made shows that'" = FI1 = Fy/(t) = cp(t).
In general 8", =f: 0 so we must try something else. When we say that M '" is holomorphically equivalent to M ",(t) we mean that there is a biholomorphic mapf: M",(t) ~ M", andf: M ~ M thus induces a diffeomorphismJ: M ~ M of the underlying differentiable manifold M. Conversely let f be any diffeomorphism such that its values and first derivatives are close to the values and first derivatives of the identity map. Cover M", with a system {Vj , 'j(z)} of local ",holomorphic coordinates and let {Vj}, Vj £; Vj be a covering of M so thatf(Uj) £; Uj • Then 'j(f(z» is a local holomorphic coordinate on Uj. Thus qJ is determined by the equation n
o'IJ(f(z» =
L qJP(z)op,IJ(f(z».
(11 )
P=l
We also know n
o'lJ(z) =
L ",P(z)op,lJ(z), P=I
that is,
~ "IJ( ) = ~ .I,P( ) o'lJ(z) !lA ~ z L... '1'1 Z _p. uZ
P=I
OZ
(12)
3.
AN EXISTENCE THEOREM FOR DEFORMATIONS II
169
Putting Equations (II) and (12) together we get
" aC"(f(z» ofP( ) + " aC"(f(z» "J). 1" af P z ~ ar 0  " Y() z y.P
 l..J qJ
ac"fJ aY fP + "l..J qJY() ac" aY J). , af z aJ)' Y.).
The matrix (aC"jap) is invertible since it is assumed that Thus,
11"'"k is close to zero.
ofP(z) + L 1/11(f(z» op'(z) ).
PROPOSITION 3.3. Let M", be the complex structure induced from M", by the map f: M ~ M. Then qJ is determined by Equation (13). We use the notation 1/1 0 fto denote cpo
Proof
We just notice that
11"'llk small implies that
(ayfP(z) + ~ "'1(f(z»oyr(Z») is an invertible matrix.
Q.E.D.
Thus to prove Kuranishi's theorem in the case 9", =F 0 it suffices to show the existence of a diffeomorphismfsuch that 9(",0 f) = O. This is our task. We need to digress for a moment to describe a way of indexing diffeomorphisms close to the identity by the use of geodesics. We shall be brief, and refer the reader to any text on differential geometry for missing details. Let an Hermitian metric (g"jJ) be fixed on M. Then we have the Christoffel symbols
r"AfJ =" gji,,(OgfJji) l..J J. ' ,.
:l uZ
170
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
and if u(z)
=
L ull(z)(O/OZIl) is a vector field, we have the covariant derivatives t7
V;.U

II
OUIl '" rll u II =;;:+ i..J ;'lI , uZ
II
V;.u =
II
OUIl
oz;"
If z(t) is a curve in M, we define
d
dz;'(t)
= d ulI(z(t» + L r~lI(z(t» d ull(z(t» , t
t
;., II
when u(z) is defined along z(t). Then the geodesics of M are the curves z(t) satisfying the differential equation
V, (
dZIl(t)) = 0 dt '
that is, d(ZIl(t» d 2
t
dz;'
+ ;.,LII r~lI(z(t»dt
dz ll (t)d (t)
t
(14)
= O.
Since M is compact, all metrics are complete, that is, the geodesics through a given point with given tangent direction are defined for all time t. Let Zll(t) = Zll(t, ZO, ~) be the solution of Equation (14) satisfying ZIl(O) = (dzll/dt)(O) = ~II. Then the following facts are easily verified by using existence and uniqueness theorems of ordinary differential equations:
zo,
(I) (2)
Zll(t, zo, e) are Coo in (t, zo, e). zll(kt, zo, e) = Zll(t, zo, ke).
We set JIZ(ZO' () = zll(l, zo, e)· Then J is COO in (zo, e) and J«(zo, te) = Zll(t, Zo, e). Differentiating this relation we get dz ll
dt
n ofa (t, Zo, e) = 1I~1 ell o~1I (zo, Ie)
n
_
Ofll
+ 1I~1 ell a~1I (Zo, te).
So
~II =
] dz n [ or ar dt (0, Zo, e) = 111;1 ell o~1I (Zo, 0) + ~II a~1I (Zo, 0) . ll
This implies
ar
or
r(zo, 0) = z~, oe ll (zo, 0) = Dp, a~1I (Zo, 0)
=
o.
3.
AN EXISTENCE THEOREM FOR DEFORMATIONS II
171
The Taylor expanison then yields (15)
where O(l~12) is a term bounded by M 1~12 for some M> 0 and for small lei. Given a vector field ~ = 2::=1 ~Il(z)(oloZIl) on M we define a diffeomorphism !c. : M + M by Zll + r(z,
~(z».
By Equation (IS) f~(z)
(16)
= Zll + ~1l(Z) + O(I~(zW).
( 16)
We wish to calculate qJ = 1/1 o!c. and we use Equation (13). We abbreviate withf~ = Zll + ~Il + ,Il. Then (13) becomes 0~{J
+ 0, + L I/I~(f~)(dz). + oe· + of).) }.
=
yt [!5~
+ Oy ~{J + or ,P + ~ I/I~(h)(ay e' + i3y fA)] qJY.
Multiplying by the inverse of the expression in brackets [ ] we get qJY = o~Y
+
L I/IHf~) dz
A
+ ...
A
= o~y
+ 2: I/IHz) dz}. + RY(I/I, O. }.
Thus,
1/1 f~ = qJ = o~ + 1/1 + R( 1/1, ~), 0
(17)
where R(tl/l, t~) = t2Rl(I/I,~, t) if t is a real number and both R, RI are COO functions of the parameters 1/I'P(z), 1/I'P(!c.(z», ~1l(Z), (a~lllozp)(z), (iJ~lllozP)(z) in local coordinates. In AO we have RO, the space of holomorphic vector fields on M. Using the L2 inner product on AO we let FO be the orthogonal complement of RO. So E F ° if and only if (~, '1) = 0 for all '1 E RO, that is, F ° is the kernel of the map R: AO + RO. Then for ~ E FO,
e
Since 8 is zero on A 0, 8~ = 0, and
yielding ~
=
G8o~.
Now give AO, Al and their subspaces the
II Ilk topology.
(I 8)
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
172
PROPOSITION 3.4. There are neighborhoods of the origin V and V in At and F 0, respectively, so that for any 1/1 E V there is a unique = 1/1) in V such that (19)
e e(
Proof
Using (17) we see (19) is satisfied if and only if
0= 8(1/1 By (18)
e=
0
h) =
Me + 81/1 + 8R(I/I, e)·
G8o~ = G81/1  G8R(I/I, ~).
Thus (19) is equivalent to
+ G81/1 + G8R(I/I, ~) = O.
~
(20)
Now choosing neighborhoods VI' VI so that R is defined on VI x VI we can define a map h: VI x VI + F O by h(I/I, e) = ~ + G81/1 + G8R(I/I, ~). By our previous remarks on R(I/I, e), we see that h is continuous if VI' VI, F o have the I Ilk topology, since R is continuous as a map from VI x VI with the II Ilk topology to A 1 with the II IIkl topology. In fact, h is even uniformly continuous and hence has a (unique) extension to a mapping h of the completion of the domain to the completion of FO. Then the partial derivative 8h
8~
I ..F+r, 0
1"0
(0,0)
where po is the completion of F O, is the identity map. Then by the implicit function theorem [see Lang (1962)] there is a COO function g on a small neighborhood of the origin in the space Al (completion of Al)such that Equation (20) is satisfied if and only if = g(I/I) for some 1/1 E U. Thus given 1/1 E V, where V is a small neighborhood in AI, there is a unique solution g(I/I) = ~ of (20) which is sufficiently small. Then 0 + 8R(I/I, ) + 8 is an elliptic secondorder equation with coo. Thus, if 1/1 E V s;;; AI, = g(I/I) satisfies
e
e
oe + 8R(I/I, ~) + 81/1 = 0 so ~ is Coo, that is,
e
E
FO.
Q.E.D.
Let us summarize our conclusions. THEOREM 3.1. (Kuranishi) (a) Let M be a given compact complex manifold, and let {'7.} be a base for W ~ Hl(M, 0). Let qJ(t) be the solution of the equation
qJ(t)
= '7(t) + !8G[qJ(t), qJ(t)],
4. where 11(t) = L~= 1 t. 11.,
STABILITY THEOREMS
ItI < p,
173
and let
OJ.
B = {t I H[ep(t), ep(t)] =
Then for each t E B, ep(t) determines a complex structure M t on M. (b) Let r/J be any vector (0, I)form satisfying or/J  1[r/J, r/JJ = O. Then r/J defines a complex structure M", on M. If II r/J Ilk is small enough, there is a unique E F O such that r/J 0 f~ = ep(t) for some t E B, and hence M", is biholomorphically equivalent to M,.
e
4.
Stability Theorems
The main point of this section is to prove that if .,I( = {M,} is a complex analytic family and Mto is Kahler, then M t is Kahler if It  tol is small. We first study elliptic differential equations depending on a parameter. Let P = {tlltl < IX, t = (t1' ... , tr )} be an open disk in cr. Let X be a compact differentiable manifold, and let PA be a differentiable complex vector bundle over X x P. Let B t = PAIXX{tl be the restriction of PA to X x {t}. Then we can consider PA = {B,} to be a family of vector bundles over X depending differentiably on t E P. Let L(PA) = the space of differentiable sections of ffI, L(B,) = the space of differentiable sections of B, . We are only interested in small deformations so we can assume that P is small and then there will be a finite covering {XJ of X such that
PA Ix,xp =CI'
X
Xi
X
P,
that is, PA is trivial over Xi x P. Let (Cf, x, t) be a local coordinate on ffllx, x p. Then the coordinate transformations on PA are written as I'
Ct = •L=1 b!.(x, t)'~ . By an (evenorder) differential operator E, : L(B,) 
L(B,),
we mean a map which can be written locally in the form I'
(E,r/J)t(x) =
L Et.(x, t, DMi;(x),
.=
1
where r/J(x) = (r/Jt<x» is a section of B, and where E~(x, t, D i ) is a polynomial of degree m =0 (mod 2) in Di = (a/ax'f). In our applications we will only need m = 2,4. For E, to be well defined we must have
L Et(x, t, Di)bikt(x, t)r/Ji.(x) = L bi1.(x, t)E;.(x, t, D,Jr/Jk(X),
"
.
"
.
174
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
where It
I/It(x) =
L1btkY(X, t)I/IZ(X).
y=
We say that E, depends differentiably on t if all of the coefficients of Ety(x, t,Dj)as a polynomial in Djare C" functions of (x, t). We assume given a Riemannian metric on M and an Hermitian metric on the fibres of B, so that
L gilv(X,t) ,tc dX A,
j
=
\I
L gkClt(X, t)C~ ,r dX k
tI, 't
is invariantly defined. We have written these expressions in terms of local coordinates for X where
,t = L Mix, t),: CI
and dX j
=
dxl'" dx~
(n = dim X).
We assume that gjAv(X, t) are Coo in (x, t). An inner product on L(B,) is defined by (1)
We will consider only those E, which are formally selfadjoint; that is, (E, cp,
1/1), =
(cp, E,I/I),
(2)
for all cp, 1/1 E L(B,). Let us see what this implies for the coefficients of E,. Let E;,;A(X, t, D j) be the terms of order m in Ej~(x, t, DJ Writing out (2) we get
fx L gjAtE'!'/(x, t, Dj)cpax)I/IRx) + lower order terms = f L gmCPt(x)E?;Y(x, t, D;)I/IXx) + lower order terms. x A,t,Y
A,
Y, t
Integrating the first term of the lefthand side by parts and using the fact that m is even we get
This is true for all cp and 1/1 so we get
L gjAvE~Yl/li = L gjYtE';/I/Ii Y, t
for all 1/1. Hence, (3)
4.
STABILITY THEOREMS
175
Replace (a/ax~) in Er';,v with YIZ' Then Er;,V(x, t, y) is a homogeneous polynomial of order m in y with coefficients which are Coo functions of (x, t). Equation (3) becomes
if Aw = "Iv giv, Er;,V(x, t, y). We have proved: PROPOSITION 4.1.
Equation (2) implies (3).
We shall assume further that Er is strongly elliptic, that is, (_l)m/2 "I Ai).t(x, t, y)w A W, > 0 A.
t
for any real y = (Yl' "', yft) =F 0, and any complex W = (WI' " ' , w,,) =F O. We need to collect some facts about such operators. We quote the following wellknown theorem which can be found, for example, in Palais(1965, p. 182): THEOREM 4.1. Er has a complete orthonormal set of eignenfunctions {erh}h": I ~ L(Bt )· Let the eigenvalues be Ah(t). Then they are real and with (erh' er)r = b jk • [Completeness means that any 1/1
E
L(B r) can be written as follows:
00
1/1
=
"I ajeth , h=l
Furthermore we can arrange the e th such that
and Iim h > 00 Ah(t) =
+ 00.
The following theorem is proved in Kodaira and Spencer (1960), and we shall not prove it here. THEOREM 4.2. REMARK.
Each eigenvalue Ah(t) is a continuous function of t E P.
Ah(t) may not be differentiable. For example, let E = ((X(t)P(t») t
y(t)b(t)
be a Othorder differential operator (just a matrix). Then 1)
AU =
(X(t) + c5(t)
± J«(X(t) 
b(t»2
+ 4p(t) b(t)
''=~~..:..:====
2
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
176
which could fail to be differentiable when (ex  (j)2 the kernel of E, . Then
1Ft
+ 4fJ(j =
O. Let 1Ft be
= {I/I IEt 1/1 = O} = {I/I I1/1 = )..(1)=0 L ah eth}.
Let F, be the orthogonal projection to 1F t , that is,
F,I/I =
L
(1/1, e'h)e'h·
)..(1)=0
The Green's operator G, is defined by
G,I/I=
L
1
1 (t)(I/I,elh)e,h .
)..(I)"'O/Lh
F, and G,are related by the equation
1/1 =E,G,I/I
+ F,I/I.
We have already investigated the case P = a point, E, = D, F, G, = G. In the general case we have the following theorem:
= H,
IFr
= IHI,
THEOREM 4.3. dim IF, is an upper semicontinuous function of t. This means that given to , there is a small enough e so that dim IF I ~ dim IF to for It  tol < e. Proof dim IF, = d, is finite since An(t) + Ah(tO) we have
...
~
00.
In the ordering of the
Aito) < 0 = Aj+ I (to) = ... = Aj+do(tO) < Aj +do +I(t)
~
By continuity, choose e so that Ait) < 0 and Aj+do+l(t) > 0 if Then in this disk d, ~ do. Q.E.D.
....
It 
tol
dim ( _d_, ,dL ,I d,
n ZI.l d,
+ £ILl) = dim (ZI,I d, '. d,LI,
By de Rham's theorem b 2 = dim (Zi,/d,L,I). We claim there is the following exact seq uence ZI.1 + £I Ll, d" d, Ll, 
o

Z 2" Z2.0 Z!l·2 d,' ill D, 1 •O + aLo. 1 · d, L ,1  o L " "
We must define 1t, and check that it has the correct kernel. Let t/I E zi" d, t/I = t/l2.0 + t/ll.l + t/l0.2. Then d,t/I = 0 yields o,t/l°·2 = o,t/l2.0 = O. So we can map ./, to ./,2.0 + ./,0.2 E Z2.0 + Z!l·2 Let ./, Ed L 1 Then 'I' 'I' 'I' 0, D,' 'I' ".
o where t/I =
t/I
= d,(cpl,O
+ cpO.l)_ O,cpl.O + O,cpl.O.
This correspondence induces the map 1.O+ 0,.°·1. Then t/l2.0 + t/l0.2
=o,u
t/I where
Thus
~
e
d,(u 1•O + .0.1) =
~ =
'1t,. To compute ker 1t"
t/ll.l _ o,u 1•0
_ 0,.0.1
suppose 1t, t/I =
= ~1.1
is of type (l, 1) Then
E
ZJ;1 and this yields
b
2 
dim(ZJ;1
d,e = d,t/I = O. t/I E d,L,1 + ZJ;l.
This exact sequence implies
;~;L:) ~ dim(o~tl~o) + dim(o~t;l)'
But Dolbeault's theorem implies dim
Z(P) (u,L, :os
1"0.1
= dim H2(M, l!J) = ,,~.2
and
hO•2 " Thus dim 1F,1.1 ~ b 2  2h~·2.
Z2.0 ) = h2 •o = dim ( _0_,_ I o
0, L , • .
Q.E.D.
184
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
LEMMA 4.2. Proof
dim IFf,l = dim IFA'o for smalliti. Let us compute dim IFA,l. We claim 1F~,1 = Il~,l ~
HI(Mo, (
1).
(10)
We use (5) and Proposition 4.3 for the proof of (10). Since
Eo cp = 0 implies Do cp = O. Conversely if Do cP = 0, then 00 cP = 0 = 8 0 cP; and Do cP = 0 so o cP = 90 cP = O. Thus Eo cP = O. This proves (10). Hence,
a
Recall that on a Kahler manifold
b2 = =
+ hA,l + hg,2 h~,l + 2hg,2. h~'o
Thus, dim
1F~,1
= b 2  2hg,2.
By the upper semicontinuity of dim 1F,I,1, for It I
1. line 10: Zl should be Z2. line 6: aZ l should be a(l. line 10: mglOzlblO should be blO  mglOZl. Add the map q> to the top arrow of the diagram. 193
ERRATA
Page 58, lines 13 and 6: E cO should be E Co. Page 61, line 3 after the diagram: ht.p should be kt.p. Page 64, line 13: n 2:: Z should read n 2:: 2. Page 64, line 6: hk(Z) = should read fik(Z) =. Page 80, line 3: There should be a space after the comma between t.p and q. Page 81, lines 12 and 13: There should be a comma after OJ "The" should be "the". Page 85, line 11: The last term in this equation should read 8 f L:Q,,B ~dza A dz,B. 2
Page 85, line 2: dZ a should read dz a . Page 86, line 10: 2 L: 9ja,B should read 2 L: 9jaij· Page 87, line 4: The second z in this equation is missing a subscript j. Page 93, line 13: '!/J ApB• should be '!/JApB. j JL1 should be JLI. Page 93, line 9: Aq should be Ap both times. Page 93, line 7: The first B in this equation should NOT have a bar. Page 95, line 5: t.p A *'!/J should read t.p A *i[;. Page 95, last line: The lefthand side of the last equation on this line should read o'!/J = instead of 'I3'!/J. Page 100, line 10: The second equation on this line should read  A 8f jk v(z) = 0. Page 108, line 12: omit the bar over 8>.. Page 112, line 7: Proposition 5.4 should read "In the Kahler case
"
Page 118, line 5: The last factor in the subscript of the R in the righthand side of the equation should be Y. Page 118, line 9: The subscript on the last R should be {3v),.. Page 120, line 4: {31 should be i31' Page 120, last line: (r)i should be (r)k. Page 124, line 10: x should be +. Page 126: line 5: {3q should be Bqj the term involving t.p after the summation sign should be t.pjB• • t.p: •• Page 126. lines 11, 12, and 13: All superscripts Bq should be Bq; in line 13 there should be a bar over the entire expression Vat.p:·.