COMPLEX MANIFOLDS JAMES MORROW KUNIHIKO I(ODAIRA
AMS CHELSEA PUBLISHING American Mathematical Society • Providence, Rhode Island ~'t~~~
~.! "BtiiYi;
,••,
2000 Mathematics Subject Classification. Primary 32Qxx.
Library of Congress CataloginginPublication Data Morrow, James A., 1941Complex manifolds I 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 1 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 v
Preface
Chapter 1.
Definitions and Examples of Complex Manifolds 1. Holomorphic Functions 2. Complex Manifolds and Pseudo group Structures 3. Some Examples of Construction (or Description) of Compact Complex Manifolds 4. Analytic Families; Deformations
11 18
Sheaves and Cohomology Germs of Functions Cohomology Groups Infinitesimal Deformations Exact Sequences Vector Bundles A Theorem of Dolbeault (A fine resolution of lP)
27 27 30 35 56 62 73
Chapter 2. 1. 2. 3. 4. 5. 6.
Chapter 3. Geometry of Complex Manifolds 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. 1. 2. 3. 4.
Applications of Elliptic Partial Differential Equations to Deformations Infinitesimal Deformations An Existence Theorem for Deformations I. (No Obstructions) An Existence Theorem for Deformations II. (Kuranishi's Theorem) Stability Theorem
Bibliography Index Errata
1 1 7
83 83 92 100 102 106 116 125 134
147 147 155 165 173 186 189 193
vii
Complex Manifolds
[1] Definitions and Examples of Complex Manifolds I.
Holomorphic Functions
The facts of this section must be well known to the reader. We review them briefly. DEFINITION 1.1. A complexvalued function J{z) defined on a connected open domain W £: C" is called h%morphic, if for each a = (a., ... , all) E W, J(z) can be represented as a convergent power series +00
L
ck , ..• kn(Zl
alt' ... (ZII


alit"
k,~O.kn~O
in some neighborhood of a.
LC
If p(z) = k ... kn (Zl  a1)k, •.• (Z/I  allt n converges at z = w, then p(z) converges for any Z such that IZk  akl < IWk  akl for 1 ~ k ~ n. REMARK.
Proof We may assume a = O. Then there is a constant C > 0 such that for all coefficients Ckl".k n ,
Ick,· .. k
n
Wk'1 ... wknl /I <  C•
Hence (1)
If
Izdwil < 1 for 1 ~ i
~
n, (1) gives Q.E.D.
1
DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS
2
We have the following picture:
Figure I
n is the region {zllz;l
VI> " ' , Un' Vn)]
o(X I , YI"", Xn , Yn)
o(u, V)
=.. o(x, y)
6 REMARK.
DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS
If/ is holomorphic, o(u, v)/o(x, Y) = Idet(o/.doz')l 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 aVA = ouA/ox. = ovA/oYv, b,;. = OVA/OX. =  ou;./oYv. Then
OUI OVI oU z oV z oX I oX I oX I OXI

OU I
OVI °YI OYI
OU z OV 2 = OYI °YI
all
b ll
al2
biz
bll
all
b 12
a 12
a21
b 21
a22
b22
b 21
aZI
b 22
a zz
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.;. = O/A/OZV = a.;. + ib v ;', we get gil
o(u, v) gZI = 0 o(x, Y) 0
glz * * gzz * * 0 gIl gl2 0 gZI gll
= Idet(g.;.)I Z
by interchanging columns 2 and 3 and rows 2 and 3.
Q.E.D.
THEOREM 1.3. (Inverse Mapping Theorem) Let/: U . en be a holomorphic map. Ifdet(o//l/oz.)lz=a =1= 0, then for a sufficiently small neighborhood N of a,fis a bijective map N + /(N);f(N) is open and/II/(N) is holomorphic
on/eN). Proof The remark gives o(u, v)/o(x, Y) =1= 0 at a. We then use the inverse mapping theorem for differentiable (real variable) functions to conclude that /(N) is open, / is bijective, and / I is differentiable on /(N). Set cp(w) = /1(11'); then z/l = 'P/l[f(z)]. Computing,
o = O~I' = oz.
f ;'=1
o'Pl' O~A OW;. OZ.
+ O~I' o~;. OW;. oz.
But det(oJ;./oz.) = det(o/A/oz.) =1= O. So by linear algebra, oCPI'/ow;. = 0 and cP =/1 is holomorphic. Q.E.D.
2.
COMPLEX MANIFOLDS AND PSEUDOGROUP STRUCTURES
7
COROLLARY. (Implicit Mapping Theorem) Let f;., A = I, ... , m be holomorphic on V s; en. Let rank (8f;./8z.) = r at each point z of V and suppose in fact that det(of;./oz');'$r i= O. If fla) = 0 for A. :::;; m for some a E V, then in a small neighborhood of a, the simultaneous equations,
have unique holomorphic solutions
A:::;; r. For 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 IC". DEFINITION 2.1. By a local complex coordinate on X we mean a topological homeomorphism z:p  z(p) E IC" ofa domain Us; 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 {z j} jEt (for some index set I) of local complex coordinates Zj: Vj such that:
en
(1)
X=UU j
•
JET
(2) The maps fjk: Zk(P)  Zj(p) are biholomorphic [that is, Zj 0 Z;1 = fjk and r;/ = Zk zjt are holomorphic maps from Zk(V j n Vk) onto z}Vj n Uk)] for each pair of indices (j, k) with U j n Uk i: fjJ. 0
DEFINITION 2.3. Two systems {Zj}jt/, {I1';J"<J' are equivalent if the maps zip)  1I',,(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 IPn. This is constructed from {OJ by identifying (p '" q)p = (pO, pI, "', pn) and q = (qO, "', qn) if and only if p). = cq). for some nonzero c E e, for 0 ::::; .A. ::::; n. Then IP n = en + I {OJ j '" is a compact Hausdorff space and one can construct a system of complex coordinates as follows: We let Vj = {p E IPnlpj::f. OJ. Then {VJj~n is an . 0 f It"". ITl>n 0 h open covenng n V j th e map Zj  (Zj ' , Zjj  I , Zjj+ I , . . . , Zjn) , were = p)./pj gives a local coordinate on V j ; in fact, Zj(V) = en. Then fjk: Zk 4 Zj is given by z~ = zt/zf for A. ::f. k, z~ = I/zf. (One simply multiplies by pkjpj.) Thus we see that {V j ' Zj} is a complex analytic system defining a complex structure on IPn.
en+ 1 
° ...
z/
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 set f( 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 nor en. Letf and 9 be local diffeomorphisms of 9. If open W s; 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)::f. fIEf. (2) fE r, 9 E r => 9 a fE r where defined. (3) fEr => fl WE r for any open W s; 9. (4) The identity map id E r. (5) (completeness) Let f be any local diffeomorphism of 9. If [) andfl Vj E r for each}, thenfE r.
= u
Vj
2.
COMPLEX MANIFOLDS AND PSEUDOGROUP STRUCTURES
9
DEFINITION 2.7. Let f (a pseudogroup on 9) and X (a paracompact Hausdorff space) be given. By a system of local fcoordinates we mean a set {z j} j d of local topological homeomorphisms Z j of X into 9 such that Zj 0 Z;:l E f whenever it is defined. {w ...} and {Zj} are equivalent (fequivalent) if 11'). 0 zjl E f when defined. A fstructure on X is an equivalence class of systems of local fcoordinates on X. A fmanifold is a paracompact Hausdorff space X together with a fstructure on X.
EXAMPLES I. 9 = en, fe = (all local biholomorphic maps of en). Then a festructure is a complex structure, and a femanifold is a complex manifold. 2. 9 = IR", fd = (all local diffeomorphisms of IR"). Then a fdstructure is a differentiable structure and a fdmanifold is a differentiable manifold. 3. Let f be the set of a local diffeomorphism / of 1R2" satisfying the following condition. The matrix (c)..) will be defined to be
0
1
0 0
\
0
0 0
0 1
1
0
where the blocks (?  b) occur on the diagonal and the rest of the entries are zeros. If x = (Xl, "', X2n) E 1R 2",f(x) = [.ft(X), ... ,f2"(X)] 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 f = (local affine transformations of IR"). These transformations have the form
j'(x) =
L" a~ XV + b\ v~
,
where the a~, b). are constants and the matrix (a~) is nonsingular. [n 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 f 2 equivalence. Hence, every f Istructure determines a
10
DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS
r 2structure. By assumption r c r d for all r. So every r structure on X determines a differentiable structure on X and every rmanifold is a differentiable structure on X and every rmanifold is a differentiable manifold. The rstructure M is defined on the differentiable manifold X. The problem of determining the rstructures on a given differentiable manifold M for given r 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. In 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)], and Kodaira and Spencer (1958). If S211 is the sphere with its usual differentiable structure, it can be shown [Borel and Serre (\ 953) and Wu (1952)] that S211 for n f= 1,3 has no complex structure
[S
2n _ {(Xl' . . . , X2n+l) 12n~ if2I Xi'2
(Xl'
•••
,X2n+1
)
E
IR
2/1 + I }] .
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 r+ be the pseudogroup of all local affine transformations,
v = 1,2 such that
We have: THEOREM 2.1. [Benzecri (1959)] If a r+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 {(xj, X])} of local coordinates such that lax;/ax~1 is constant on Vj 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 mappingj: W + C l is holomorphic (difjercl1tiahle, and so on) if I zj I is holomorphic (d(fJerentiable, and so on) for each j where defined. Let N be another complex manifold with coordinates {1I' .d and I: W + N. Then I is holomorphic (differentiable, and so on) if II'A a I 0 zj I is holomorphic where defined. 0
DEFINITION 2.8. A subset S £; M 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 {; 3 S, I ~ A~ r, such that S n U = {p Ilip) = 0,
3.
CONSTRUCTION OF COMPACT COMPLEX MANIFOLDS
11
I ~ A ~ r}. Then f). = 0, 1 ~ A ~ n, are the local equations defining 5 at s. The subvariety 5 is called a submanifold if 5 is defined at each s E 5 by local equationsf). = 0 such that
l
Of;.(P)] . d d f ran k  = r. IS In epen ent 0 s.
ozi(p)
Suppose det(af)./ozj)1
V 2 :::> V 3 ••. be a base of rela
3.
13
CONSTRUCTION OF COMPACT COMPLEX MANIFOLDS
tively compact neighborhoods at q. Then Fm = {g IgVm n Vm =f. cp} is a finite subset of G and Fm;2 Fm +S' ;2 •.•. If 3gmE Fm, gm =f. 1 for all m, then since each Fm is finite, n Fm 3 g, 9 =f. 1. Therefore, gVm n Vm =f. cp, for all m and Vrn + q, gives g(q) = q, contradicting the nonexistence of fixed points. Hence we cover M with open sets Vj such that p" P2 E Vj implies pi =f. pi and thus, Vj ~ V; = {p* I p E V j } is 1  1. We give V; the complex structure that Vj has. That is, if Zj: P + zip) is a local coordinate on V j ' then zj: p* + zj(p*) = ziP) gives a local coordinate on M*. The system {zj} then defines a complex structure on M* and the topology of M* is just the quotient topology for the map M + M*. Q.E.D. EXAMPLES Complex tori. Let M = Take 2n vectors {w" ... , W 2n }, n ••• , w kn ) E C so that the Wj are linearly independent over ~. Let
cn.
1. (Wk"
Wk
=
2n
G = {g I g: z + g(z) = z
+
L mkWk , mk E l}.
k='
Tn = en jG is a (complex) torus of complex dimension n. Let n = 1 and arrange it so that WI = I, W2 = w, where the imaginary part of W is positive. Then T=C'jG.
w
Figure 3 Z I
exp 2"j
We have a map C   C*, z + w = e "z where C* = {z I z =f. OJ. If we first take g(z) = z + m,w + m 2 and then exponentiate, we get e 27ti (z+m l w). So exp 21ti 0 9 = ami. exp 21ti where a = eZ"iw and g(z) = z + m,w + m2' and o < lal < 1 since Im(w) > O. Looking a little closer we see we have the diagram If'
exp 2"i
IVC
*
·cc 1 ." ,., 1:' which commutes. Hence, if we let G* = {g* I g*: w + am l1', mEl}, we see T = C/G = C*jG*.
14
DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS
Figure 4
Hop! manifolds. Let W = eN  {O} and G = {gin I m E Z, g(wl' .. " = «(XIIV I , " ' , (XNWN), where < l(Xvl < I}. Then WIG is a compact com2.
°
wN ) plex manifold since it is easy to see that G is properly discontinuous and has no fixed points on W. It is also easy to see that WIG is diffeomorphic to Sl x S2NI. 3.
Let M be the algebraic surface (complex dimension 2) defined:
M
= {(
I (g + n + (~ + (~
=
o} £
[FD3.
Let
G = {gml m = 0, 1,2,3,4 where g«(o,"', (3) = (p(o, p2(1' p3(2' p4(3) and p
= e 2"i/5}.
Then 9 is a biholomorphic map [FD3 ~ [FD3 and g5 = 1. Consider the fixed and the points of gm on [FD3. They satisfy (0 = v ~ 3), (pln(v+l)  c) (v = fixed points are (1, 0,0, 0), (0, I, 0, 0), (0, 0, I, 0), and (0, 0, 0, I). These points are not on M so there are no fixed points on M and MIG is a complex manifold. We saw before that M is simply connected and X(M) = d(d 2 4d + 6) where d = 5. Therefore, the Euler number of M is 55. Then the fundamental group 1t 1 (MIG) ~ G and x(MIG) = II. 4. Last we have the classical examples of Riemann surfaces and their universal covering surfaces. If S is a compact Riemann surface of genus 9 ~ 2, the universal covering surface of S is the unit disk D = {2 E ell 121 < I}. Then S = DIG where each element of G is an automorphism of D and hence of the form
°
g(2)
.
2 
(X
= el8   , (X;: 
I
I(XI < I.
3.
CONSTRUCTION OF COMPACT COMPLEX MANIFOLDS
15
Finally we consider surgeries. Given a complex manifold M and a compact submanifold (subvariety) ScM, suppose we also have a neighborhood W::::> S and manifolds S* c W* with W* a neighborhood of S*. Suppose I: 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 = I(z*).

J
[ry
Figure 5
EXAMPLE I. Hirzebruch (1951) Let M = pi X pl. In homogeneous coordinates, pi = {(I ( = «(0' (I)); = {C u {(Xl in inhomogeneous coordinates, (= (d(o E C u roo}. M = pi X pi = {(z, 01 z E pi, (E pi} contains S = {OJ X pi and W = D X pi where D = {zllzl < E} is a neighborhood of Sin M. Let W* = D X pi' = {(z, (*)Iz ED, (* E pi'} and S* = {OJ x P*. Fix an integer m > 0 and define I: W*  S* ~ W  S as follows:
I(z, (*) ~ (z, 0 = [z,«(*/z'")J
where 0 < Izl
o} [some sort of a generalization of 1m W > in Example (1)]. Let CfJ = the set of all transformations
°
n + (An + B)(Cn + D)l = n', where
(~ ~) E SL(n, Z), the invertible integral matrices of determinant + 1.
This group does not really act on H since it is possible for cn + D to be singular; one should consult KodairaSpencer for more details. H should be extended to something more general on which SL(n, l) acts. In any case,
Tn = Tn,
ifn'
= gn, 9 E r§.
We would like to form H/r§. But it turns out that CfJ is not discontinuous. In fact, for any open set U c H, there is a point n E U such that {gn I9 E CfJ} n U is infinite. Hence, the topologial space H/CfJ 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 It E B} such that M t = M for t =i' to and M to =i' M. EXAMPLE 3. A Hopf surface is a compact complex manifold of complex dimension two which has W = 1[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: (Zl' where
°< lal < I and t
LEMMA 4.1.
{M t It
E
Z2) +
E
(az 1
+ tz 2 , az 2 ),
that is, (::)
+
(~ :)(:~),
C. Then M t is a compact complex manifold.
C} is a complex analytic family.
24
DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS
Proof
= {M t 1 t E C} = (: x W/f, where f = {I'm 1m E~}, and
v(i;) ~ (~ ~ DGJ
Q.E.D.
We claim (I) (2)
M t = Ml (complex analytically) for t Mo =1= MI'
Proof of (1).
=1=
O.
We make the following change of coordinates:
Then the equation
implies that Ml = M t when t Proof of (2).
=1=
O.
First we prove a special case of Hartog's lemma.
LEMMA 4.2. Any holomorphic function defined on W = (:2  {CO, O)} can be extended to a unique holomorphic function on (:2. Proof Let f(zl' define the function
Z2)
be the function on W. Pick a number r > 0, and
F(zl'
1 1
few,
Z2)
'j dw, 2m Iwl=r w  ZI
Z2) =  .
for IZII < rand Z 2 arbitrary. Then F(zl' Z 2) 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 that few, Z2) is holomorphic if Z2 =1= o. So Cauchy's theorem gives for IZll < r, Fix ZI' 0 < IZII < r. Then F(zl' in Z2; therefore,
Z2)
= f(zl'
Z2)
for
Z2
Z2 =1=
=I
o.
o. Both are analytic
F(zl' 0) = f(z(, 0). Hence they agree where defined, proving the lemma. Now let us suppose M, = Mo. I =r= o. Then there is a biholomorphic map + Mo· Wis the universal covering manifold of M, and M o , sofinduces
f: M t
4.
25
ANALYTIC FAMILIES; DEFORMATIONS
a map I: W + W which is biholomorphic, such that
W~W
commutes. It follows that Gr = 1' Go! Hence for generator g, of G"
g, =1'g"5'!
(9)
Write the map I in coordinates as
I(z, , Z2) = Chez"~ Z2)'/2(Z" Z2)]. Then by Hartog's lemma extend liz" Z2) to a holomorphic function F;. (z" Z2) on C 2 • Then F maps C 2 into C 2 [F = (F" F2)J, and F(O) = O. For if not, extend 1' to F which satisfies F'[F(z)] = z on Wand by continuity, F[F(O)J = O. But if F(O) '" 0, F[F(O)] = 1' [F(O)J '" O. This contradiction gives the result. Now expand F;.,
F;.(z, , Z2) = F;.,z,
+ F).2Z2 + F;'JzT + F)..Z,Z2 + ....
We know that/[gr(z)J = g"5 1 [fez)] so
( 0)
F[grCz)J = ~ a
±,
F(z).
Rewriting this gives
F,(az, +lz 2 ,az 2 ) =a±'F,(z"Z2), F2(az, + tz 2 , O:Z2) = a±'F,(z" Z2)' Expanding these and taking the linear terms yields
( FI1 F , 2
I)
F'2)(a = F22 0 a
(IX
0
O)±' a
This can only happen when t = O. Hence M, '" Mo.
Q.E.D.
EXAMPLE 4. Ruled Surfaces (examples of surgery) will be !P' bundles over !P'. Let !P' = {( I( E C U {oo}} coordinates). M(m) = V, x !P' U V 2 x.!P' where V, u V 2 = !P'  {O}, and identification takes place as follows Let (z" (,) E V, x p', (Z2' ~2) E V 2 X pl. Then
REMARK.
M(III) ' " M(I)
Our ruled surfaces (nonhomogeneous V 2 = p', V, = C, (recall Section 3):
for m '" {' (not to be proved now).
26
DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS
THEOREM 4.2. M«() is a deformation of M(m) if m  t == 0 (mod 2). Assume that m > f. Then there is a complex analytic family {M tit E C} such that Mo = M(m) and M t = M(f) for t '# 0.
= VI
where (ZI, (I) (Z2, (2) if ZI = l/z 2 , (I = Z;(2 + IZ~ where k = 1(111  t). Then it is easy to see that {M tIE C} is a complex analytic family and that M 0 = M(m). Suppose t '# O. Introduce new coordinates on the first [PI by
Proof
Define M, as follows: M,
C1On the second
V2
X [pI
k, ZI
Ie
I
I 
(linear fractional transformation).
[pI,
r'
':.2
Then, using
X [pI U
ZIZ2
= I, and
(I
'2 +
= I~mkv "'"2
1,2
12'
= Z~'(2 + tz~, we get r' m 2/.. ... , ':.1=Z2 ~2'
Hence, in the new coordinates,
Z1Z2
= I,
for
PROBLEM.
t
,~
=
i= O.
z~(;; so
Q.E.D.
Findapairofcomplexanalyticfamilies {Mtlltl < 1},{Ntlltl < l}
such that (a) (b)
Mo '# No,
(c)
Nt = Mo
M,=No
for for
1'#0, 1'# O.
(not complex analytically homeomorphic)
There are no known examples of this type.
[2] Sheaves and Cohomology I.
Germs of Functions
Let M be a complex (or differentiable) manifold. A local holomorphic (differentiable) function is a holomorphic (differentiable) function defined on an open subset U s; M. We write D(f) for the domain of f Let p E M and suppose given local functions f, 9 such that D(f) n D(g) 3 p. We say that f and 9 are equivalent at p if fez) = g(z) for z E W S; D(f) n D(g), Wa neighborhood of p. By a germ of a function at p we mean an equivalence class of local functions at p. Denote by fp the germ off at p, (!J p the set of germs of all holomorphic functions at p, and:?) p the set of germs of all differentiable functions at p. The definitions
afp
+ flgp = (af + flg)p
a, fl
E
C,
fp' gp = (fg)p, are well defined, hence,
(!) p'
:?) p become linear spaces over C. We also define,
(!) =
U (!)p,:?) = U :?)p. peM
peM
We put a topology on (!J and:?) as follows: Take any cp E (!J (or :?); then cp E (!J p (or ~ p) for some p. Take any holomorphic (differentiable) f with fp = cp and define a neighborhood of cp as follows:
where p E Us; M, U is an open set in D(f). It is easy to see that the system of neighborhoods iJ7J(cp;f, U) defines a topology on (!) (or :?).
EXAMPLE. (!) on the complex plane C. Let p E C. Then if f and 9 are holmorphic at p we have expansions valid in some neighborhood of p, 00
I(z)
= L Ik(Z 
00
p)k, g(z)
= L gk(Z 
k=O
p)\
k=O
so f and 9 are equivalent at p if and only if fk
9k for all k. Hence, the germ at = ring of convergent power series. And an element cp E (!) p can be represented by cp = fp = {p; fo ,fl' ... } where limk_oo Ifkll/k < + 00 and the radius of convergence is r(cp) = II lim. =
p is represented by a convergent power series;
27
(!J p
28
SHEAVES AND COHOMOLOGY
We define ~(q>;
{t/!I t/!
s) =
= f q, Iq  pi
< s where 0 < s < r(q>)}.
In terms of our representation we calculate 00
fez) =
I
OQ
fk(Z  p)k =
k=O
I
fm(Z _ q
+ q _ p)m
",=0
Hence
~(q>; s) = {"'It/! =(q; go, ... , gk' ... ), I(q gk
=
m~J;)fm(q 

p)1 < s
p)mk}.
We note that t/I E If//(q>; e) means that t/I is a direct analytic continuation of q>. The case of E0 on IR is not so simple. If q> = fp where f is a Coo function at p, In
j(x) =
I
fk(x  p)k
+ O(x
_ p)m.
k=O
But f is not determined by the fk'S since there exist Coo functions f which are not identically zero, but which have all derivatives zero at some point. Define w: (!) (or E0) . M by w«(!)p) = p. (1) wis a local homeomorphism (that is, there exists ~ such that w: ~(q>;J, U) > U is a homeomorphism). (2) wl(p) = (!)p (or E0 p) (obvious). (3) The module operations on wl(p) are continuous (that is, IXq> + IN PROPOSITION
1.1.
depends continuously on q>, t/I).
Proof (1) ~(q>;J, U) = {fqlqE U} and w:!q >q is certainly 1 1. It is obvious that wis continuous. To show that w1 is continuous, let ~(w; g, V) be a neighborhood of t/! = f . We want to find a neighborhood W of q so • that w = WI(W) E ~(t/!; gY) for wE W. We know that gq = t/I = f q, so f and 9 are equivalent at q. Hence, f = 9 in some neighborhood N of q. Let W = N n V. Thenfw =gw on W, so fw E ~(t/I; g, V) for WE W. This proves that the w 1 is continuous. (3) Let q>=fp,t/!=gp. Then IXq>+f3t/!=(rx/+f3g)p. Let ~(IXq>+f3t/!; h, U) be a neighborhood of IXq> + f3t/!. Then IXq> + f3t/! = hp = (IXf +f3g)p so
r
1.
29
GERMS OF FUNCTIONS
h =af + fJg in some neighborhood V£ U of p. Then if (J E cf/1(cp;f, V), , E cf/1(l/J; g, V), we have a(J
+ f3,
+ fJgq = (af + fJg)q
= afq
= hq E cf/1(acp Since cf/1(acp
+ f3l/J; h, V)
£ cf/1(acp
+ f3l/J; h,
V).
+ fJl/J; h, U) we are done.
Q.E.D.
We now give a formal definition. Let X be a paracompact Hausdorff space. A sheaf!/ over X is a topological space with a map w: X onto X such that
DEFINITION
!/ t
1.1.
(1) wis a local homeomorphism [that is, each point s E !/ has a neigh. borhood cf/1 such that w: i1lt t w( cf/1) C X is a homeomorphism onto an open neighborhood of w(s)]. (2) wlex), x E X is an Rmodule where R = 7l., IR, 1[:, or principal ideal ring. (3) The module operation (s, t) t as + f3t is continuous on WI(X) where a, f3 E R.
(The reader can easily generalize this definition, but for our purposes it suffices.) The set!/x = WI(X) is called the stalk of!/ over x.
EXAMPLES. (1)
(2) (3) (4)
(of sheaves)
on a complex manifold. on a differentiable manifold. The sheaf over X of germs of continuous (IR. or I[: valued) functions. The sheaf over X of germs of constant functions. (!)
~
In Example (4) !/ = X x I[: with the following topology: Let s = (x, z); then cf/1(s) = {(y, z) lyE U, z fixed}. If r t f(r) is a continuous map into !/ of I = {ria < r < b}, thenf(J) = {(y, z) Iz fixed andy = w(j(r»rE l}. In other words we give X x I[: the product topology where X has its given topology and I[: has the discrete topology. DEFINITION 1.2. Let U be a subset (usually open) of X. By a section (] of !/ over U we mean a continuous map x t (](x) such that w(](x) = x. Suppose X = M, a complex (or differentiable) manifold; and suppose !/ = (!) (or ~). If fez) is a holomorphic (or differentiable) function on U, then (]: p t fp, P E U is a section.
30
SHEAVES AND COHOMOLOGY
1.2. Let a: V +!J? be a section (!J? as above). Then a determines a holomorphic (or differentiable function) 1= I(z) on V such that
PROPOSITION
u(p) = Ip.
Proof a(p) E (1)p (or £2,,). Hence there is a holomorphic (or differentiable) g(z) defined on some neighborhood of p so that a(p) = gp' Since g depends on p we write, g(z) = gCP)(z). Define I as follows: I(p) = g(")(p). Then I is obviously well defined. Then (1)
I(p) is a holomorphic (differentiable) function on U.
Proof Take Wa neighborhood of p, W ~ V. Let I1IJ = l1IJ[a(p); g("), W] {(g("\ Iq E W}. Since a is continuous, for any small neighborhood N of p, N ~ W, we have u(N) ~ illt. Hence u(q) = (g(P\. But we also know u(q) = (g(q»q. Thus, (g(q»q = (g(P\, and g(q)(z) = gC")(z) for z in a small neighborhood V of q, V c N. But I(q) = g(ql(q) = g(Pl(q) for q E V. So I(z) = g(P)(z) for z E V and g(P) holomorphic (or differentiable) in V implies that I is also. =
(2)
By definition u(p) = (g(P»p = Ip for each p E V.
Q.E. D.
Hence we have the maps: local holomorphic (differentiable) functions ~ germs ~ sections = holomorphic (or differentiable) functions. f( V, !J?) will denote the Rmodule consisting of all sections of !J? over U. We remark that f( V, (1) are all holomorphic functions over V and f( V, f:&) are all differentiable functions over U. Let {V ).11 ::;; A ::;; n} be a finite family of open sets in X such that n V). =1= 4>. Let u). E f( V). ,!J?) and IX). E R. Then L IX). a). E f(V,!J?) where V = n V).. Let W be an open set and u E f( V,!J?) for some open set V. Then x + O'(x), x E W n V defines a section of f( W n V, !J?). We denote this section by rwa and call it the restriction of 0' to W n U.
2.
Cohomology Groups
Let X be a Hausdorff paracompact space and let !J? be a sheaf over X. Fix a locally finite covering ()li = {V j } of X. A Ocochain CO on X is a set CO = {a j } of sections 0') E f( Vj ,). A Icochain C l = {O'jk} is a set of sections
2.
q
COHOMOLOGY GROUPS
31
Uk ,.9') such that O"jk =  O"kj (skewsymmetric). A qcochain is a set of sections O"jo .. ·jk E qUja II ... II Ujk ,9') which are skewsymmetric in the indices Jo ... Jk. Let C(JU) be the Rmodule of all qcochains. We define a map C(JU)~Cq+l(JU), the coboundary map as follows: For Ocochains, JCo = {'rjk} = {O"k O"j} where CO = {O"k}; for lcochains C l = {O"jd, JC l = {rjk(} where 'rjk( = O"k( O"j( + O"jk = O"jk + O"u + O"(j. In general, Jcq = {rjo ... j.+ ,} if Cq = {O"jo ... j.}, where O"jk E
Uj
II
cq = {O"ja ••• Jk}
+ ( l)q+ l(Jjo ... j. = L (_l)kO"jo··· j~ ... jq+,' where
A
(1)
means "omit."
We denote the qcocycles by zq(JU) = {C q I JC q =O}.
The qcohomology group (with respect to JU) is Hq(fI) = Well, 9') = zq(Olf)/JCql(Olf).
(2)
We should remark that JC is always skewsymmetric and JJ = 0 so that JCql(Olf) £; zq(JU) and Equation (2) makes sense. The qth cohomology group of X with coefficients ill the sheaf 9' is defined to be Hq(X, 9') = lim Hq(Olf, 9'). Vfi
This limiting process will now be explained. We say that the open covering "f" = {VJAeA of XisarefillementofJU = {Uj}jeJ ifthereisamaps:A + Jsuch that V). c: Us ().) = Uj ().) , where we setJ()") = s(),.). We define a homomorphism n~~
n~:
: (q(OU) + Cq("f"),
{O"jo ... j.} +
{r Ao ···;.• },
where
(3) It is easy to check that
(4) so that n~ maps Zq(Ulf) into zq("f") and JCq1elf) into JCql("f"). Hence n~ induces a homomorphian n~: Hq(~71) > Hq("f"). LEMMA 2. J.
n$: Hq(:Jlf) > W("f")
is independent of the choice of map
s : A > J in the definition of refinement.
32
SHEA YES AND COHOMOLOGY
Proof
First some notation: fix indices ao, ... , aq E A. Let V = V,~ n'" n V,~, V t
= V,
~
.A.
n'" n V,~ n'" n V,~,
/'.... /'.... Uj( = UJ(ao) n ... n U naj ) n Ug(a J) n ... Ug(at) n .,. n Ug(a q) ,
and Ui
=
UJ(a,) n ... n UJ(a j ) n Ug(a J) n ... n Ug(a q )
,
where/, 9 : A + J are two refining maps. Define a function (kU)A""A q by q
(kuh, ... Aq
=
L (1 y I rv ocrJU,)' .. J(Ap)g(A p)"
.g()'q)
(5)
p~O
Let us call the maps n~, defined by f and g,J*, and g*. We claim that the following equation holds: (6)
The function kcr is not necessarily skewsymmetric in its indices; so we skewsymmetrize
r~t ... ). = (k'r)A, .. , A =~, L sgn(AI111 q. q
q
Next we use (6) to see that [(Dk'
+ k'D)U]aO"'a q =
(g*cr  j*cr)ao"'aq '
Hence, if Dcr = 0, Dk' cr = g*cr  f*u E DCq  I (1""). Hence,f* and g* induce the same map, Hq(o/i) + W(1""). Therefore we prove (6). The reader can easily check the following calculations:
=
t ( 1/ [ttl (
+
crJ(ao) ... J(ajg(aj) ... g(a;) ... g(aq)
i=O
t (
i~(+
(Dkcr)ao'" aq =
1)ir v t
rv
t~O
1)i  I rVI cr J(ao) ... j(;;)J(aj)g(aJ)'" g(a q)]
I
L (1)(+
j
rv UJ(ao) '"
j«
J(aJ)g(aJ)'"~)''' g(aq)
+ 'L ( 
1)(+j+1 rv
= L (
l)i+t rv cr J(.o) '"
crf(ao) .. , J(at} ... f(aj)g(a J ) ••• g(aq) .
(7)
j>t
Similarly, (kDcr)ao'" a.
j(;;) .. ,J(aj)g(aj}
., g(a q)
lsj
(8)
33
COHOMOLOGY GROUPS
2. Equations (7) and (8) give
q
 L ry
(1/(ao) "'/(aj)g(aj) '" g(aq)
j=O
= ry (1 g(ao) ," g(aq)

rv (1/(ao) " '/(aq) ,
(9)
Q.E.D.
proving Equation (6).
Knowing that the map n~ depends only on r1IL and "1"', we proceed to the definition of the limit. We write r1IL < "fr if "fr is a locallyfinite refinement of r1IL. Then < is a partial order and given r1IL, "I'" there is "fr so that r1IL < "fr and "I'" < "fr. Hence the set of all locally finite coverings of X forms a directed set with respect to r1IL such that n~ 9 = the equivalence class of 9 by
n!, h.
Denote
g. Let
Hq(r1IL, !/) = {g I9 E Hq(lJlt, !/)}. The map 9 ~
g defines a homomorphism Il tlJi , n~:
Hq(r1IL, !/) ~ Hq("I"', !/),
and n~ induces a homomorphism TI~, n~
TI~ is injective.
LEMMA 2.2.
Proof
o and g =
: Hq(lJlt, !/) ~ HQ("I"', !/).
TI~g
O.
= 0 if and only if n~
0
n~ 9 = 0 for some W. So n~ 9 =
Q.E.D.
Hence, identifying HQ«()71,!/) with n~HQ(r1IL, !/), we may consider Hq(r1IL, !/) c HQ("I"', !/) provided that r1IL < "1"'. Then by definition,
W(X, !/) =
UHq(r1IL, !/), tIJi
34
SHEA VES AND COHOMOLOGY
and nUll: Hq(uu, f/)
+
Hq(uu, f/)
£;
Hq(X, f/) is a homomorphism of
Hq(dI!, f/) into Hq(X, f/).
PROPOSITION
Proof
2.1.
HO(X, f/)
= reX, f/).
By definition C I = 0 so H°(OlI, f/) = ZO(UU, f/).
= [0' 10' = {O'j},O'j E rcUj,
ZO(UU, f/)
f/),
DO'
= OJ.
But (j0' = 0 means O'iz)  O'k(Z) = 0 on Uj n Uk' Hence O'(z) E reX, f/), defined by O'(z) = O'/z) when Z E Uj , is meaningful. This proves HO(dI!, f/) = reX, f/) and implies HO(X, f/) = f/). Q.E.D.
rex,
PROPOSITION
COROLLARY.
2.2.
Ho/I: Hl(UU, f/)
H'(X, f/)
+
HI(X, f/) is injective.
= U Hl(UU, f/). 0/1
Proof (of the proposition). Suppose hE H'(Ulf, f/) = Zl(Ulf)jDCO(i1l/). Then h = {O'ld, O'jk Ere Uj n Uk' f/) where O'ij + O'jk + O'kj = O. We want to show that no/lh = 0 implies h = O. no/lh = 0 means Ii = 0 and this is true if and only if n~h = 0 for some "Y, "Y > Ulf. Let "If! = {Wo.1 Wi;. = U j n Vl}' Then "If! is a locally finite refinement of "Y and n~h = n::;:. 0 n~h = O. Also "fII > Ulf since "fII jl c U j and we can use the maps(iA) = i in the definition of refinement. Then we have where
= 'iljll = "Wi" n Wj,. O'ij. Then n~ h = 0 implies {, jljll} = D{, il}, that is, 'iljll = 'jll  ' i l ' Since 'Wil = rW,,,nWi,,O'jj = 0, we obtain 'ill = 'U on W il n Will' U i = UlWil , and ' i = 'ilL on Will defines an element, jEre U i' f/). Then the equation 0' ij = ' j 'i implies h = O. Q.E.D. '(il)(jll)
Consequently, in order to describe an element of H'(X, f/), it is sufficient to give an element of Hl(Ulf, f/) for some Ulf.
EXAMPLE. dime HI(M, (1) Proof
Let M
=
{(Zl' z2)llzti < 1, IZ21 < 1, (Zl' Z2) i= (0, O)}. Then
= + 00.
Set
= {(ZI' Z2) I (Zl' Z2) EM, Zl i= OJ, U 2 = {(z" Z2) I (z" zz) E M, Z2 i= O}. UI
3.
35
INFINITESIMAL DEFORMA nONS
I n this case M = VI U U 2 so chose as covering Ill! = {V I, U 2}' Then HI(Ill!, &) = ZI(Oll, &)/tJCo(lll!, (9) where ZI(Ill!, (9) = {0"121 0"12 E nV I n V 2 , &)} ,CO(<J/, (9) = {T IT = (T I, T2), TILE n V Il' &)}, and tJCO(<J/, &) = {T2  Til TIL E r(VIl' (9)}. We note that V, n U 2 = {(ZI' a Laurent expansion for 0"12
Z2)
+00
O"dz) =
L m=co
10< IZII < 1,0 < IZ21 < 1}, so we have +00
L
"=00
amnZ~Z2'
'I IS holomorphic on VI = {(ZI' Z2) 10 < IztI < I, IZ21 < I} so T,(Z) = L'~~<XlL:=obmnz'rz~. Similarly for '2' T2(Z) = L~=oL:=<Xl<Xlcmnz7z2' and T2  TI = Lm?;Oorn?;O amn Z7 z2' Then H'(Ill!) ~ {0"121 0"12 = L;;;l<Xl L;:  <Xl amnz'rz;}. Hence dim H'(OlI, f/') = +00 and since HI(<J/, f/') S; H'(X, f/'), dim HI(X, f/') = + 00. Q.E.D.
2.3. If H'(Vj,f/') =0 for all H'(X, f/') where <J/ = {V j }.
PROPOSITION
VjEIll!, then H'(<J/,f/')~
Proof We already know that H'(Ill!, f/') s; H'(X, f/'). Hence we only need to show the following. Let 1/ = {V).} be any locally finite covering. HI(Oll) + Let 1f/ = {Wj).1 Wj). = U j n V;.}. Then it suffices to show that H'("If/") is surjective. Take a lcocycle {O"P,kv} of Hl(1f/) where O"Ujll + O"jllkv + O"kvi). = O. Then {O"WIl} for each fixed i is a Icocycle on the covering {Wi).} of Ui' Since HI(U;, f/') = O,HI({Wi).},f/') s; HI(V;, f/')givesHl({Wi)'}, f/') = 0 for each i. This implies the existence of Ti). E r( Wi)., f/') such that 0" WIl = Till Ti).. Let, be the Ocochain {Ti).} on "If/". Then {O";)'kJ = {O"i).kv}  (;, defines a Icocycle on "If/" which defines the same cohomology class in HI ("If/") as 0". From the definition of T we see that O";).ill = O. So O"i).ill + O"illkv + O"~vi)' = 0 yields O"illkv = O"i)'kv' Similarly, 0" i)'k. = O";llkw' Hence, O"ik = O"i)'kv = O"ivkv' and O"ik E r( Vi n Uk, f/'). Now we have found O"ik so that n~'(O"ik) = O";)'k., and {O"i)'kv} is cohomologous to {O"mv}' Hence n~ is surjective. Q.E.D.
n: :
3.
Infinitesimal Deformations
Using cohomology groups we will give an answer to the following problem: Let .A = {M r/t E B} be a complex analytic family of compact complex manifolds Mr and let t = (tl, "', tn) be a local coordinate on B. The problem is to define (aMr/at·). For this we define the sheaf of germs of holomorphic vector fields. Let M be a complex manifold and let Wbe an open subset of M. Let <J/ = {Vj, z)
36
SHEAVES AND COHOMOLOGY
be a covering of M with coordinates patches with coordinates p + Zj(p) = [zf{p), ... , z7(p)]. A holomorphic vector field 0 on W is given by a family of holomorphic functions {OJ} on W 11 U j where
a L (}j(p)ex a= OZj n
() =
I
on W 11 Uj • These functions should behave as follows: On W
Uk,
0
n
() =
11
L O~(p) OZk
{J •
{J=I
We want
so the transition equation (1) should be satisfied on W 11 Uj 11 Uk. 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 E> 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 rlotJ. First we consider the case B = {tlltl < r} ~ C. vi! is a complex manifold and w: vi( + B is a holomorphic map satisfying the usual conditions
Mr=wI(t); (2) the rank of the Jacobian of w = 1 = dim B. We can find an B > 0 small enough so that wI(Ll), Ll = {tlltl < e} looks as follows: (1)
J
wI(Ll) =
UlJlI j= 1
j
(a union of a finite number of open sets).
On each OIJ j there should be a coordinate system
p + [zJ(p), ... , zj(p), t(p)], where t(p) = w(p) and such that OIJ j = {pllzj(p)1 < Bj. It(p)1 < B}. We write p = (Zj' t) = (zJ, ... , z'j, t). This construction is possible because rank =I These charts are bolomorphically related so
w
zj(p) = fjk[Z~(P)' ... , z~(p), t(p)] on Uj
11
Uk. Let Urj
=
Mr
11
IJlI j , It I
;,.(,., t), t] = gj(C, t) = zj =
/'}k(Zk> t)
= /'}k[gk(C, t), t] on "f/"). n "f/" •• Differentiating we obtain
I
ogi oq>~v + agi = I a/}k agf + a/jk . a,~ at ot oze at at
(2)
Then (2) implies [multiplying by (a/ozj)]
0)
" azi ( ag~ "a/}k a " azj ( a ) aq>t L.. oze azj . at + L.. at azj = L.. o'~ azj
"ogj a
at + L.. at azj'
(3)
Hence,
'l;'v
[_a_J 0
[_0_]
= " ag~(\.) + "L.. iJg~().) "L.. p +).v, at OZS(;') at aZS(v)
(4)
on "f/";, n "f/" v' Therefore if we let
O;.(t) =
L ag~().) r~l' at
oZso.)
Q.E.D. So we see that the infinitesimal deformation, dM,/dt E H 1(M" 0,} is determined uniquely by the family vii = {M, I t E B} and is thus well defined. If we introduce new coordinates on B. t = t(s) so that t'(s) ¥ 0 then the relation (5)
is obvious. Now to return to the more general case, let {M, 11 E B} be a family where B is now a general connected complex manifold. Let Ll be a coordinate neighborhood around bE B and let (t 1, ..•• t m ) be local coordinates. Then
3.
INFINITESIMAL DEFORMAnONS
39
we may assume l:!. so chosen that wI(l:!.) = U/'ltj' a union of finitely many coordinate neighborhoods on each of which there are coordinates (z), "', zj, tl, "', t m), where rJlt j = {(Zj' t)llzjl < Bj' tEl:!.}. Again we have transition functions fjk
zj = fjizk' tl, "', tm ) on rJlt j n rJlt k . DEFINITION 3.1. {8 jk I.(t)} where
(aMr/at') E HI(Mr' 0 r)
8.
Jklv
is the
cohomology
class
of
(t)= ~ afjizk' t)(~) ~t'
L...
«=1
~ « • uZj
u
If (a/at) denotes the tangent vector
a
m
a
at = V~I c. at
V '
then we define
aMr = ~ c aMr at V~I' at' . We make the following definition: DEFINITION 3.2. .A = {M r 1 t E B} is locally trivial (complex analytically) if each point b E B has a neighborhood l:!. such that wI(l:!.) = Mb x l:!. (complex analytically). This means that we can choose coordinates (zj, t) such that, zj = fMzk' b) (independent of t). If .A is locally trivial, then each Mr is complex analytically homeomorphic to M 0; hence M r is independent of t. PROPOSITION 3.2.
If .It is locally trivial then (aMr/at V ) =
o.
Proo}: Trivial. We mention here a theorem of W. Fischer and H. Grauert (1965).
THEOREM. If each Mr is complex analytically homeomorphic to Mb , then .It is locally trivial. We now study some examples:
EXAMPLE I. Let R be a compact Riemann surface. Fix a point a E R. Let w be a coordinate in a neighborhood of a point bE R such that w(b) = O. We define a family {Mr) as follows: M r 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: r = O?" Define the following neighborhoods on R: Wb = Wo = WI =
{wllwl < r}, {wllwl < r/2}, {wi r/4 < Iwl < r}.
We can write Mr = Uo U UI U U2 ••• Uj '" where Uo = nI(Wo), UI = nI(WI ), n(U j ) n Wo = 4> for j # 0, 1 and nis the map 1t: M, + R defined by the covering map n : Rp + R. We introduce local coordinates as follows on M,,(tE!Wo): Zo
= J w  t on Uo ,
ZI
=Fw
on U I ,
and Zj on Uj can be an arbitrary coordinate which should be fixed and independent of t. Then we have Zj = jjiZk' t) for holomorphic!jk' In fact, Zo
= 101(zl' t) =
Jw  t = Jz~  t,
and Zj
= jjk(Zk) (independent of t)
for (j, k) ¢ {CO, 1), (1,0)}. Then OCt) = {Ojk(t)} has only one nonzero component,
00l(t) = 0101 . (~) ot ozo
=_
1 2J zi

(~) = __ 1 (~)
t ozo
2zo ozo .
Let Vo = Uo , VI = UJ ~ I Uj • Then OCt) is a 1cocycle on the covering "f/ = {Vo, Vd; OCt) E HI("f/, 0/) ~ HI(M" 0/). Suppose dMrldt = O. Then there are holomorphic vector fields Oy(t) on Vy such that
so
(6) We make the definition
3.
INFINITESIMAL DEFORMATIONS
41
Then y/(t) is a vector field on M, which is holomorphic on M,  {p} and has a simple pole at p. If the genus 9 of R is
LEMMA 3.2.
~
1, then such vector fields '1 do not exist.
COROLLARY. If 9 ~ 1, then dM,ldt =1= 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 holomorphicdifferential o. Since Y/ = y(z)(dldz) 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'l U U'2 where each V'v = C X ~I and (ZI, (1)(Z2, (2)
if and only if '1
= Z2"2 + tz~,
and
ZI
= l/z 2
•
We are assuming m ;::: 2k, k ;::: 1. Then M, is independent of t
dM,ldt = 0 for t
=1=
=1=
0 for t =1= 0 so
o.
(For this, one could use the theorem of Fischer and Grauert.) What is dM Iidt 1'=0 ? Consider the covering of M 0, 6lf = {U 01, U02}; then
dM,ldtl,=o = 0(0) E Hl(6lf, 0 0)
£;
HI(Mo, 0 0 ).
Then
so that
012 (0) = Suppose dM ,Idt
(a/:2) at ,=0 (a) a'i = k(a)
= 0 at t
Z2 0'1
=
E
f(U ol n V 02 ' 0).
O. Then 012 (0) = O2

01
where each O. is a holomorphic vector field on V o•
= C x IP' 1.
42
SHEAVES AND COHOMOLOGY
LEMMA 3.3.
Any holomorphic vector field on C x IPI is of the form
0= g(Z)(:z)
+ [a(z)(2 + b(z)( + C(z)] (:,),
where g, a, b, care holomorphic functions on C. Assume Lemma 3.3. We have the following relations:
(a~J = z~(o;J, (o~J = mZI'I(a~J  ZT(o~J,
(7)
where (zv, C) are coordinates on UOv = C X IPI. Let us compare the coefficients of (0/0(1) in O2  01 and 0 12 (0). From Equations (7), we get z~
=
Z~C2(ZZ)  CI (ZI)
= Z2 c2 (zz) In

C I ( ;1 ;) ,
where the dz..) are entire functions. Expanding,
and 0
< k < m. This is impossible. Hence, dM t I/1;0 =1= O. dt
For the lemma we have:
Proof Let (z, 0 E C X IPI, where ( is a nonhomogeneous coordinate on At (= 00., the local coordinate on IPI is 1] = 1/(. Restrict the vector field to C x IPI  C X {oo} = C 2 • Here
IPI.
0= g(z, ()(:z) + h(z, ()(:,), where 9 and hare holomorphic on C 2 . At 00 we have
() = y(z, 1])(:z) + (J(z,
1])(:1])'
where' = 1/1] and y, {J are holomorphic. Then O(fOI] = 1/1]2 so (0/01])  (z(%O. Hence at 00,
0= y(z,
IJ)(:J  (2{J(Z, 1])(:,),
=
3.
43
INFINITESIMAL DEFORMATIONS
since g(z, ') = y(z, 1'/), g(z, ~) is holomorphic on IC x [F»1. So g(z, as a function of , g(z,
0 is constant
0 = g(z).
Finally, h(z, 0 = _,2{3(Z, 1'/) 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) = VI U V 2 , Vv = IC X IPI, and
if and only if
We must count the number of parameters involved HO[M(m), 0]. By the lemma,
In
representing a
() E
on each Vy , and (}1 = (}2 on VI n V 2 • Changing coordinates,
(a~J = z~(a~J, Hence
+ (a2 zim'i + b 2ZT'1 + C2)ZI2(~) aZI
=
Zfg2(a~J + [a 2zT'f + (b 2 + mz lg2)'1 + C2Z~m](a~J
= (}1
=
gl(a~J + (al'i + bl'l + CI)(a~J·
Equating coefficients,
g\(z\) = Zigz(Z2)' al(zl) = ZTa2(Z2), b 1(z\)
= b 2(Z2) + mz t gz(z2), Ct(ZI) = Z~mC2(Z2)'
SHEA YES AND COHOMOLOGY
44
These functions are all entire functions of ZI. Let us investigate their behavior at ZI = 00. Since Z2 = lizi. we see that gl 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
= glo z~ + gl1 Z 1 + g12, a l = aloz'~ + ... + aim, C I = 0 (by Liouville's theorem).
gl
Consider the b terms: co
bl
1
co
= L blnZ~ = L b2n ;; n=O
()
mg ll

ZI
.=0
So bl(zl) =  mg lo zlb lO . blO). Hence,
mgloz i

1 mg I2  · ZI
depends linearly on (glo, gil' g12,
alO, ... , al lll
,
(8)
We therefore have: THEOREM 3.1.
M(m) =1= M(n)
(complex analytically) if n =1= m.
REMARK 2.
M(2n) =1=
REMARK 3.
Let {M t It E C}, M t given by
as before. Then Mo
M(2nl)
=
Mt lll ),
topologically.
M t
=
M(m2k)
dMtldt = { 0, =1=0,
for t =1= O. And we have shown
for t =1= 0 for t = o.
Suppose we "reparametrize" and consider {Ms21 SEC}. Ms2 is defined by
1
ZI
Then Mo
= M(III),
Ms2
=.
= M(III2k), S =1= 0 as above.
Z2
But
dMs2 dM t ds 2 2s dM t =_·==0 ds dt ds dt for all sEC. We know that M t independent of t implies dM tldt = O. We have just seen that dM tldt = 0 does not imply that M t is independent of t. However, we have the following theorems:
3.
INFINITESIMAL DEFORMA nONS
45
THEOREM cx. If dim Hl(Mn 0 t) is independent of t and if aMt/at V = 0 for all v and t, then {M t t E B} is locally trivial and hence M t is independent of t. THEOREM p. The function t function of t. That is
+
Hl(Mt, 0 t) is an upper semicontinuous
dim Hl(Mt, 0 t) ~ H 1(M., 0.), if t is in a sufficiently small neighborhood of s; that is, lim dim H 1(M" 0 r ) ~ Hl(M .. 0.).
t.
THEOREM y. of s.
If H 1(M., 0.) = 0, then M t = M. for t in a small neighborhood
Theorem IX 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 IX 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 t It E B} such that Mto = M, we can find a neighborhood N of to such that M t = Mto for tEN. (More precisely, if w:..It + B is the family {M t}, then w 1(N) = N x Mto complex analytically.) The following theorem follows from Theorem y. 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) =/; O.
REMARK. IP n is rigid. For n 2 2 the only known proof is to show H 1 (lP n , 0) = 0 [Bott (1957)]. Let us proceed to the proof. Proof (of Theorem 3.2) sists of two elementary ideas: (1) (2)
The proof will be elementary in that it con
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
SHEA YES
46
AND
COHOMOLOGY
dim B = 1. The result is local so we may assume B = {tlltl < r} and to = O. We can cover Wl(~£), ~. = {tlltl < e} with coordinates flItj
= {(Zj' t)llzjl < ej , It I < e}.
Then
zj = fjk(Zk, t) on flit j n flit k' M is covered by u UJ = M where j
UJ = {zjllzjl < ej}X{O} s; flIt j .
Then M x B = u(UJ x B) where for (w., t) E U? x B, j
if and only if
that is,
wj = gjk( wk),
(9)
We can rephrase our result: THEOREM. If b is sufficiently small there is a biholomorphic map cp of Wl(~6) onto M x ~6 such that cp: maps wl(t) onto M x t and cp: M = (i)1(0) ~ M x 0 is the identity map. Suppose we choose b so that cp maps
u2
u2
into x B, cp(flItf) s; x B. Let (Zj' t) E flIt~. Then, cp(Zj, t) = (w j , t) = [cpiZj, t), t] so on each flIt~, cp is represented by holomorphic functions cp~(Zj' t) where Cpj(Zj' 0) = zj. On flIt~ n flItf,
zj = nk(Zk, t) ; so
implies
(10) Therefore we see that we can prove the theorem if we can construct holomorphic functions cpj(zJ' t) on flIt~ satisfying (10) and (11)
3.
INFINITESIMAL DEFORMATIONS
47
For simplicity we may as well assume that IlIi j is of the form IlIi j
= {(Zj' t)llzjl
O} and U~ ~ M n IlIi j • If expand lPiZj, t) into a power series, we get lPiZj, t)
= z} + ({JJ/1(Zj)t + ({JJ/z(z})t Z + ... + ({JJ/m(z)t m + ... ,
(12)
where each ({J iI m(z) is a holomorphic vector valued function. If we expand both sides of (10) we get 00
00
I Fm«({Jm, "', ({Jjim) = m=O I Gm«({Jkll' "', ({Jklm)t m, m=O
(13)
where Fm and Gm are polynomials. We introduce some notation: If pet) = Pn t" and Q(t) = L Qn t n are two power series, PCt) == Q(t) means Qn = P n m
I
uij to n = m [that is, pet) == Q(t) mod (tm+l)J. Therefore, to solve (formally) Equation (13) we need only solve ({Jj[Jjk(Zk, t), t]
== gjk[({J~(Zk' t)], m
for each m, where ({Jj(Zj, t) =
Zj
+ ... + ({Jjlm(Zj)t m.
(14)
First consider m = 1. We have 00
Zj
= !jizk> t) = 9 jk(Zk) +
I
!jklm(Zk)t m •
m=1
Using (I 3)10 gjiZk)
+ !jkll(Zk)t + ({Jjll[gjk(Zk)]t == gjk[Zk + lPkll(Zk)t] 1
So
Now ejk =
~ (a~:k)t=o(a:j) = L !jk l l(a:j)
belongs to H 1 (M, 0). By assumption Hl(M, 0) = 0 so {Ojk} is cohomologous to zero. But Equation (13)1 says we must find {({Jk (1} so that ()jk = ({Jk 11 ({J j II' HI (M, 0) = 0 allows us to do this so the first step in an induction proof is completed.
48
SHEAVES AND COHOMOLOGY
Assume that ({>j(Zj' t) are determined so that Equation (13)m holds, that is, ({>m,,(!jk)gjk«({>'k) == rjkt m+I , We must show that ({>jlm+I(Zj) can be m+1
' d so t h at d etermme
({>jm+l
= ({>jm + ({>j/m+lt m+I' satls fies (13) m+l> W h'IC h'IS
({>j+ 1 [Jjk(Zk , t), t]  gjk«({>'k+ I ) == O. m+l
This is equivalent to ({>j(fjk)
+ ({>jlm+l[Jji Zk, t)]t m+1 ==
gjk[km) gjk
OZa.
+ k.:;p({>klm+l ~ j (J ()tm+ 1 Zk ' vZk
Here we have used
and ({>'k(Zk, t) = Zk
+ ....
So if we can solve oza.
rjk(Zj) =
L vZk~ ({>klm+I(Zk) :l
({>jlm+l(Zj),
(15)
(J
for ({>Jlm+l we will have
(l3)m+l'
r jk
({>jllll+1
Let
= ~ rMZ)(o~j)' =
~ ({>jlm+l(Z)(O~j)'
Then we want to solve r jk = ({>k I m+ 1 ({>j Im+ I' We claim that {r jk } is a cocycle [belongs to Hl(M, 0)]. Then we would be done as before, since Hl(M, 0) = 0 and {rjd must be a coboundary, r jk = Ok  OJ, and set Ojlm+l = OJ, LEMMA 3.4, That is,
{r jk} is a cocycle, that is, r
ik
=
r ij + r jk on 0lI i n
IIIJ j n CIlI k n M,
3. Proof
49
INFINITESIMAL DEFORMAnONS
By definition
rik(Zj)t m+ 1 == q>~[fik(Zk' t), t]  gik[q>r(Zk, t)], m+ 1 so
gik(q>r) = giJgjk(q>r)] == giJq>j(fjk)  r jk t m+ 1J. m+1
Now rJk t m+ 1
== q>j(jjk)  gjk(q>r) and
m+1
So
By assumption
q>rEfij(zj, t), t]  YiJq>j(Zj, t)] == rij(z)t m+ 1, m+l
== rij[fjk(Zk, t)]tm+ 1 
••••
m+l
Hence
Q.E.D. This finishes the construction of the formal power series
q>j(Zj, t) zjEiJ{tj
n M,
iJ{tj
=
Zj
+ q>j\1(z)t +"',
n M = {zjllz}' < I}
such that q> Jjjk(Zk, t)] = gjk[ q>k(Zk' t)] as formal power series. LEMMA 3.5.
The power series q>j(Zj, t) converges for
It I < ()
for some small
() > O. Proof We dominate q>j with a convergent series. We fix some notation. Let t/I(z, t) = L~=o t/lm(z)t m be a power series where t/lm(z) =[t/I~,(Z), "', t/I:,(z)] Z E U. Let a(t) = L~=o am t m , am :c: 0 be a series with real, positive coefficients. We write t/I(z, t) ~ aCt) and say that a(t) dominates t/I(z, t) if
50
SHEAVES AND COHOMOLOGY
It/I~nCz)1
::; am for all
max sup a
It/I~(z)l.
Z E U and all C( Consider the series
=
1, ''', n. The norm of
t/lm is It/lml =
zeU
b
(et)m
ro
L
A(t) = 16 2 ' e m=1 m
where band e are constants to be determined later. Then b {
A(t)
converges for
emItm}
ro
= 16 t + ml;2 ~
It I < I/e. In Lemma 3.5 it suffices to prove t) 
Illt. Then H'J(Uli, [Il)~Hq(OZt, [Ill!)
jn'l
1V
jn'll
iV
Hq("lf!, [Il)~ Hq( "/f/ [Ill!) commutes. Hence h induces a homomorphism h : Hq(X, [Il)
+
Hq(X, [Il").
58 THEOREM 4.1.
SHEAVES AND COHOMOLOGY Assume that
o+ fI"
~ fI' ~ fI'"
+
0
is exact. Then there is a homomorphism b* such that
is exact.
Proof i is injective so fI" ~ i(fI") c fI', and i(fI") = ker h. Thus we consider fI" c fI' where fI" = ker hand i is the inclusion map. Recall that HO(X, fI') = Zo(X, fI') = reX, fI'). Since reX, fI") c fI'), we see that
rex,
0+ HO(X, fI") ~ HO(X, fI') is exact. If a E reX, fI'), then ha = 0 if and only if a E reX, fI"); so HO(X, fI") ~ HO(X, fI') ~ HO(X, fI'") is exact. LEMMA 4.5. HO(X, fI') ~ HO(X, fI'")~ H1(X, fI") is exact (where we must define b*).
Proof Let a" E reX, fl'1I)' Since h is a local homomorphism there is a section Ty Ere Uy, fI') 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 Olt = {Uj } of {Uy}, that is, there is a map j + y(j) such that Uj £ Uy(j)' Set t j = rVj ty(j) E reUj, fI'). Then h t j = a" where defined. Let eO = {tJ E eO (Olt, fI'). Then: DEFINITION (4.4)1' b*a" = [beO] E H1(X, fI") where for any cq E zq(Olt, fI'), Ceq] denotes the cohomology class in Hq(X, fI') of cq. (One should check that b* is well defined.) Since beo = {Tk  tj} and hTk  htj = a"  a" = 0, we see that beo E ZI(01I, [/') so Definition (4.4)1 makes sense. Exactness means b*a" = 0 ifand only if a" = ha for some a E reX, [/). So suppose t5*all = [beO] = O. Then &0 = be o' where eO' = {tj} E CO(Olt, fI"). SO CO  co' = a E ZO(X, fI') = reX fI'), and ha = hco = {ht j } = a". Now suppose a" = ha. Then h(t j  a) = 0, so t j  a E reUj, [/'). Set c~ = {t j  a} = Co  (J E CO(Olt, fI"). Then bc~ = bC o since (J E ZO(X, fI') and hence b*a" = [bco] = [bc~] = O. Q.E.D. We now turn to check that
4.
59
EXACT SEQUENCES
is exact. Take e l ' EZI(OU, f/'). If [el'] = (j*(J" = [(jeO], then i[e l '] = 0, and if i[e l '] = 0, then e l ' = {)co; so 0 = he l ' = Meo = (jheo. Thus heo defines an element (J" E r(X, f/"). By definition, [el'] = (j*(J". We want to prove exactness
LEMMA 4.6. Given e4 " E Cq(OU,f/"), then we can find a locallyfinite refinement "fI/ and eq E C q("fI/, f/) so that eq" = heq.
n:,.
Proof We give proof for q = 2. Let OU = {U), eq" = {(J;:"d, where (J;:"k E r( U i n U j n Uk' f/"). Choose a covering "f/ = {Vj} such that Vj c UJ • Since OU is locally finite, a given Y E X belongs to only finitely many Uj • We choose a neighborhood Ny of y sufficiently small so that if Y E Uk n Uj n Uk there is T E r(Ny , f/) with (J;:"k(X) = hT(X) for Ny (remember h is a local homeomorphism), (2) for each y there is Vj such that Ny c Vj , (3) if Ny n Vj =F
OU. Define T = {T).l'v} E C 2 ("fI/, f/) as follows: we have W). c Vi' WI' C Vj , Wy C Ck where i =j.. , j =j.. , k =j•. By (3) if Ny). n Vj =F G. the cocycle {h jk} where
This is a
hjk(z) =
en+ m
bundle which is defined by
(~k g~J
(2) Tensor Product F® G. This is a h jiz) = Jjk(Z) ® g jk(Z). Recall that •..
enm bundle defined by hll nl
••.
{h jk } where
hll) nm
, h~:
and here h'j;PIl = /jkP g7kll. A point in F ® G has coordinates (z") I, •.• , m , ••• , 'T), where (z, '1) and (z, 'k) are identified for Z E Vj n V k if and only if
'1
,j). = L fjkp(z)gfkll(zK~Il. Dual bundle F* of F. This is the bundle defined by {fjt} where (z, '1 ..) is identified with (z, if and only if
(3)
fjt = (fjk I)' = (fkj)' which is the transposed inverse of Ijk· Then
,:P)
'r
Sometimes we write
=
L fJMzK: P = L ffj..(zK: p•
'T. for n". Then we have '1.. = P=I L" nk. (zK:p •
(4) {fjk}.
Complex Conjugate F of F. This bundle is defined by the co cycle
Let us now define subbundles and quotient bundles. Suppose that, by a suitable choice of Olt = {U j} and of fibre coordinates ,;, the matrices Jjk in the lcocycle {fjd defining F can be written as follows:
66
°°
SHEA VES AND COHOMOLOGY
Hencefj~p(z) = for I ~ f3 ~ m, m + I ~ ex ~ n. Thus(j = Lp=m+t!jk/l(f for ex> m, and if (f = for f3 > m, then (j = 0 for ex > m. Let F' = u U j X em where em = {«(J ' .. " (j, 0, .. " O)} !;; and we identify (z, () and (z, (k) if (j = A jk(Z)(k' Then F' is a subbundle of F. The quotient bundle F" = F/F' m bundle defined by the lcocycle {C }. is a jk
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 ncp(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 (9(F) (or !,)(F» for sheaf over M of germs of hoi om orphic (or differentiable) sections of F. Then locally, (9(F) = (91 U j EEl" . EEl (91 V j (sum ntimes), where {!} 1Vj means (9 restricted to Vj and (!}z(F) = {!}z EEl'" EEl {!}z (ntimes). We now review tangent bundles and tensor bundles. Let M be a complex manifold and {VJ an open covering of M with coordinate patches with coordinates (zj, .. " zj) on V j • A (holomorphic) tangent vector at z is an element of the form v = L:= 1 (j(%zj). It is easy to see that the set Tz(M) of all complex tangent vectors at z is a complex vector space TiM) ~ en. If z E V k another chart at z, then we identify Lp= 1 W%zf) with L (j(%zj) if n
(j =
L fjk/l(zK~,
P=l
This is a linear identification so the vector space structure of Tz(M) is well defined. The set T(M) = UzeM T.(M) is a complex analytic vector bundle defined by the Icocycle {fj~p(z)}. T(M) is the holomorphic tangent bundle of M. T(M) is the conjugate (holomorphic) tangent bundle of M. And !Y(M) = T(M) EEl T(M) is the (complexified) tangent bundle of M. Then 0, the sheaf of germs of holomorphic vector fields, is (!}(T(M». If M is a differentiable manifold with local coordinates (xj, .. " xj), then the (complex) tangent bundle !Y(M) = UxeM!Y iM), where
The real tangent bundle !Y R(M)
=
UXEM !Y xR(M) , where
The relation is !Y(M) = !YDiM) ®R e, where bundle over M considered as a real bundle.
e is
the trivial (complex) line
5.
67
VECTOR BUNDLES
Let T*(M) be the dual bundle of T(M). If Tz*(M) the transition relations are
= {(C;t> "', C;n)}, then
We use the following notation: An element U E Tz*(M) shall be written = L~ C;~ dzj, where dz'j(%z1) = op as an element of Tz*(M). Briefly let T = T(M), T* = T*(M). A tensor bundle is a bundle of the form
U
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 holomorphic bundle so (® T)P ®(® T*)q is a holomorphic bundle but (® T)P ®(® T*)q ®(® T)r ®(® 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, cp j~, ... ~pPI ... P.(z)] of (® T*)P ®(T*)q over W such that the fibre coordinates cP jll, ... ~pPI ... P. are skewsymmetric with respect to 0(1 ••• O(p PI ... Pq • If p = 1, q = 0, Proof
for Izl < R  e,
°and R  e > 0. (of Theorem 6.1 ) We may as well assume U
= UR = {zllzl < R, where
lal
= maxlz"l}.
Then we want to prove that if oq> = 0,
1 q.
q> = ,
L q>P! ... p
dzP!
1\ •.. 1\
dzP,
q
is a Coo(O, q)form on U R , then for any e > 0, R  e > 0, there is a COO(q  1)form l/J on UR  t such that
ol/J
=
q> on U Rt
(q ~ 1).
Suppose
By this we mean that form q> 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
Hence, (ojoz")q>, ... q =
°for
(X
~
q + 1. Define
If
g ( z ' , ... , z") = 1
q>,oo.i('Z2'···'Z")dJ'd y , .. ",. \,z
TC
~2 + ~2
< R  (tl")
78
SHEA YES AND COHOMOLOGY
ThengisC'" on UR(£/R) and og/OZI = (P!. .. iz).SoletljJ(z)= g(z)dz 2 dz q • Since O(p/oz« = 0 for IX 2 q + 1, og/oza = 0 for IX 2 q + 1. Thus,
±
aljJ(z) =
a= 1
0:« dz a OZ
/\
dz 2
/\ ••• /\
/\ ••• /\
dz q
= <po Now assume the lemma is proved for m
= 1, p.
"En L.
"1'(11 '" /J.
~
k  1 and consider m = k. So,
dZ(11 /\ ... /\ dz(1.
/J1';'kl
and
Thus, 0 = (%z«)
is holomorphic and if dq> = oq> = O. The rest of the proof uses the observation made just before the statement of the lemma. The details are essentially the same as in the previous proofs and are left to the reader. d
THEOREM 6.4. is exact.
OIC
REMARK.
is not a fine sheaf.
QP
(!)
d
0' 0 2 
••• +
0"+ 0
Now we consider holomorphic vector bundles F over M. Let F be defined by the lcocycle {fjk(Z)} wherejjk(z) = [fjkP(zHz,p= ' ..... m' On each coordinate patch (!)(F) = (!) E9 ... E9 (!) (m times). Let q> be a section of (!)(F) over an open set W ~ M. On W n UJ '
(p(z) = [cpJ(z), "', cpj(z)], where CPJ is a holomorphic function on W n U j' and
cp7(z) = L JJkl'(Z)CP~(z)
for z E W n U j n Uk'
I'
By a (p, q)form cp with coefficients in F over W we mean cp(z) = [cpj(z), .. " cpj(z)] where each q>1(z) is a (p, q)form over W n Uj such that m
cp;(z) =
L fMz)q>~(z)
1'=1
SHEA YES AND COHOMOLOGY
82
as differential forms for Z E W n U j n Uk' We define Ap,q(F) to be the sheaf over M of germs of (p, q)forms with coefficients in F. At each point x E M, the stalk A~,q(F) = Af,q EB ... EB A~,q (m times). We have: iJ
THEOREM 6.5. 0 ~ (!)(F) ~ AO(F)  Ao,I(F) ~ '" ~ oAO,n ~ 0 resolution of (!)(F).
IS
a fine
Proof We remark on the definition of 0 and leave the proof to the reader. The functions !jk are holomorphic so (O/OZ)Jjk(Z) = O. Hence,
So 0 is well defined and if cp is a (p, q) form with coefficients in F, ocp is a (p, q + l)form with coefficients in F. We note that Ocp~(Z) =
2: f;k,,(Z) ocpHz) + 2: af;k"(Z)CP~(z), "
11
and ocp is not well defined on AO,p. We also notice that Ap,q(F) = Ap,q®(9(!)(F). Remember that cp(p,O) is a differentiable section of T* ® ... ® T* which is skewsymmetric. We define T* /\ ... /\ T* to be the subbundle of T* ® ... ® T* consisting of those (z, (. Let {p j} be a partition of unity subordinate to {U j}'
J 0 hence M is oriented and (x] , ... , xJn) is a positively oriented chart.
Proof
(of Theorem 1.4)
Let w = iL9"1l dz"
A
dzlJ. Then
90
GEOMETRY OF COMPLEX MANIFOLDS
We claim
f
w" > O. For
M
w" = (i)"
11) sgn (If3 L sgn (a1I ... ... all I
a. fJ
... f3n) g~ PI ... 1
..."
and
1 ...
n)
= sgn ( f3 ... f3 I
=
where 9
II
gIlT ... g"ii
n)
1 ... sgn ( f31 ... f3" g,
= det(gaP). Therefore, w" =(i)"n!g dz l
1\ ••• 1\
dz ii •
Since
dz l
1\
dz T = (dx l
+ i dx 2 )
= 2 i dx 1 w"
=
1\
2" n! 9 dx 1
1\
(dx l

i dx 2 )
dx 2 , 1\ ••• 1\
dx 2 ".
We have assumed (gap) is positive definite so 9 > O. Thus SM of > O. If w" = dlji then M w" = 0 by Proposition 1.3. Thus w" =f. drj;. In fact, we claim w 1\ ..• " w = wk =I drj; for any Iji. Since dw = 0, d(w k ) = 0 and if w k = drj; then w" = wk 1\ W" k = drj; 1\ W,,k = d(rj; 1\ W" k ). Now recall that
J
.
2k
•
HO(M, dA zk 
b u = dime H (M, IC) = dime dHo(M, A 2k 
1)
1)"
The facts just proved show that w k E HO(M, dA 2k 
wk
Thus, bZk
I ),
rt dHo(M. A lk I).
z I.
THEOREM 1.5. If M is compact Kahler and if N eM is a compact complex submanifold, then N is not homologous to zero in M. Proof
(sketch)
We first prove:
I.
HERMITIAN METRICS; KAHLER STRUCTURES
91
PROPOSITION 1.4. (Stokes' theorem) If W,,+I is a compact differentiable manifold with boundary aw,,+1 = M", then Jw"+! dlj; = JMn Ij; for any nform Ij; on W= W,,+I.
Proof We cover W with a locally finite family of coordinate patches {U j } such that U j = {xjllxjl < r j } if Uj ~ int(W) and if Uj n M =I cp, then Uj={xjllxjl I, rj<x)::;rj }, and MnUj={xjlxJ=,). We choose a partition of unity {Pj} with the following properties: (I) supp p) = {x I pix) > O} c Uj if Uj n M = cp. (2) supp Pj £:; {xjlrj < x) ::; r j , Ixjl < r j , ex ~2}. (3) Pj is Co, Pj ~ 0 and L Pj = I.
Then Jwdlj; = LjJwd(pjlj;) = LjJuJd(Pjlj;). From Proposition 1.3 JVjd(Pk Ij;) = 0 if Uk n M = cp. Suppose Uj n M =I cp and Ij; = L::: Ij;j dx), /\ ... dxjl /\ dx a + 1 /\ ••• /\ dxj+l on U j • Then
and
For ex
= I,
Hence, f
VJ
d(p"")=f p )'1') .. I,I(r.) 'x~ ... ' x~+I)dx~···dx"+1 )'1' )' ) ) ) Vj
Thus,
where M =aw. For the proof of the theorem, suppose M" is compact Kahler and N m is a complex submanifold N m eM". Suppose N = aWe M for Wan embedded suhmanifold. Then if w is the Kahler form on M, 0 < IN w m = Jw dw'" = 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 QP 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, 1jJ) for cp, IjJ E r(AM), which makes r(AM) into an (incomplete) inner product space. We introduce an Hermitian metric 2 L gja.fJ dzj dzq = 2 L gap dz a dz P on M. Associated to this metric we have the form w = i L gap dz a 1\ dz P and w" = 2" n! g dx l 1\ ••• 1\ dx 2n as before where x 2 L... a j a j'ji * q> V). j ,
and since aj is nonsingular (8q»j=O Conjugate this to get
that is
Thus
ifandonlyif
o(~ajvjJ*q>j)=O.
106
GEOMETRY OF COMPLEX MANIFOLDS
and
So
and
# : .ifnp,nq(F)   + .ifp,q(F). Lemma 4.1 implies that # is a (conjugate linear) isomorphism. EXAMPLE.
Q.E.D.
Let M be a compact Riemann surface. Then HI(M, @) = HI(M, QO) ~ HO(M, QI)
and HO(M, Q I) is the space of holomorphic differentials
on M. Thus,
Further HI(M, 0)
= Hl(M,
@(T» ~ HO(M, QI(T*»
= HO(M, @(T* ® T*» which is the space of holomorphic quadratic differentials on M. Thus, dim HI(M, 0) = (
0, 1, 3g  3,
°
if genus (M) = if genus (M) = 1 if genus (M) :2: 2.
[For example, see Teichmliller (1940).]
s.
Covariant Differentiation on Kahler Manifolds
In this section we want to exhibit some of the special facts that a Kahlerian structure imposes on the Hermitian geometry of M, a complex manifold. For instance, 0 = [] = 16 holds on a Kahler manifold. First we must review the idea of a covariant derivative. Suppose ~j(z)(a/azj) = ~k(z)(a/azn is a given COO section of T. If Zj E Vi " V j " V k =1= cp, then in general,
La
La
5.
COVARIANT DIFFERENTIATION ON KAHLER MANIFOLDS
107
because
and thus, 01'~
_"'_J
= '\
OZ~
_"'_k + '\ __J_
L... OZ~ OZ;
OZ;
0 2 Z~
01'/!
_J
L... OZ~ OZ~
~e
.
We would like to define a "correction" term rj;.p(z) such that
where V
a1'~
t.
;. t.,j
"'i = ;;: + '\ L... uZ,
r.j;'fJ "'j1'P •
(J
We temporarily fix the following notation: !2(T), !2(T*) will be written
eX!
sections in !2(T), !2(T*)
L ~j(a~j). L CPj~ dz}, L I]~(a~~)' L I/Ij. dz~, respectively. We suppose we have fixed an Hermitian metric ds 2 = L 9Jali dz} /\ dz~ on M. Later we will assume that this metric is Kahler. Let V be a coordinate patch with coordinates (ZI, ... ,z"). Let A;. = %z\ 0;. = D/ai;'. The '1~ transform as follows:
'1~(z) =
t ( oz~ '1~(z). OZ~)
Since
o;.'1~(z) = t
(!;D
o;.lJf(z).
Similarly for 1/1 j~' We define
V;. '1~ =
a;. 1]1' V;. 1/1 ja. = 0;.1/1 jii. •
Let (g~~) = (g j~p) I. Then p(~) =
L" gjy/i~j dz P E f2(f*) y= I
(1)
108
GEOMETRY OF COMPLEX MANIFOLDS
and we define
that is,
Thus,
and
where G = (9«p). Similarly,
n
L q", cP jy ,
= a;. cP j« 
y= I
since "g L ",p gPY
=
fJ
{)Y
""
and 0= L a;. g«pgllY fJ
+ Lg«j1 a;.gllY. II
We also notice that if (Zl, ... , zn) are different coordinates on
so
Suppose now that
Zj
= z. Then we see that V;.
and
e" dz;' ® (~) E ~(T* ® T) az'"
5.
COVARIANT DIFFERENTIATION ON KAHLER MANIFOLDS
109
that is, ~ _ " oz~ ozj
Vjl~j 
f..., ;:) ;.;:) p /I,P uZ j uZk
P
Vk/l~k'
This remark is used in differential geometry as motivation for defining V : ~(E)  ~(T* ® E), where E is a Coo vector bundle over M a differentiable manifold. We also define V, Vl~~
=
o),~~, V),qJ~
=
o),qJ~,
V),r,jj = o),r,jj + L r~pr,P, p
V), t/fjj = 0), t/fjj 
Lp r~", t/fp.
In fact, we could define V;. qJ analogously for any tensor field qJ
E ~(T
® ... ® T ® ... T* ® ... T*),
by raising or lowering indices until p(qJ) E ~(T ® .,. T*) taking 0), and then P 1. We will not write out the result in local coordinates here.
Proof
V;.g",P=Lg"'tio;.(gtiYgyp)
= L g"'ti 0i p •
Since we also have n
V).((>a
= O).((>a 
L r~a((>p,
P=I
we get,
V V).((>IZ = rMV).((>a) y
= OYO).((>IZ 
Lp ;\rt((>p  L r~aOy((>p. p
Thus Q.E.D.
Q.E.D.
6.
PROPOSITION
Proof
6.6.
CURVATURES ON KAHLER MANIFOLDS
119
[V A' V.]CPii =  III Rl VA CPp .
Conjugate Proposition 6.5 to get [VA' V.]qJa =
I
R~VAqJp
Q.E.D. We could similarly prove [VA' V.J~a/l1
= 
I
R~J..CIl'i
+I
t
Rp,\.~at'i
t
+I
Rh.~alli·
t
THEOREM
6.1.
For any (p, q)form cP = ljp!q!
(OCP)a, .··P.
= 
I
a.1I p
I
CPa,"'p. dz a, A .• , A dz Pq .
gPaVaVp CPa, ,,·P. q
+ i=lk=lt,a L I IRta'Pkiicpa,."a,_,ta,+,·"Pk_,iiPk+'''·P.
Proof
As usual let A denote (OCP)A/lo" P.
and
0(1 •••
a p ' Then
= (1)P{Vpo CPAP'P2'"

VII, CPAPOP2'" + ... }
120
GEOMETRY OF COMPLEX MANIFOLDS
Also
so
Thus
= 
L"
a., /1= 1
gPa.Va. V/I ep AP
j
.. ,
P.
(1)
Let us calculate the second term on the righthand side of (1), For a form ep of type (l, 0)
[V,t, Vv]epa. =
L R'a.v;,ept t
and
,t For a form ep of type (0, 1) [V;" VJepp=
'I Rp'v,tept t
so
We also have ~
L. /I
R Pt p, P =
~
L.
a.,/I,y
gta.gPy R pa.pjY
L g'a.Rp,a. = Rp,t. II
Similarly, we see
L gfJa.[Va.' Vp...]ep APPj ,, PA  P.
a.,/I
(2)
6.
121
CURVATURES ON KAHLER MANIFOLDS
Since Rpktp/ is symmetric in is zero. Thus, using (2)
i/3 and
qJ is antisymmetric in
i/3, the last term (3)
p
=
L LfJ Rta,p}qJa, ...
(t), ...
app ... 71,. ···P.
i= I t,
 L Rp,:q>a, ... aptP,···iA·'·P., t
where (r)i means that 1: occurs in the ith place. Multiplying (2) by (1)). and plugging into (I) yields the theorem. Q.E.D. We want to derive a similar theorem for Do acting on r(AM(F)) where F is a complex line bundle defined by the Icocycle C(;k}' (As usual in this section we are assuming that M is Kahler.) A form q> E r(AM(F)) is given locally by a family of (p, q)forms {q>j} on {Uj} where {Uj} is a covering of M with coordinate patches over which F is trivial such that q>j =fjkqJk on U j n (Tk'
(4)
Let w = i L gap dz a 1\ dz P be the Kahler form of M and suppose we have chosen an Hermitian form (,) on the fibres, so
(C 0 = aj "jI2, where Cis a fibre coordinate of, and aiz) is a real positive Coo function on U j Then aj "j12 = ak "k1 2 implies
•
(5)
For two forms qJ, IjJ E r(AM(F)) their inner product (q>, 1jJ) is then (q>, 1jJ) =
f ajq>j
1\
* i/ij.
M
The integrand is well defined since by (4) and (5), ajqJj
1\
* i/ij =
akqJk
1\
* i/ik on
Uj n Uk'
Recall that we defined
and by Equation (l), Section 3
* (a i loa j 1\ * q» = 9q> j  * (I a j lOa a j d za 1\ * qJ j ) =  * (0 * qJ)  * (L aj10aa j dz a 1\ * q>j)'
(9 a qJ) j = 9qJ j

122
GEOMETRY OF COMPLEX MANIFOLDS
Recall also by Proposition 5.2 and Theorem 5.2
9cpj
* (0 * cp) =
= 

*
Ct,Va dz
a /\
* cPj)
and
Thus we have: PROPOSITION
6.7.
(9 a CP)a, ... p, ... P._, = ( l)p Proof
L gl1a(Va + oa log a)CPja, ... a I1P, ... P._,' p
a,p
We need only show
To do this we assume gpa(zo) = lJPa and cP j = dz Ap
/\
dz l1•. Then
* cP j = sgn(Bq Bn q ) inc 1 )tn(nI )+np dz Bn ApAn_p
q /\
dz An  p.
Next
* (dz a
/\
dz Bn .
/\
dz An  p ) = IJdz Ap
/\
dz x,
where 17 and X are as follows: X is the increasing set of numbers (x, ... x q _ J) complementary to the set rxBn  q £;; (1 .. ·n). If we order rxBn _ q in increasing order Y let
s = sgn
(BY)' rx nq
Then 17 = sine _l)tn(n 1)+n(nq+ 1) sgn
_ 'n(_I)tn(n')+n(nq+ I) I
Thus,
Hence,
sgn
(~p}) (An_pAp) rx B nq X'
6.
123
CURVATURES ON KAHLER MANIFOLDS
where X = (Xl'" Xql) is an increasing set of numbers from (I ... n) so that X is the complement of a in B q • But we also have
L
(1)P IX
ca({JjApax, ... xq_,dzAp/\dzx
v X=Bq
= (1)P
L Ca sgn (BaXq )
dz Ap /\ dz x
ae Bq
corresponding to the righthand side of (6). Letting Ca = aj laa. aj we see (6) is verified at Zo in the case gPa(zo) = o/1 a, ({J j = dz Ap /\ dz p•. The general case Q.E.D. follows easily from this. We want to define covariant differentiation of sections ({J E nAP·q(F)). Let ({J = {({J E qA(p,q)(F)). Then ({J jAP = !jk ({JkA/J' One can easily check the following fact: If ({J is a form and! is a COO function,
J
and
But in our case
Ojjk = 0 so
Thus we define (7) However,
Va ({J JAB = !jk Va ({JkAB SO
+ aa!jk ({JkAB
we must make a different definition of Va. which depends on
Hence,
aj
1
({J JAB
=
J ak ({JkAB' jk
Now
so that
aj'
We know
124
GEOMETRY OF COMPLEX MANIFOLDS
thus proving
We define
v~a)
0
7. Then as in Theorem 7.2 dcp is a (1, I)form and
129
VANISHING THEOREMS
['J = c(F)c so ,
 y = dcp, where cp is a Iform. Thus,
dcp = ,  y = '1
+ Ot/!,
where '1 and t/! are (I, I)forms and 0'1 = O. But then
A'1 = 2 0'1 = 0
d'1 = (j'1
so
= o.
Also ddcp = 0 so
dcp = '1
+ t(d(j + (jd) t/!
implies
0= d(jdt/!
(1)
and hence
«(jdt/!, t5dt/!) = (dt/!, dt5dt/!) =
o.
Thus,
(jdt/! = 0 so
dcp = '1
+ t d(jt/!.
Then
('1, dcp) = «(j'1, cp) = 0
= ('1, '1) + t ('1, d(jt/!) = ('1, '1). Hence '1 = O. Using Equation (I) 0= «(jdt/!, t/!)
=
(dt/!, dt/!)
so
dt/! = O.
(2)
From (2)
o = dt/! = iN + at/!. Thus, at/! = ot/! = 0 since t/! is of type (1, 1). So
dcp =
e
y = 09t/! =  i( aAat/! 
= i 0 aAt/!. Thus,
0 aAt/!)
130
GEOMETRY OF COMPLEX MANIFOLDS
where I is a Coo function on M. But
~  y= ~ 
y= 
2~ 0 aJ
i 0= 2n: a If
since ~  y is a real form. Hence, ~  y = Uj2n:)a 0(1)(1 + j) and thus we may assume that ~  y = (ij2n:)a 0 I where I is real valued. Finally
Y=
i
_
aoj = 2n:
~  
i_ a(log 2n:

a
a.  f). J
i
y=aologa.
(1)
2n:
J
Q.E.D.
REMARK.
Perhaps we should explain this proof a little more clearly. We
claim: PROPOSITION
7. I.
If
and if [1/1] = 0 (that is, IjJ = d({J), then there is a Coo function 1/1 = 0I when M is a Kahler manifold.
a
Proof
Let 'II
I
such that
= {I/I 11/1 = dcp, 1/1 of type (l,l)}. Then a 0{0 s; 'II, where Y EB a0 {0, where
{0 is the space of differentiable functions, and'll =
Y = {I]I I]
E
'II, (I], aof) = 0, for all IE 22}.
We note that (I], a 0I) = 0 if and only if 8!}. I] = O. We claim that if M is Kahler, then Y = {O}. For Kahler implies !d = 0 = 0 and a8 + 8a = 0 = 2 1'/ = DOl] = (08 + 8o)(a[}. + 9a)I] for 1'/ E Y. Since I] is of 08 + 8e. Thus type (I, I) and I] = dcp, 0 = dl] = al] = 01]. Thus,
±d
±d 2 1]
= (o8a[}. + 9a(9)I] = 
009!}. I]
+ 9080 I]
=0. Thus ,121] = 0 and (,121].1]) = (,11], ,11]) = O. So ,11] = 0 and hence (jl] Finally, ('1, 1]) = (dcp, 1]) = (cp, (jl]) = 0 and I] = 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/2rri') X"il dz" 1\ 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 ()) = i X"il dz" 1\ dz il is a Kahler form. Hence M is a Kahler manifold. Rewording Theorem 7.1 gives: THEOREM 7.5.
If F  K is positive, then Hq(M, (D(F)) = 0 for q
THEOREM 7.6.
If F is positive, then Hq(M, (D(F»
~
I.
= 0 for q::; n 
l.
Proof Serre duality gives Hq(M, OP(F)) ~ Hnq(M, onp(F)) where dim M = n. Notice that on ~ (D(K) and let p = O. Then
Hq(M, OO(F» ~ H"q(M, 0"( F» ~
H"q(M, (D(K  F»
= 0 for n  q ~ I if K  F  K
= F is positive.
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»
~
.Yfp·q(F)
= {cp I OaCP
For cP
E
= 0, cP of type
(p, q)}.
.yt(P·q)(F) we let the reader check the following inequality:
Thus if X'ij is sufficiently positive definite, then the integrand is positive for cP # 0 we see .ytp.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 (9 acp)j = {(I/a)9(aj cp)}, and so forth. LEMMA 7.2.
(ooa
r (AP,q(F». Proof
+ oao)cp
X /\ cP, where X = 00 log aj and
=
We have (oa cp)j =
cP
E
{~ o(aj' cPj)}
= {ocpj
+ alog aj
/\ CPj}.
Thus, O(OaCP)j={Oocpj+oologaj /\ cpologaj /\ oCPj}.
Add o/Jcp j = {o()cp j
+ 0 log aj
/\
()cp j} to get (ooa
+ oa o)cp = X
/\ cpo
Q.E.D. THEOREM 7.8. [Nakano (1955); Calabi and Vesentini (1960)] r (AP,q(F» be such that ocp = 8acP = O. Then
Let cP
E
Os )=1(X /\ Acp  A(X /\ cp), cp).
s Fl (oaCP, OaCP) = (9o acp, cp) (ot/!, tIt) = (cp, 8at/!) and (9cp, tIt) = (cp, oat/!).
Proof
0
since  J=19 = oA  Ao. Hence
By Proposition 5.4
O:s; (Boacp, cp) = J=1 (oAoacp  Aooacp, cp)
(3)
=)=1 (Aoacp, 8acp)  )=1(A(ooa
=
)=1 (A(X
+ oao)cp, cp)
/\ cp), cp).
But we also have Os (Bcp, Bcp)
= (oaBcp, cp) = )'=1 (oaoAcp =
 oaAocp, cp)
)=1 (oaoAcp, cp)
= )=1 (oaOI\'I'+ oOaACP, cp) since ocp = 8acp = O. Thus,
J=1 (X /\ Acp, cp) ~ O. Now as Equations (3) and (4) to get the theorem. THEOREM 7.9.
[Nakano (1955)J
Q.E.D.
If F is negative, then
Hq(M, QP(F» = 0 when n = dim M.
(4)
for p
+q s
n 1
7. Proof
133
VANISHING THEOREMS
By the harmonic theory
Hq(M, OP(F» where P·q(F)
=
~
{
(0,2).
As in the argument of Theorem 7.2, we can find differentiable functions Aij such that Cijk = c5(A};jk = Ajk + Aki + Ajj' Then we can find differentiable Iforms t/lj such that dAjk = t/lk  t/lj. Then t/I in Diagram (I) is obtained by t/I = dt/lk = dt/lj. For the Dolbeault isomorphism, OA jk = q>k  q>j' where the q>j are (0, I)forms. Then q> = Oq>k' We can split up t/lj = t/I/I,O) + t/lP'O) into forms of type (1,0) and of type (0, I). We know that d = a+ 0 so we compute ;) 1 _ UIL jk 
OA jk
./,(1,0) 'I' k 
./,(1,0) 'I' j ,
= t/llo,t)  t/ljo.I).
Thus we may assume that q>k = t/lk(O,I). Then q> = ot/l/O,I) = t/I(O, 2) [the (0, 2) part of t/I]. Thus, if Cc  t/I, J.lC  t/I(O, 2). Now we have assumed Cc '" w which is of type (1,1). Thus t/I = W(I.I) + dl], with I] = I]li. O) + 1](0.1). Thus t/I(0,2) = 01](0,1) which means J.lC = O. Q.E.D. With the obvious definition of elements of type (I, I) in H2(M, 71.) we have: Let M be a compact complex manifold. Then the il1'age of the map HI (M, (9*) ~ H2 (M, 7L) is the set of elements of type (I, 1). COROLLARY.
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 IP N ). We first outline the idea. We know there is a positive line bundle
E E H 1(M, (!)*). Let F = mE where m is a large positive integer. Let dim HO(M, (!)(F»
=N +1
choose a basis {f3o, ... , f3N} for HO(M, (!)(F», and let F be defined by the Icocycle {fjk} with respect to some covering {Uj} of M (remembering that the !jk are never zero). By definition
f3v = {f3v/z)}, f3vi z) = fik(Z) . f3viz), where the f3vj(z) are holomorphic on U j . Consider the candidate for a map 0
(10)
Then
c(F  K  n[C]  nED]) my 
K + Me
+ mlD,
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 .Nt and is positive definite on.Nt  C  D. But (1e > 0 near C and (1D > 0 near D. Then Equation (10) follows. This proves Proposition 8.1, and thus part (l) 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 EM, P =1= q. The proof of C is almost the same as the proof of B. We want to show that is biholomorphic at each p EM. Consider Y = (!J(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 Y z and write down the exact sequence
OYQ(F)Y"O, Y
z
= (!J(F)..
z
=1= p
Yp= {CfJICfJ = CfJj,CfJj(z) = kl
L
+"'+k"~2
akl"'k"Z;l"'Z~"'CfJE(!J(F)p}.
Then if z =1= p if z = p. Thus HO(M, Y") ;;;; C+ I. We write down the exact cohomology sequence
6
1
H(M,Y)···. It is easily seen that to prove is biholomorphic at p we need only show HI(M, Y) = O. To prove this we once again use Nt = QiM), C = Qip).
8. LEMMA 8.2.
HODGE MANIFOLDS
143
If HI(M, (!J(t  2[C]»
= 0,
then
Hl(M, (!J(F  2p» = O. Proof The proof is the same as that of Lemma 8.1. One only has to notice that if cp has a zero of order 2 at p, P*cp has a zero of order 2 (at least) on C and vice versa.
LEMMA 8.3. Proof
HI(M, (!J(l'  2[C]» = 0 if m is large enough where F
= mE.
Using Proposition 8.1. we find
F  2[C]  K(M) = ml!. 
= mE 
K
(n  1)[C]  2[C]
K  (n
+ 1)[C].
Hence
e(l  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, (!J) = 0, then M is projective algebraic. Proof
The exact cohomology sequence of 0+ 7L + (!J + (!J* + 0
yields
... ., HI(M, (!J*)~ H 2 (M, 7L)" O. Thus everything in H 2 (M,7L) is the Chern class of some bundle. Let {bl' ... , bm } be a basis for the free part of H 2 (M, 7L) so that
H2(M, IC) = Cb l
+ ... + Cb m •
144
GEOMETRY OF COMPLEX MANIFOLDS
Each b;. = c(F;.) and hence is cohomologous to a real 2form of type (I, 1). Let
w=
iL gall dza 1\ 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)
w'" L p;.b;. where P;. E ~ (w is real and the b;. are real). Given e, we can always find integers k;., r E 11 such that
A=l, .. ·,m. But then for a small enough e
,
W =
(P).  k).) w  '" L... r Y;.
defines a Kahler form on M where y;. '" b;. is a real (1, 1) form. Hence w= rw' is also a Kahler form. But
Thus
en defines a Hodge metric on M, and M
is algebraic.
Q.E.D.
Theorem 8.4. [Kodaira (1954)J Let M be a compact complex manifold. If the universal covering manifold Sf is complex analytically homeomorphic to a bounded domain rJI c:;;; cn, then M is algebraic. Proof We make use of the Bergmann metric on rJI [see Helgason (1962)]. We have M = rJIjG where G, the set of covering transformations of rJI, is a collection of biholomorphic maps from rJI to rJI. Let ds 2 = L gall dza dz(J be the Bergmann metric on f!4. We claim
L
(I) ds 2 is invariant under G and hence induces a metric gall dzlJ dz(J on M = rJIjG. (2) If w = (ij2n) gall dzlJ 1\ dz(J, w'" c(  K); so we have a Hodge metric on M.
L
This gives the theorem, thus we need only prove (I) and (2). Let :It' be the Hilbert space of all holomorphic functions f on rJI which have bounded norm
IIfl12 =
f If(zW dX, 1M
8.
145
HODGE MANIFOLDS
where dX = dX 1
dX 2n and
•••
Za = X2al
+ iX2a'
Let {I.} be any orthonormal base of .Yr. Then the Bergmann kernel K(z, z) is given by 00
K(z, z) =
_
L f.(z)f.(z)
.=
[= K(z)].
1
The kernel K(z) is actually independent of the choice of orthonormal basis {I.} [see HeJgason (1962)]. Then L ha/1 dza. dz P is a positive definite Hermitian metric where 2 h ( ) = 8 log K(z)
8Zex 8zp
a./1 z
Let y : f!A LEMMA
+
.
f!A be a biholomorphic map, y(z) = z'.
8.4.
I
K(z) = det
o(z~, ... , z~) 12 K(z'). 8(ZI' "', zn)
Proof 00
_
K(z) = L f.(z)f.(z) .=1 and
f f.(z')jiz') dX' = f f.(z)f;'(z) dX = D.).. ~
~
Let F.(z) = f.(z')
det(~~~;)
and notice
Thus,
and {F} gives a new base. Hence, K(z) =
L Fv(z)Fv(z) = v= 00
1
I
0(Z,)\2 det  8) Lfv(z')Jv(z') (z
I
o(z') 12
= det 8(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 TC : B + BIG = M. Let Vj be an open set in M on which a local inverse of TC is defined, and choose one Jl j = TC 1 to use as a coordinate chart for V j (Jl j(p) E en if P E V). Suppose P E V j n V k' Then there is Yjk E G such that Jlip) = Yjk(Jl j(p». The canonical bundle K on M is defined by the lcocycle
and we have K(z)
= Ifj kl 2 K(Zk)
by the lemma. Recall that if we have positive Coo functions aj on V j such that aj Ifj kl 2 = ak, then c(  K)
i ~ =

2TC 0
0 log a j
.
Therefore, let aj = K 1(zj)' Then j
c(  K)
i
= 2TC aa log K(zj) = 2TC L gall dzj
1\
dz~.
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 Br = {til tl < r, tEem} is an open disk around the origin of em. We want a manifold .JI and a holomorphic map w: .JI + Br with maximal rank so that w is proper and each fibre Mr = w1(t) has the structure of a complex manifold which varies analytically with t. We want a covering {Oft i} of vft so that 1,
It I < r}
e = (C,"', ej), w(e
t) = t,
Oft i = {(e i
,
t)IIU
... ,Pm} is a base
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
156
for H1(M, 0), then we can define
(aMr) at. Proof
(1)
vi{
so that
= fJ r=O
forv=l,···,m. v
We will accomplish the proof in the following two steps:
Construction of a vector (0, I)form
qJ(t) =
L qJk
1 '"
km t~1 ... t~~"
such that
qJ(O)
=
0,
oqJ(t) l[qJ(t), qJ(t)] = 0, and = fl. E H1(M, 0). ( a~(t)) ot. 1=0
(2) Show that qJ(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
qJ =
L qJ~ dza(a~/I )
a vector (0, I)form on U. Let
L~ = (a~a)  ptqJ~(Z)(o~p), 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 L~ and L~ are (complex) linearly independent, and if
oqJ  t[qJ, qJ] = 0, then Equation (1) has n Coo solutionsiI(z), .. ·,f,,(z) such that
[1' ... , [n)) ¥ °
det(O(f1' "', in' a( z 1, ... , Zn, Z l'
•.. , Zn)
(that is, iI, ... ,f" 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~,
REMARK
2.
Linear independence is needed; for if
then
La.! =
IS7
°
implies f is independent of xa.
REMARK 3. If M is a complex manifold and cp 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 cpo We say the almost complex structure cp is integrable, and hence associated to a complex structure.
In order to construct our form cp(t) we need to do some more potential theory. We want to define the Green's operator on
.!l'q = r(AO,q(T» = the space of vector (0, q)forms. To do this we introduce an Hermitian metric product,
where the
* operator
gajJ
on M, and define an inner
has been defined before. We have the adjoint 9 of 0, + 09. Then the space of
(9cp, t/I) = (cp, ot/l); and the Laplacian 0 = 90 harmonic forms
w= ~
{cp I cp E .!l'q, OCP = o} Hq(M, 0),
defines a Hodge decomposition,
.!l'q = W EEl O.!l'q
:::I
W EElo.!l'ql
+ 9.!l'q+ 1
into an orthogonal direct sum of subspaces. Thus, for cp 1'/ E IHlq,
t/I E .!l'q.
Since
E
.!l'q, cp = 1'/
+ Ot/l,
t/I E .!l'q, t/I = , + t/l1'
'E
W, t/l1
E
O.!l'q,
and
Thus (2)
158
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
LEMMA
2.1.
Proof
The decomposition in Equation (2) is unique. Surely 11 is unique. If tjl', t/I are both orthogonal to Wand
= 11 + ot/l', qJ = '1 + Ot/l,
qJ then
O(t/I'  t/I)
=
°
and t/I't/lEW. But t/I'  t/I .1 W so t/I'
=
Q.E.D.
t/I.
DEFINITION 2.1. Given qJ, the unique t/ll making Equation (2) true is denoted GqJ, and the mapping qJ + Gcp defines G : .!l'q + O.!l'q. G is called the Green's operator, and is a linear map. We write" = HqJ and call H the harmonic
projection operator. Then (3) PROPOSITION
oH
2.1.
= Ho = 0, 9H = H9
GH
= 0,
=
0, oG = Go,
= 9H = H9
is analogous.
=
HG
9G = G9.
Proof
oHqJ
=
0 since HqJ E W =
{t/I! ot/l =
9t/1
HOqJ = 0 since oqJ E o.!l'q .1 W. The proof that 0 HGqJ = 0 since GqJ .1 W. For GH = notice
°
HqJ
=
HHqJ
=
O}.
= HqJ + DGHII'
and uniqueness yields GHqJ = 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 = 092. Thus 00 = Do and
OqJ =
aD GqJ = ooGqJ
= HaqJ
= OGoqJ.
2.
AN EXISTENCE THEOREM FOR DEFORMATIONS J
159
Since oGq> ..l.. IHlq and Hoq> = 0, we use uniqueness of decomposition Equation (3) to see oGq> = Goq>. Q.E.D. To proceed further, we need to introduce the Holder norms in the spaces !f q • To do this we fix a finite covering {Uj} of M such that (Zj) are coordinates on V j . Let cP E !f q ,
).
q>"
1 = q!
). I. q>j5oI"
_a
al
50 q
dZ j
Let k El, k ~ 0; aE IR,O < a < I. Leth where n = dim M. Then denote
1\'"
11q>IIk+a
=
I q> Ilk +.
+
h
=
x;a 1 + ixj.
is defined as follows:
m~x{ L )
dz/.
= (hi' "', h 2n ), hi ~O'L?~l hi = Ihl z]
Then the Holder norm
1\
(sup
jI.1 :sk
ID1 q>;a.I" •.(Z)I)
ZEU;
I
ID~ q>J"1 ... ".(Y)  DJ q>;ii.
sup
Iy  zl
J1,%IiV j Ihl =k
'" ,.(Z )I}
a '
(4)
where the sup is over all A., ai' ... , a q . We have the following a priori estimate of Douglis and Nirenberg (1955).
1Iq>IIk+a where k
~
~
C(II Dq>lIk2+a + 11q>llo),
(5)
2, C is a constant which is independent of q> and
11q>llo
=
max
sup
j
ZE
A, al.·'
I
1q>7a., ... ".(z)l.
Vj
aq
REMARK. One can see that two norms defined as in Equation (4) for two different coverings {Vj}, {Vj} induce equivalent topologies on !f q •
PROPOSITION
dent of
2.2.
q> and I/J.
Proof
II [q>, I/J] IlkH s
C
11q>11k+ 1 +a 11q>11k+ 1 +a'
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. I Gcpllk+a :$; C Ilcpllk2+a, k Z 2, where C depends only on not on cpo
PROPOSITION
k and
r:1.,
Proof
We use Equation (5) to get
IIGCPIIk+a:$; C(IIDcpllk2+a + IIGcpllo) :$; c(llcp  Hcpllk2+a + IIGcpllo) :$; C(IICPllk2+a + IIHCPllk2+a + IIGcpllo)· The space IHI q is finite dimensional, so let {b.};= 1 be a base. Then m
Hcp =
L (cp, flv){Jv, v= 1
so m
Ilcpllk2+a:$;
L \' =
Ilflvllk2+a max I(cp, flv)1 v
1
:$; C1
0:: I fJvllk 2+a) I cp 110
:$; C21Icpllk2+a' Thus,
(6) and we need only prove
that is,
IIGcpllo < C Ilcpllk2+a  4'
(7)
Suppose (7) is not true. Then there is a sequence cp(v) such that
.
11m v
IiGcp(V)il o () I cp v II k  2 + ~
=
+ 00.
By multiplying cp(v) by a constant we may assume that IIGcp(vlll o = I, and then 11q>(V)llk_2+a + O. Then Equation (6) implies that
iIGcp(Vlllk+a:$; K
(is bounded for k
z 2).
Write
G'rfIl(vl
=
9~! "L.. (GfIl(Vl)A_ 'r Ja I
..
_ dZ~'J
a.
II. ..• II.
d~'J • to
(8)
2.
AN EXISTENCE THEOREM FOR DEFORMATIONS I
161
Then Equation (8) implies that each of (G cp (V»).ja.l ... a..
and all of its partial derivatives up to order k are uniformly bounded and equicontinuous. We are in a position to use Ascoli's theorem. We can choose a subsequence {cp(v,,) I n = 1,2, ... } such that Gcp(Y n ) and D~ Gcp(v n ) converge uniformly to 1/1 and D~ 1/1 for Ihl ::; k. For simplicity let us replace Vn by v. Then we get
Y .... 00
(self adjointness)
=
lim(DGcp(V), GI/I) Y .... GO
V .... 00
(since cp(Y) + 0) = O. Thus 1/1 = O. But we should have 111/1110 = I, since 111/1110 = lim IIGcp(v)ilo = 1. V .... 00
This contradiction proves the proposition. Let us now begin to construct the cp(t) of Part (a). We use power series techniques but we notice that we could also use the implicit function theorem for Banach spaces [compare Kuranishi (1965)]. We want to construct cp(t) = I:'=l cpit), where
and each cp" • ... "n, E r(Ao,I(T) such that
ocp(t)  ![cp(t), cp(t)] = 0,
(9)
m
CPI(t) =
I
"= I
I1v tv.
(10)
where {I1J is a base for W ~ HI(M, 0). We use a method due to Kuranishi. Consider the equation (II)
where CPI(t) is given by (10). We first show that (II) has a unique formal
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
162
power series solution cp(t). In fact, this is clear since CP2(t) = !.9G[CPl(I), CPl(t)] CP3(t) = !9G([CPl(t), cpit)]
+ [CP2(t), CPl(t)]) (J 2)
PROPOSITION 2.4. IIk+a' Proof
For small Itl, cp(t) = 'L:'= 1 cpit) converges in the norm
Let
P
A(t) = 161'
L y"(tl + ... + tmY 00
,,= 1
" A \' I  L..
.. "'m
t VI t
... t"on m •
As usual, Ilcp(t)llkH ~ A(t) means IICPvI" Vno IIk+a ~ AVI .. Vm' and cp"(t) = CPt(t) + ... + CPJt). Then (12) is equivalent to (13) We want to choose P and y. Suppose they are chosen so that Ilcp" l(t)lIkH ~ A(t). Since .9 is a linear differential operator of first order, II~ 9G[cp, 1jJ] ilk +a ~ C 1 il G[cp, 1jJ] 11k+ 1 +a
~ C1Ck,all[CP, 1jJ]likl+a
:s; C 1 Ck ,,, C IIcpllk +" IltiJ Ilk+a' by Propositions 2.2 and 2.3, where Cl' Ck. a' and C are constants independent of cp and tf;. Hence by (13) I cp(" >c t) Ii ~ + a ~ C 1 C~. a C il cp(" 1 l( t) II k+ a II cpU' 
1 l( t) II k+ a
~ C 1 Ck , a C(A(t»2
~ CICk,aC(~)A(t) as in Section 3, Chapter 2. Thus choose p and l' so that C1 Ck,a C(Pjy) < I. Then IlcpP(t)llk+a ~ A(t). These constants are all independent of /1. So if P and yare chosen so that Ilcpl(t)llkH ~ A(t), which is clearly possible, then C1 Ck,a CWly) < I yields Ilcp(t)llk+a ~ A(t).
So for small Itl, cp(t) converges.
Q.E.D.
2.
AN EXISTENCE THEOREM FOR DEFORMATIONS
I
163
PROPOSITION 2.5. The q>(t) of Proposition 2.4 satisfies oq>(t)  ![q>(t), q>(t)] = 0 if and only if H[q>(t), cp(t)] = 0, where H: r(AO. 2(T» ~ 1HJ2 ~ H2(M, 0) is the orthogonal projection to the harmonic subspace of 22 = r(AO,2(T».
Proof If oq> = 1[q>, q>], then 0 = Heq> = !H[q>, q>], since Ha = 0, Conversely let H[q>, cp] = 0 and set t/I(t) = oq>(t) 1[cp(t), q>(t)]. Then each /'). E lH]1 so 01}. = 0 and
2t/1(t)
= o8G[cp(t), cp(t)]  [q>(t), q>(t)].
Recall that any w can be decomposed w = Hw + DGw. Since H[q>, q>] we get 2t/1(t) = (u9G  DG)[q>(t), q>(t)]. Because D ='09 + 9'0 we get
= 0,
2t/1(t) =  9OG[q>(t), q>(t)] =  9Go[q>(t), q>(t)]
=  29G[ocp(t), q>(t)]. This last equality is true because
a[q>, q>] = [acp, q>]  [q>, aq>]
= [ocp, q>] + [aq>, cp] = 2[oq>, q>]. Then
q>(t) =  9G[oq>(t), q>(t)]
= 9G[t/I(t) + 1[cp(t), q>(t)], q>(t)] =  9G[l/t(t), q>(t)] by the Jacobi identity. Estimating, we get
Choose It I so small that 11q>(t)IIk+~ C, Ck,~ C < 1. Then we get the contradiction 1It/1(t)IIk+~ < 11t/I(t)llkh unless t/I(t) = 0 for all small t, Q.E.D. With the assumption H 2(M, 0) = 0 we have completed Part (a) of our task. We should notice the dependence of q> on z and t. PROPOSITION 2,6.
q>(z, t) is COO in (z, t) and holomorphic in t.
Proof It is immediate that q> is C k since the series converges in II 11k+~' Coo dependence is not so obvious. To give the proof we refer to the regularity theorem for quasilinear elliptic operators [Douglis and Nirenberg (1955)].
164
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
Since the '1. are harmonic,
O l,r
°
for any real y = (Yl' "', Yn) # 0, and any complex w = (WI' "', wit) # 0. 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. EI has a complete orthonormal set of eignenfunctions {eth}h"=1 ~L(BJ Let the eigenvalues be Ah(t). Then they are real and
[Completeness means that any t/J
E
L(B 1) can be written as follows:
00
t/J =
L= ajeth •
h
1
Furthermore we can arrange the e1h such that
and lim h _ oo 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 = (r:x(t)P(t)) 1
y(t)b(t)
be a Othorder differential operator (just a matrix). Then
1)
/1.( t
+ c5(t) ± J(r:x(t) ~ b(t))2 + 4P(t)b(t) = r:x(t) ,,,""":",,,,,,,,:,,2
176
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
which could fail to be differentiable when (tX  15)2 the kernel of E r • Then
IFr={I/IIEtl/l=O}={I/III/I=
I
+ 4f3t5 = O.
Let IFr be
aherh }.
).h(r)=O
Let Fr be the orthogonal projection to 1F t , that is,
I
Ft 1/1 =
(1/1, eth)eth'
).h(t)=O
The Green's operator G r is defined by
Gtl/l =
I
1
1 (t)(I/1, erh)eth ·
"h(t)*O Ah
F t and Gr are related by the equation 1/1 = E t G r 1/1
+ Ftl/l·
We have already investigated the case P = a point, E t = D, Fr = H, 1Ft Gt = G. In the general case we have the following theorem:
= !HI,
THEOREM 4.3. dim 1Ft is an upper semicontinuous function of t. This means that given to, there is a small enough e so that dim 1Ft ~ dim IFro for It  tol < e.
Proof dim 1Ft Ah(to) we have
...
~
= d t is finite since An(t) + 00. In the ordering of the
A)to) < 0 = Aj+ I(tO)
= ... =
Aj+do(tO) < Aj +do +let)
~
By continuity, choose e so that Ait) < 0 and Aj+do+l(t) > 0 if Then in this disk d t ~ do. Q.E.D.
....
It  tol
,
= (0/8zj)*. Our transformation law for (8/8zj) is iJ ) (iJz%8) IP= 8z~ J( iJz% . n
8z~
I
Thus the law for dzj is n
iJz'"
dz~J = "L...a~ _ J dz P k'
P= 1 Zk
This implies that dzj is not the differential of zj. If it were, it would transform according to the law n
8z~
m
az~
dz'"jL...;:,p  " _ J dz P _ J dt), k + " L...;:,).' P=I uZk ),=1 ut Next, let fJI be a vector bundle on vi{ and let B t = rt(fJI). Then L r , S(fJI) is the space of COO sections of fJI ® !Y*(r, s) and C, S(B r) is the space of COO sections of Br ® T,*(r, s)(M t ). If f!4 is given by the transition equation 11
(7 = I
v=1
and IjJ
E
L r , S(f!4) is given locally by IjJ
b7k.(Z,
tm
= (1jJ J ' .. " 1jJ), then
180
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
where
We have assumed an Hermitian metric on ~ depending differentiably on t; hence, if CPt> I/It are in L(Et) and are Coo in t, then (cpl' I/It)t depends differentiably on t. We can introduce the operators dt! ot! Ot, 9t , b t , !J. t acting on L~' S = L" S(Et). Then the operators
Ot = Ot 9 t + 9t Ol' 6 t = dtb t + btdt , Ot depend differentiably on t. We now state the main theorem of this section. w
THEOREM 4.6. Let.A    + P be a differentiable family. If M to = (i)l(tO) is a Kahler manifold, then M t = (i)I(t) is Kahler for It  tol small enough. REMARK I. A good problem would be to find an elementary proof (for example, using power series methods). Our proof uses nontrivial results from partial differential equations (Theorem 4.1). REMARK 2. Hironaka (1962) has given an example of a nonKahler deformation of a Kahler manifold. Hence the theorem is only true for It  tol small. Proof (of the theorem) Assume P Kahler we have a Kahler form Wo on M 0
Wo =
=
iL gap(z) dz a
{tlltl < I} s;; IRm. Since Mo is 1\
dz P•
We extend this to an Hermitian metric on all of the fibres as follows: Let .. " tm ) be local coordinates on 0/1 j ~ Uj x P s;; en x IR m, such that (i)(ZIJ" ... tm) = (11 " ... . tm) Then on 0/1.J
(Z}, .. " zj, tl,
i
L gjap(z) dzj 1\ dz~ = Wj
a,p
is a Coo Hermitian form which is independent of t. Let {p iz, t)} be a partition of unity subordinate to {o/1j}. Then we define Wt = Pj(z, I)W/Z) = i gtiZP(Z) dz iZ 1\ dz P• j a,p
L
L
The metric Wt depends differentiably on t and Wo is the Wo we started with. Let our inner products (cp, I/I)t be defined with respect to the metric Wt where cP, 1/1 E L;' s. Then the operator 0 t = 9 t 0t + 0t 9 t is strongly elliptic and depends differentiably on t so our theorems apply. In fact, one can check that the principal part of 0 t is
4.
ST ABILITY THEOREMS
181
in a local coordinate where (gfa) is the inverse matrix to (grajJ) (see, for example, Chapter 3, Section 2). We define the following spaces: Z~;s = {q> I q> E L~,sol q> =
Zb;s
O},
= {q> I q> E Cr", a, q> = O},
Lqt
= "~ Lr,s t,
Z:J,
=
r+s=q
{q> I q>
E L~,
dl q> = O},
= {q> I q> = L~'" 0, q> = O}, 1Hl~ = {q> I q> E L;, LlI q> = O}.
1Hl~'s
The theorems of Dolbeault and de Rham yield ~ 
01 L~' s
1Hl;
~
Hq(M I , C),
I
where
!1~
Z~'s 0,
IHlr,s
~ 1 
HS(M !1r) , ,
I'
(4)
is the sheaf of germs of holomorphic rforms on M,. As usual, let
h;'s = dim
1Hl~'s,
bq = dim IHI;.
Then bq is the qth Betti number of MI (which is independent of t since all of the MI are diffeomorphic to each other). M 0 is Kahler, so 1::.0
= 20 0 = 2Do .
(5)
This was proved in Chapter 3, Section 5. We have for each t 0101
+ 0101
= 0 = 9101 + 91 81 ,
(6)
since Kahler is not necessary for this. However, if the reader will consult the proof of (5) he will find that in the Kahler case (for example, t = 0) (7) Now we define EI = 01019,91+ 91910101+ 91018101 + ,91019101+ 9101 + 910,.
Then (EICP, 1/1)1
= (9, 01q>, 91911/1)1 + (0101q>, 01011/1)1
+ (91 VI cP, 91011/1)1 + (01q>, 011/1)1 + (01q>, 011/1)1' PROPOSITION 4.2. EI is a strongly elliptic selfadjoint differential operator of order 4 acting on L~'s.
Proof
We clearly have (Elq>, 1/1)1 = (q>, EII/I)I'
(8)
182
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
Thus E, is selfadjoint. We let the reader check that the principal part of E, is " P(1. by L. gj gj ;) (1., p, y, b VZj
(1.
04
::I P ::I Y ::Ib VZj VZj VZj
Q.E.D.
in a local coordinate system.
4.3. Eo = Do Do + 8 0 00 + 80 00 and E, cP = 0 if and only if 0, cP = 0, cP = o.
PROPOSITION
8, 9, cP =
Using Equations (5), (6), and (7) we get
Proof
Do Do
= (9 0 00
=
+ 00 80 )(8 0 00 + ( 0 80)
90 00 80 00 + 00 90 00 80 + 00 80 90 00 + 9000 00 80
= 80900000 + 00 00 8 0 80 + 80009000 + 80 00 80 00 . This proves the first statement. For the second statement we use the obvious inequality 2
(E,CP, cp) ~ 118,8, CPII,
2 + 1101cpll, + 110Icpll"2
Q.E.D.
As before, let 1F~'s = {cp I E, cP = 0, cP E L~'S}. Let F, : L~'s orthogonal projection to 1F~'s and G, be the Green's operator, so 1/1 = E,G,I/I PROPOSITION
r
zr,s  0 0 d,'II
4.4.
J
+
1F~'s
be
+ F,I/I.
,sl t.:p Fr,s where w"
t.:p W
means orthogonal
direct sum. Proof If cP E IF~'" then d,CP = O. If 1/1 E oIO,L~l,sl then, d,CP = 0 since dlo,O, = (0 , + or)o,or = O. Thus oIO,L~l,Sl + 1F~'s s;;; z~;'. The sum is orthogonal since
(0 1 0, cP, 1])r for 1] E 1F~'s. Next take 1/1
E z~·s.
~,
= (cp, 8r 8, 1]), = 0
Then
= E, G, 1/1 + F,I/I = 0, or IX + 9, 13 + ij, Y + 1],
where 1] = Fr 1/1. Since d r1/1 = 0 dl (8, f3
+ ij, y) = o.
Let a = 8, 13+ ij, y. Then a E L~'s and Or : L~" = 0, a = O. We claim a = O. For
+
(9) L~+ 1 ,s, Or : L~' s + L~' s+ I, SO
0, a
(a, a), = (8, 13
+ ij, y, a), = (13, 0, a), + (y, 0, a), = o.
Thus
183
STABILITY THEOREMS
4.
= O. Hence IjJ = 0,0,(1. + 1], and
0'
Q.E.D. dim [Fl·I ;::: b 2  2h~·2 where b 2 is the second Betti number of
LEMMA 4.1. M,.
Proof
[FI.I , Since
a (j L I
,
~ 
O ZI.I/ (j , L. O" d,
O
c dL ,I n ZI.1 d,
,
ZI.I ) dim [FI.I > dim ( ~ n Z1.1 ,£ILl, d,
+ £ILl) = dim (ZI.I d, '. £I LI
, ,
By de Rham's theorem b 2 = dim (ZJ,/d,L,I). We claim there is the following exact sequence
o
~
ZI.I + d L I Z2 n Z2.0 Z()·2 d" I d,' il, 0, d LI ~ d e~oLI.o+ J LO. I ·
, ,
, ,
"
"
We must define 1[, and check that it has the correct kernel. Let IjJ E Z}" d, IjJ = IjJ = 1jJ2.0 + IjJI.I + IjJO.2. Then d,1jJ = 0 yields 0,1jJ0.2 = 0,1jJ2.0 = O. So we can map IjJ to 1jJ2.0 + IjJO.2 E + Z~:2. Let IjJ Ed, Li. Then
o where
Z;:o
IjJ = drC.
ofjkv(z) =
o.
Page 108, line 12: omit the bar over 0>.. 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 1/. Page 118, line 9: The subscript on the last R should be {3i1>... Page 120, line 4: {31 should be i31. Page 120, last line: (f)i should be (f)k. Page 124, line 10: x should be +. Page 126: line 5: {3q should be Bq; the term involving cp after the summation sign should be CPjB• . CP:·. 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 ~ ",cp:•.