2
Cohomology Chapter of (Mostly) Constant Sheaves and Hodge Theory
2.1 Real and Complex
Let X be a complex analytic manifold of (complex) dimension n. Viewed as a real manifold, X is a C manifold of dimension 2n. For every x X, we know TX,x is a C-vector space of complex dimension n, so, TX,x is a real vector space of dimension 2n. Take local (complex) coordinates z1 , . . . , zn at x X, then we get real local coordinates x1 , y1 , . . . , xn , yn on X (as an R-manifold), where zj = xj + iyj . (Recall, TX is a complex holomorphic vector bundle). If we view TX,x as a real vector space of dimension 2n, then we can complexify TX,x , i.e., form TX,x C = TX,x R C, a complex vector space of dimension 2n. A basis of TX,x at x (as R-space) is just , ,..., , . x1 y1 xn yn These are a C-basis for TX,x C , too. We can make the change of coordinates to the coordinates zj and z j , namely, zj = xj + iyj , z j = xj iyj , and of course, 1 1 (zj + z j ), yj = (zj z j ). 2 2i has a basis consisting of the /zj , /z j ; in fact, for f C (open), we have xj = f f f = i zj xj yj and f f f = +i . z j xj yj
So, TX,x C
More abstractly, let V be a C-vector space of dimension n and view V as a real vector space of dimension 2n. If e1 , . . . , en is a C-basis for V , then ie1 , . . . , ien make sense. Say ej = fj +igj (from C-space to R-space), then, iej = ifj gj = gj + ifj . Consequently, the map (e1 , . . . , en ) (ie1 , . . . , ien ) corresponds to the map J ((f1 , g1 ), . . . , (fn , gn )) ((g1 , f1 ), . . . , (gn , fn )) where V is viewed as R-space of dimension 2n. The map J an endomorphism of V viewed as R-space and obviously, it satises J 2 = id. 67
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CHAPTER 2. COHOMOLOGY OF (MOSTLY) CONSTANT SHEAVES AND HODGE THEORY
If, conversely, we have an R-space, V , of even dimension, 2n and if an endomorphism J EndR (V ) with J 2 = id is given, then we can give V a complex structure as follows: (a + ib)v = av + bJ(v). In fact, the dierent complex structures on the real vector space, V , of dimension 2n are in one-to-one correspondence with the homogeneous space GL(2n, R)/GL(n, C), via class A AJA1 . Denition 2.1 An almost complex manifold is a real C -manifold together with a bundle endomorphism, J : TX TX , so that J 2 = id. Proposition 2.1 If (X, OX ) is a complex analytic manifold, then it is almost complex. Proof . We must construct J on TX . It suces to do this locally and check that it is independent of the U is trivial. By denition of a patch, we have an coordinate patch. Pick some open, U , where TX isomorphism (U, OX U ) (BC (0, ), OB ) and we have local coordinates denoted z1 , . . . , zn in both cases. On TX U , we have /z1 , . . . , /zn and /x1 , . . . , /xn , /y1 , . . . , /yn , as before. The map J is given by J ,..., , ,..., ,..., , ,..., . x1 xn y1 yn y1 yn x1 xn We need to show that this does not depend on the local trivialization. Go back for a moment to two complex manifolds, (X, OX ) and (Y, OY ), of dimension 2n and consider a smooth map f : (X, OX ) (Y, OY ). For every x X, we have an induced map on tangent spaces, df : TX,x TY,y , where y = f (x) and if, as R-spaces, we use local coordinates x1 , . . . , xn , y1 , . . . , yn on TX,x and local coordinates u1 , . . . , un , v1 , . . . , vn on TY,y , then df is given by the Jacobian u u xj yj . JR (f ) = v v xj yj If f is holomorphic, the Cauchy-Riemann equations imply u v = xj yj Now, this gives v yj JR (f ) = u yj and v u = . xj yj u yj = v yj
A B
B . A
Going back to our problem, if we have dierent trivializations, on the overlap, the transition functions are holomorphic, so JR (f ) is as above. Now J in our coordinates is of the form J= 0n In In 0
and we have JJR (f ) = JR (f )J when f is holomorphic (DX). So, an almost complex structure is a bundle invariant. Question: Does S 6 possess a complex structure?
2.1. REAL AND COMPLEX
69
The usual almost complex structure from S 7 (= unit Cayley numbers = unit octonions) is not a complex structure. Borel and Serre proved that the only spheres with an almost complex structure are: S 0 , S 2 and S6. Say we really have complex coordinates, z1 , . . . , zn down in X. Then, on TX R C, we have the basis ,..., , ,..., , z1 zn z 1 z n and so, in this basis, if we write f = (w1 , . . . , wn ), where w = u + iv , we get w w zj z j , JR (f ) = w w zj z j and, again, if f is holomorphic, we get w w = = 0, z j zj which yields w zj JR (f ) = 0 Write J(f ) = and call it the holomorphic Jacobian. We get (1) JR (f ) = J(f ) 0 0 , so, R-rank JR (f ) = 2C-rank J(f ). J(f ) = w z j w zj 0 A 0 . 0 A
(2) We have det(JR (f )) = | det(J(f ))|2 0, and det(JR (f )) > 0 if f is a holomorphic isomorphism (in this case, m = n = the common dimension of X, Y ). Hence, we get the rst statement of Proposition 2.2 Holomorphic maps preserve the orientation of a complex manifold and each complex manifold possesses an orientation. Proof . We just proved the rst statement. To prove the second statement, as orientations are preserved by holomorphic maps we need only give an orientation locally. But, locally, a patch is biholomorphic to a ball in Cn . Therefore, it is enough to give Cn an orientation, i.e., to give C an orientation. However, C is oriented as (x, ix) gives the orientation. Say we have a real vector space, V , of dimension 2n and look at V R C. Say V also has a complex structure, J. Then, the extension of J to V R C has two eigenvalues, i. On V R C, we have the two eigenspaces, (V R C)1,0 = the i-eigenspace and (V R C)0,1 = the i-eigenspace. Of course, (V R C)0,1 = (V R C)1,0 . Now, look at
l
(V R C). We can examine
p,0 p 0,q q
(V R C) =
def
[(V R C)1,0 ] and
(V R C) =
def
[(V R C)0,1 ],
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CHAPTER 2. COHOMOLOGY OF (MOSTLY) CONSTANT SHEAVES AND HODGE THEORY
and also
p,q p,0 0,q
(V R C) = Note that we have
l
def
(V R C)
p,q
(V R C).
(V R C) =
p+q=l
(V R C).
D Now, say X is an almost complex manifold and apply the above to V = TX , TX ; we get bundle decomD positions for TX R C and TX R C. Thus, D (TX R C) = l=1 p+q=l 2n p,q D (TX R C).
is multiplication by (1)q ip+q . Therefore, J does not act by scalar multiplication in Note that J on l general on (V R C). Say X is now a complex manifold and f : X Y is a C -map to another complex manifold, Y . Then, for every x X, we have the linear map Df : TX,x R C TY,f (x) R C.
p,q
1,0 The map f wont in general preserve the decomposition TX,x R C = TX,x 1,0 1,0 However, f is holomorphic i for every x X, we have Df : TX,x TY,f (x) .
0,1 TX,x .
Let us now go back to a real manifold, X. We have the usual exterior derivative
l l+1 D TX,x R C D TX,x R C,
d:
namely, if 1 , . . . , 2n are real coordinates at x, we have aI dI
|I|=l |I|=l
daI dI .
here, the aI are C-valued function on X near x and dI = di1 dil , with I = {i1 < i2 < < il }.
D In the almost complex case, we have the p, q-decomposition of TX R C and consequently p,q ip,q l D (TX R C) d l+1 D (TX R C) = r+s=l+1 r,s D (TX R C).
D (TX R C)
We let
p,q p+1,q D (TX R C) D (TX R C)}p,q
= {p,q = prp+1,q d ip,q : and
p,q
p,q+1 D (TX R C) D (TX R C)}p,q .
= { p,q = prp,q+1 d ip,q :
2.1. REAL AND COMPLEX
71
Note that d = + + other stu. Let us take a closer look in local coordinates. We can pick 1 , . . . , n , 1,0 0,1 some coordinates for TX , then 1 , . . . , n are coordinates for TX (say x1 , . . . , x2n are local coordinates in p,q D (TX R C) has the form the base). Then, any =
|I|=p e |I|=q
aI,I dI d I , e e
and so d =
|I|=p e |I|=q
daI,I dI d I + e e
|I|=p e |I|=q
aI,I d(dI d I ) = + + stu. e e
If we are on a complex manifold, then we can choose the j so that j = /zj and j = /z j , constant over our neigborhood and then, d =
|I|=p e |I|=q n
daI,I dI d I e e aI,I e zj aI,I e z j
=
|I|=p j=1 e |I|=q
dzj dzI dz I + e
dz j dzI dz I e
= + = ( + ). Consequently, on a complex manifold, d = + . On an almost complex manifold, d2 = 0, yet, 2 = 0 and = 0 in general. However, suppose we are lucky and d = + . Then, 0 = d2 = 2 + + + , and we deduce that 2 = = + = 0, in this case. Denition 2.2 The almost complex structure on X is integrable i near every x X, there exist real 1,0 0,1 coordinates, , . . . , n in TX and 1 , . . . , n in TX , so that d = + . By what we just did, a complex structure is integrable. A famous theorem of Newlander-Nirenberg (1957) shows that if X is an almost complex C -manifold whose almost complex structure is integrable, then there exists a unique complex structure (i.e., complex coordinates everywhere) inducing the almost complex one. Remark: Say V has a complex structure given by J. We have V = V R R V R C V 1,0 . The vector space V 1,0 also has a complex structure, namely, multiplication by i. So, we have an isomorphism V V 1,0 , as R-spaces, but also an isomorphism V V 1,0 , as C-spaces, where the complex structure on = = V is J and the complex structure on V 0,1 is multiplication by i. Therefore, we also have an isomorphism V V 1,0 , where the complex structure on V is J and the complex structure on V 0,1 is multiplication by = i.
1,0 0,1 For tangent spaces, TX is spanned by /z1 , . . . , /zn , the space TX is spanned by /z 1 , . . . , /z n ; D D also, TX 1,0 is spanned by dz1 , . . . , dzn and TX 0,1 is spanned by dz 1 , . . . , dz n . pr1,0 2 2 2
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CHAPTER 2. COHOMOLOGY OF (MOSTLY) CONSTANT SHEAVES AND HODGE THEORY
2.2
Cohomology, de Rham, Dolbeault
2 D TX d D TX d d 2n D TX ,
Let X be a real 2n-dimensional C -manifold and let d be the exterior derivative, then we get the complex
(d2 = 0). The same is true for complex-valued forms, we have the complex
2 D TX R C d D TX R C d d 2n D TX R C,
D (d2 = 0). Here, there is an abuse of notation: TX denotes a sheaf, so we should really use a notation such as D D TX . To alleviate the notation, we stick to TX , as the context makes it clear that it is a sheaf. These maps induce maps on global C -sections, so we get the complexes 2 D (X, TX ) d D (X, TX ) d d 2n D (X, TX )
and
D (X, TX R C) d
2 D (X, TX R C) d d
2n D (X, TX R C).
Dene
l l ZDR (X) l ZDR (X)C l BDR (X) l BDR (X)C l HDR (X) l HDR (X)C l+1 D (X, TX ) l
= = = =
Ker d, Ker d, Im d, Ker d,
where where
d: d:
l1
D (X, TX ) l+1 D (X, TX R C)
D (X, TX R C) l D (X, TX ) l1
where d : where d:
D (X, TX ) l D (X, TX R C)
D (X, TX R C)
l l = ZDR (X)/BDR (X) l l = ZDR (X)C /BDR (X)C .
l l Note: HDR (X)C = HDR (X) R C. These are the de Rham cohomology groups. For Dolbeault cohomology, take X, a complex manifold of dimension n, view it as a real manifold of dimension 2n, consider the D complexied cotangent bundle, TX R C, and decompose its wedge powers as l D (TX R C) = p+q=l p,q D (TX R C).
Since X is a complex manifold, d = + and so, 2 = = 0. Therefore, we get complexes by xing p or q: (a) Fix q: (b) Fix p:
p,q H (X). 0,q D (TX R C) D (TX R C) 1,q D (TX R C) D (TX R C) n,q D (TX R C). D (TX R C).
2
p,0
p,1
p,n
p,q The above are the Dolbeault complexes and we have the corresponding cohomology groups H (X) and p,q Actually, the H (X) are usually called the Dolbeault cohomology groups. The reason for that is if f : X Y is holomorphic, then df and (df )D respect the p, q-decomposition. Consequently, p,q p,q D (TY,f (x) R C) D (TX,x R C)
(df )D :
2.2. COHOMOLOGY, DE RHAM, DOLBEAULT for all x X and
73
(df )D Y = X (df )D
p,q p,q imply that (df )D induces maps H (Y ) H (X).
The main local fact is the Poincar lemma. e Lemma 2.3 (Poincar Lemma) If X is a real C -manifold and is actually a star-shaped manifold (or e particularly, a ball in Rn ), then p HDR (X) = (0), for all p 1. If X is a complex analytic manifold and is a polydisc (P D(0, r)), then
p,q (a) H (X) = (0), for all p 0 and all q 1. p,q (b) H (X) = (0), for all q 0 and all p 1.
Proof . Given any form (P D(0, r)) with = 0, we need to show that there is some p,q1 (P D(0, r)) so that = . There are three steps to the proof. Step I . Reduction to the case p = 0. Say the lemma holds is
0,q
p,q
(P D(0, r)). Then, our is of the form =
|I|=p |J|=q
aI,J dzI dz J .
Write I =
|J|=q
0,q
aI,J dz J
(P D(0, r)).
Claim: I = 0. We have =
|I|=p
dzI I and 0 = =
|I|=p
(dzI I ) =
|I|=p
dzI I .
These terms are in the span of dzi1 dzip dz j dz j1 dz jq and by linear independence of these various wedges, we must have I = 0, for all I. Then, by the 0,q1 (P D(0, r)), so that I = I . It follows that assumption, there is some I =
|I|=p
dzI I =
|I|=p
(dzI I ) = (
|I|=p
dzI I ),
with
|I|=p
dzI I
p,q1
(P D(0, r)), which concludes the proof of Step I. > 0, there is some
0,q1
Step II : Interior part of the proof. We will prove that for every (P D(0, r)) so that = in P D(0, r ). Let us say that depends on dz 1 , . . . , dz s if the terms aJ dz J in where J {1, . . . , s} are all zero, i.e., in , only terms aJ dz J appear for J {1, . . . , s}.
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CHAPTER 2. COHOMOLOGY OF (MOSTLY) CONSTANT SHEAVES AND HODGE THEORY
0,q1
Claim: If depends on dz 1 , . . . , dz s , then there is some only on dz 1 , . . . , dz s1 in P D(0, r ).
(P D(0, r)) so that depends
Clearly, if the claim is proved, the interior part is done by a trivial induction. In , isolate the terms depending on dz 1 , . . . , dz s1 , call these 2 and 1 the rest. Now, 1 = dz s , so = dz s + 2 and we get 0 = = ( dz s ) + 2 . Examine the terms aJ dz s dz J , z l ()
where
l > s.
Linear independence and () imply aJ = 0 if J {1, 2, . . . , s 1} z l If s J, write J = J {s}. Look at the function J (z1 , . . . , zn ) = 1 2i aJ (z1 , . . . , zs1 , , zs+1 , . . . , zn ) d d . zs () and l > s.
||r
We have the basic complex analysis lemma: Lemma 2.4 Say g() C (r ) (where r is the open disc of radius r), then the function f (z) = is in C (r ) and f = g on r . z aJ (z1 , . . . , zn ) = and if l > s, J 1 = z l 2i by the above. So, if we set = Step III : Exhaustion. Pick a sequence, { t }, with t monotonically decreasing to 0 and examine P D(0, r t ). Write rt = r t , then the sequence {rt } monotonically increases to r. Claim. We can nd a sequence, t
0,q1 J ||r
1 2i
g()
||r
d d z
By this lemma, we have J z s on r (zs)
aJ d d = 0, z l zs
J dzJ , then depends only on dz 1 , . . . , dz s1 in P D(0, r ). e
(P D(0, r)), such that
(1) t has compact support in P D(0, rt+1 ). (2) t = t1 on P D(0, rt1 ). (3) t = on P D(0, rt ). We proceed by induction on q, here is the induction step. Pick a sequence of cuto C -functions, t , so that
2.2. COHOMOLOGY, DE RHAM, DOLBEAULT (i) t has compact support in P D(0, rt+1 ). (ii) t 1 on P D(0, rt ). Having chosen t , we will nd t+1 . First, by the interior part of the proof, there is some 0,q1 (P D(0, r)) with t+1 = in P D(0, rt+1 ). Examine t+1 t on P D(0, rt ), then t+1 (t+1 t ) = t+1 t = = 0. By the induction hypothesis, there is some
0,q2
75
(P D(0, r)) with on P D(0, rt ).
= t+1 t Let t+1 = t+1 (t+1 ) = t+1 t . We have
(1) t+1 C0 ( 0,q1
(P D(0, rt+2 ))).
(3) As t+1 1 on P D(0, rt+1 ), we have t+1 = t+1 and so, t+1 = t+1 = on P D(0, rt+1 ). (2) t+1 t = t+1 t = 0 on P D(0, rt ). Now, for any compact subset, K, in P D(0, r), there is some t so that K P D(0, rt ). It follows that the t s stabilize on K and our sequence converges uniformly on compacta. Therefore, = lim t =
t
and
= lim t = .
t 0,1
Finally, we have to deal with the case q = 1. Let (P D(0, r)), with = 0. Again, we need to nd some functions, t , with compact support on P D(0, rt+1 ), so that () t = on P D(0, rt ). () t converges uniformly on compacta to , with = . Here, t , C (P D(0, r)). Say we found t with t t1
,P D(0,rt2 )
1 . 2t1
Pick t+1 C (P D(0, r)), with t+1 = on P D(0, rt+1 ). Then, on P D(0, rt ), we have (t+1 t ) = t+1 t = = 0. So, t+1 t is holomorphic in P D(0, rt ). Take the MacLaurin series for it and truncate it to the polynomial so that on the compact P D(0, rt1 ), we have t+1 t
,P D(0,rt1 )
1 . 2t
Take t+1 = t+1 (t+1 ). Now, t+1 has compact support on P D(0, rt+2 ) and on P D(0, rt+1 ), we have t+1 1. This implies that t+1 = t+1 , so t+1 t and t+1 = t+1 + = t+1 = on P D(0, rt+1 ), as is a polynomial. Therefore, the t s converge uniformly on compacta and if = limt t , we get = .
,P D(0,rt1 )
1 2t
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CHAPTER 2. COHOMOLOGY OF (MOSTLY) CONSTANT SHEAVES AND HODGE THEORY
p,q
e (U ), where U X is an open subset of a complex manifold, X, Corollary 2.5 ( Poincar) Say and assume d = 0. Then, for all x U , there is a neighborhood, V x, so that = on V , for some p1,q1 (V ). Proof . The statement is local on X, therefore we may assume X = Cn . By ordinary d-Poincar, for every e p+q1 (V1 ), so that = d. Now, x X, there is some open, V1 x, and some
p+q1 r,s
(V1 ) =
r+s=p+q1
(V1 ),
so, = (r,s ), where r,s
r,s
(V1 ). We have = d =
r,s
dr,s =
r,s
( + )r,s .
p,q
Observe that if (r, s) = (p 1, q) or (r, s) = (p, q 1), then the r,s s have dr,s (V1 ). It follows that r,s = 0 and we can delete these terms from ; we get = p1,q + p,q1 with d = 0. We also have = d = ( + ) = p1,q + p,q1 + p1,q + p,q1 = + p1,q + p,q1 , that is, p1,q + p,q1 = 0. Yet, p1,q and p,q1 belong to dierent bigraded components, so p1,q = p,q1 = 0. We now use the and -Poincar lemma to get a polydisc, V V1 and some forms e p1,q1 1 and 2 in (V ), so that p1,q = 1 and p,q1 = 2 . We get (1 ) = p1,q and so, (1 2 ) = p1,q + p,q1 = , which concludes the proof. Remark: Take C = the sheaf of germs of real-valued C -functions on X, then
1,1
and (2 ) = (2 ) = p,q1
H = Ker
: C
(X)
is called the sheaf of germs of pluri-harmonic functions. Corollary 2.6 With X as in Corollary 2.5, the sequences
p,0 p,1
0 p X (when p = 0, it is 0 OX
0,0
X
X
X
q
0,1
X ),
0,q 1,q
0 X and 0 H C X
1,1
X
2,1
X
1,2
X
q
d
X
X
d
are resolutions (i.e., exact sequences of sheaves) of p , X , H, respectively. X
2.2. COHOMOLOGY, DE RHAM, DOLBEAULT e Proof . These are immediate consequences of , , and d-Poincar.
77
In Corollary 2.6, the sheaf p is the sheaf of holomorphic p-forms (locally, = I aI dzI , where the aI X q are holomorphic functions), X is the sheaf of anti-holomorphic q-forms ( = I aI dz I , where the aI are anti-holomorphic functions) and H is the sheaf of pluri-harmonic functions. If F is a sheaf of abelian groups, by cohomology, we mean derived functor cohomology, i.e., we have : F F(X) = (X, F), a left-exact functor and H p (X, F) = (Rp )(F) Ab. We know that this cohomology can be computed using asque (= abby) resolutions 0 F G0 G1 Gn , where the Gi s are asque, i.e., for every open, U X, for every section G(U ), there is a global section, G(X), so that = U . If we apply , we get a complex of (abelian) groups 0 (X, F) (X, G0 ) (X, G1 ) (X, Gn ) , and then H p (X, F) = the pth cohomology group of (). Unfortunately, the sheaves arising naturally (from forms, etc.) are not asque; they satisfy a weaker condition. In order to describe this condition, given a sheaf, F, we need to make sense of F(S), where S X is a closed subset. Now, remember (see Appendix A on sheaves, Section A4) that for any subspace, Y of X, if j : Y X is the inclusion map, then for any sheaf, F, on X, the sheaf j F = F Y is the restriction of F to Y . For every x Y , the stalk of F Y at x is equal to Fx . Consequently, if S is any subset of X, we have F(S) i there is an open cover, {U }, of S and a family of sections, F(U ), so that for every , we have S U = S U . Remark: (Inserted by J.G.) If X is paracompact, then for any closed subset, S X, we have lim F(S) = F(U ),
U S
()
where U ranges over all open subsets of S (see Godement[5] , Chapter 3, Section 3.3, Corollary 1). [Recall that for any cover, {U } , of X, we say that that {U } is locally nite i for every x X, there is some open subset, Ux x, so that Ux meets only nitely many U . A topological space, X, is paracompact i it is Hausdor and if every open cover possesses a locally nite renement.] Now, we want to consider sheaves, F, such that for every closed subset, S, the restriction map F(X) F(S) is onto. Denition 2.3 Let X be a paracompact topological space. A sheaf, F, is soft (mou) i for every closed subset, S X, the restriction map F(X) F S(S) is onto. A sheaf, F, is ne i for all locally nite open covers, {U X}, there exists a family, { }, with End(F), so that (1) (2)
c Fx = 0, for all x in some neighborhood of U , i.e., supp( ) U .
= id.
We say that the family { } is a sheaf partition of unity subordinate to the given cover {U X} for F.
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CHAPTER 2. COHOMOLOGY OF (MOSTLY) CONSTANT SHEAVES AND HODGE THEORY
Remark: The following sheaves are ne on any complex or real C -manifold: (1) C (2) (3)
p p,q
(4) Any locally-free C -bundle (= C -vector bundle). For, any open cover of our manifold has a locally nite renement, so we may assume that our open cover is locally nite (recall, a manifold is locally compact and second-countable, which implies paracompactness). Then, take a C -partition of unity subordinate to our cover, {U X}, i.e., a family of C -functions, , so that (1) 0. (2) supp() << U (this means supp() is compact and contained in U ). (3)
= 1.
Then, for , use multiplication by Remark: If we know a sheaf of rings, A, on X is ne, then every A-module is also ne and the same with soft. Proposition 2.7 Let X be a paracompact space. Every ne sheaf is soft. Say 0 F F F 0 is an exact sequence of sheaves and F is soft. Then, 0 F (X) F(X) F (X) 0 Again, if 0 F F F 0 is an exact sequence of sheaves and if F and F are soft, so is F . Every soft sheaf is cohomologically trivial (H p (X, F) = (0) if p > 0). Proof . Take F ne, S closed and F(S). There is an open cover of S and sections, F(U ), so that U S = U S. Let U0 = X S, an open, so that U0 and the U cover X. By paracompactness, we may asume that the cover is locally nite. Take the Aut(F) guaranteed as F is ne. Now, we have ( ) = 0 near the boundary of U , so ( ) extends to all of X (as section) by zero, call it . We have F(X) and = exists (by local niteness).
is exact.
As
U S =
U S, we get = ( ) = ( ) on U S
and we deduce that =
=
( ) =
( ) =
( ) = ;
on S.
2.2. COHOMOLOGY, DE RHAM, DOLBEAULT Therefore, is a lift of to X from S. Exactness of the sequence 0 F F F 0
79
implies that for every F (X), there is an open cover, {U X}, and a family of sections, F(U ), so that ( ) = U . By paracompactness, we may replace the U s by a locally nite family of closed sets, S . Consider the set S= (, S) (1) S = S , for some of our S (2) F(S), S = , for each S as in (1).
The set S is, as usual, partially ordered and it is inductive (DX). By Zorns lemma, S possesses a maximal element, (, S). I claim that X = S. If S = X, then there is some S with S S. On S S , we have ( ) = = 0, where ( ) = , by (2), and ( ) = , by denition. By exactness, there is some F (S S ) so that () = on S S . Now, as F is soft, extends to a global section of F , say, z. Dene by = on S + (z) on S .
On S S , we have = = + (z) = + () = , so and agree. But then, (, S S ) S and (, S S ) > (, S), a contradiction. Therefore, the sequence 0 F F F 0 has globally exact sections. Now, assume that F and F are soft and take F , with S closed. Apply the above to X = S; as F is soft, we deduce that F(S) F (S) is onto. As F and F are soft, the commutative diagram F(X) F(S) 0 implies that F (X) F (S) is surjective. For the last part, we use induction. The induction hypothesis is: If F is soft, then H p (X, F) = (0), for 0 < p n. When n = 1, we can embed F in a asque sheaf, Q, and we have the exact sequence 0 F Q cok 0. If we apply cohomology we get 0 H 1 (X, F) H 1 (X, Q) = (0), since Q is asque, so H 1 (X, F) = (0). () / F (X) / F (S) / 0
80
CHAPTER 2. COHOMOLOGY OF (MOSTLY) CONSTANT SHEAVES AND HODGE THEORY
For the induction step, use () and note that cok is soft because Fand Q are soft (Q is asque and asque sheaves are soft over a paracompact space, see Homework). When we apply cohomology, we get (0) = H j (X, Q) H j (X, cok) H j+1 (X, F) H j+1 (X, Q) = (0), (j 1)
so H j (X, cok) H j+1 (X, F). As cok is soft, by the induction hypothesis, H j (X, cok) = (0), so = H j+1 (X, F) = (0). Corollary 2.8 Each of the resolutions (p > 0)
p,0 p,1
0 p X (for p = 0, a resolution of OX ),
0,q
X
X ,
1,q
0 X and 0
0
q
X
X ,
1
R d d X = C X , C p,q p X, X vanishes). is an acyclic resolution (i.e., the cohomology of Proof . The sheaves
p,q
X,
p
X are ne, therefore soft, by Proposition 2.7.
Recall the spectral sequence of Cech cohomology (SS):
p,q E2 = H p (X, Hq (F)) = H (X, F),
where (1) F is a sheaf of abelian groups on X (2) Hq (F) is the presheaf dened by U H q (U, F).
Now, we have the following vanishing theorem (see Godement [5]): Theorem 2.9 (Vanishing Theorem) Say X is paracompact and F is a presheaf on X so that F (= associated sheaf to F) is zero. Then, H p (X, F) = (0), all p 0. Putting the vanishing theorem together with the spectral sequence (SS), we get: Theorem 2.10 (Isomorphism Theorem) If X is a paracompact space, then for all sheaves, F, the natural map H p (X, F) H p (X, F) is an isomorphism for all p 0. Proof . The natural map H p (X, F) H p (X, F) is just the edge homomorphism from (SS). By the handout on cohomology, Hq (F) = (0), all q 1. Thus, the vanishing says
p,q E2 = H p (X, Hq (F)) = (0),
all
p 0, q 1,
which implies that the spectral sequence (SS) degenerates and we get our isomorphism.
2.2. COHOMOLOGY, DE RHAM, DOLBEAULT Comments: How to get around the spectral sequence (SS).
81
(1) Look at the presheaf F and the sheaf F . There is a map of presheaves, F F , so we get a map, H p (X, F) H p (X, F ). Let K = Ker (F F ) and C = Coker (F F ). We have the short exact sequences of presheaves 0 K F Im 0 and 0 Im F C 0,
where Im is the presheaf image F F . The long exact sequence of Cech cohomology for presheaves gives H p (X, K) H p (X, F) H p (X, Im) H p+1 (X, K) and H p1 (X, C) H p (X, Im) H p (X, F ) H p (X, C) , and as K = C = (0), by the vanishing theorem, we get H p (X, F) H p (X, Im) H p (X, F ). = = Therefore, on a paracompact space, H p (X, F) H p (X, F ). = (2) Cech cohomology is a -functor on the category of sheaves for paracompact X. Say 0 F F F 0 is exact as sheaves. Then, if we write Im for Im(F F ) as presheaves, we have the short exact sequence of presheaves 0 F F Im 0 and Im = F . Then, for presheaves, we have H p (X, F) H p (X, Im) H p+1 (X, F ) and by (1), H p (X, F) H p (X, F ), so we get (2). = (3) One knows, for soft F on a paracompact space, X, we have H p (X, F) = (0), for all p 1. Each } is an eaceable -functor on the category of F embeds in a asque sheaf; asque sheaves are soft, so {H sheaves and it follows that {H } is universal. By homological algebra, we get the isomorphism theorem, again. In fact, instead of (3), one can prove the following proposition: Proposition 2.11 Say X is paracompact and F is a ne sheaf. Then, for a locally nite cover, {U X}, we have H p ({U X}, F) = (0), if p 1. Proof . Take { }, the sheaf partition of unity of F subordinate to our cover, {U X}. Pick Z p ({U X}, F), with p 1. So, we have = (U0 Up ). Write =
( (U U0 Up )).
Observe that exists as section over U0 Up as is zero near the boundary of U ; so can be extended from U U0 Up to U0 Up by zero. You check (usual computation): d = . Corollary 2.12 If F is ne (over a paracompact, X), then H p (X, F) = (0), for all p 1.
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CHAPTER 2. COHOMOLOGY OF (MOSTLY) CONSTANT SHEAVES AND HODGE THEORY
Figure 2.1: A triangulated manifold Theorem 2.13 (P. Dolbeault) If X is a complex manifold, then we have the isomorphisms
p,q H q (X, p ) H (X) H q (X, p ). = X = X
Proof . The middle cohomology is computed from the resolution of sheaves
p,0 p,1 p,2
0 p X
X
X
X .
p,q X are acyclic for H (X, ) and for H (X, ). Yet, by homological algebra, we can Moreover, the p q q (X, p ) by any acyclic resolution (they are -functors), compute H (X, X ) and H X
To prove de Rhams theorem, we need to look at singular cohomology. Proposition 2.14 If X is a real or complex manifold and F is a constant sheaf (sheaf associated with a constant presheaf ), then there is a natural isomorphism
p H p (X, F) Hsing (X, F), =
provided F is torsion-free. Proof . The space, X, is triangulable, so we get a singular simplicial complex, K (see Figure 2.1). Pick a vertex, v, of K and set St(v) = { K | v }, the open star of v (i.e., the union of the interiors of the simplices having v as a vertex). If v0 , . . . , vp are vertices, consider St(v0 ) St(vp ) = Uv0 ,...,vp . We have Uv0 ,...,vp = if v0 , . . . , vp are not the vertices of a p-simplex a connected set if v0 , . . . , vp are the vertices of a p-simplex. / if (v0 , . . . , vp ) K if (v0 , . . . , vp ) K.
Observe that {Uv X}vvert(K) is an open cover of X and as F is a constant sheaf, we get F(Uv0 ,...,vp ) = 0 F
Let be a Cech p-cochain, then (Uv0 ,...,vp ) F and let ( )((v0 , . . . , vp )) = (Uv0 ,...,vp ), where (v0 , . . . , vp ) K. Note that ( ) is a p-simplicial cochain and the map ( ) is an isomorphism
p C p ({Uv X}, F) Csing (X, F) =
that commutes with the coboundary operators on both sides. So, we get the isomorphism
p H p ({Uv X}, F) Hsing (X, F). =
We can subdivide K simplicially and we get renements of our cover and those are arbitrarily ne. Subdivision does not change the right hand side and if take we right limits we get
p H p (X, F) Hsing (X, F). =
As a consequence, we obtain
2.3. HODGE I, ANALYTIC PRELIMINARIES Theorem 2.15 (de Rham) On a real or complex manifold, we have the isomorphisms H p X, R C H p X, R = C R Hp = sing X, C R Hp = DR X, C
83
Proof . The isomorphism of singular cohomology with Cech cohomology follows from Proposition 2.14. The isomorphism of derived functor cohomology with Cech cohomology follows since X is paracompact. Also de Rham cohomology is the cohomology of the resolution 0 R d C C
1 2
X
d
X ,
d
and the latter is an acyclic resolution, so it computes H p or H p . Explicit Connection: de Rham Singular.
Take a singular p-chain, j aj j , where j = fj (); fj C(); = the usual p-simplex (aj Z, or aj R, or aj C, ... .) We say that this p-chain is piecewise smooth, for short, ps, i the fj s actually are C -functions on a small neighborhood around . By the usual C -approximation (using convolution), each singular p-chain is approximated by a ps p-chain in such a way that cocycles are approximated by ps cocycles and coboundaries, too. In fact, the inclusion
ps sing Cp (X, R) Cp (X, R)
is a chain map and induces an isomorphism
ps sing Hp (X, R) Hp (X, R). ps X, a de Rham p-cochain, i.e., a p-form. If Cp (X, R), say = Say then dene () via: p j
aj fj () (with aj R),
()() =
=
def j
aj
fj ()
=
def j
aj
fj R.
ps p The map () is clearly a linear map on Cp (X, R), so we have () Cps (X, R). Also, observe that
(d)( ) =
=
(by Stokes) = ()( ),
from which we conclude that (d)( ) = ()()( ), and thus, (d) = (). This means that
p
: is a cochain map and so, we get our map
p (X, R) Cps (X, R)
p p HDR (X, R) Hsing (X, R).
2.3
Hodge I, Analytic Preliminaries
Let X be a complex analytic manifold. An Hermitian metric on X is a C -section of the vector bundle 1,0 1,0 (TX TX )D , which is Hermitian symmetric and positive denite. This means that for each z X, we have 1,0 1,0 a map (, )z : TX,z TX,z C which is linear in its rst argument, Hermitian symmetric and positive denite, that is:
84
CHAPTER 2. COHOMOLOGY OF (MOSTLY) CONSTANT SHEAVES AND HODGE THEORY (Hermitian symmetric)
(1) (v, u)z = (u, v)z
(2) (u1 + u2 , v)z = (u1 , v)z + (u2 , v)z and (u, v1 + v2 )z = (u, v1 )z + (u, v2 )z . (3) (u, v)z = (u, v)z and (u, v)z = (u, v)z . (4) (u, u)z 0, for all u, and (u, u)z = 0 i u = 0 (positive denite). (5) z h(z) = (, )z is a C -function.
1,0 0,1 Remark: Note that (2) and (3) is equivalent to saying that we have a C-linear map, TX,z TX,z C. 1,0 In local coordinates, since (TX )D = 0,1 D D TX,z and TX,z , we get 1,0 1,0 0,1 D TX and TX = TX and since {dzj }, {dz j } are bases for
1,0
h(z) =
k,l
hkl (z)dzk dz l ,
for some matrix (hkl ) Mn (C). Now, (, )z is an Hermitian inner product, so locally on a trivializing 1,0 0,1 cover for TX , TX , by Gram-Schmidt, we can nd (1, 0)-forms, 1 , . . . , n , so that
n
(, )z =
j=1
j (z) j (z).
The collection 1 , . . . , n is called a coframe for (, ) (on the respective open of the trivializing cover). Using a partition of unity subordinate to a trivializing cover, we nd all these data exist on any complex manifold. Consider (, )z and (, )z . For R, (1), (2), (3), (4), imply that (, )z is a positive denite 1,0 bilinear form, C as a function of z, i.e, as TX,z real tangent space TX,z , we see that (, )z is a C = Riemannian metric on X. Hence, we have concepts such as length, area, volume, curvature, etc., associated to an Hermitian metric, namely, those concepts for the real part of (, )z , i.e., the associated Riemannian metric. If we look at (, )z , then (1), (2), (3) and (5) imply that for R, we have an alternating real 2 1,0 1,0 1,0 D (TX,z C). bilinear nondegenerate form on TX,z , C in z. That is, we get an element of (TX,z TX,z )D D In fact, this is a (1, 1)-form. Look at (, )z in a local coframe. Say k = k + ik , where k , k TX,z . We have k (z) k (z) =
k k
(k (z) + ik (z)) (k (z) ik (z)) (k (z) k (z) + k (z) k (z)) + i
k k
=
(k (z) k (z) k (z) k (z)).
1,0 Now, a symmetric bilinear form yields a linear form on S 2 TX,z = S 2 TX,z ; consequently, the real part of the Hermitian inner product is (, )z = k (k (z)2 + k (z)2 ). We usually write ds2 for k k k and 2 1,0 (TX,z )D : (ds2 ) is the associated Riemannian metric. For (ds2 ), we have a form in n
(ds2 ) = 2
k=1
k k .
We let ds2 = =
1 2
(ds2 )
2.3. HODGE I, ANALYTIC PRELIMINARIES and call it the associated (1, 1)-form to the Hermitian ds2 . If we write k = k + ik , we have
n n n
85
k k =
k=1
(k + ik ) (k ik ) = 2i
k=1 k=1
k k .
Therefore,
n
=
k=1
k k =
i 2
n
k k ,
k=1
which shows that is a (1, 1)-form. Remark: The expession for in terms of (ds2 ) given above depends on the denition of . In these notes, = but in some books, one nds = (ds2 ). Conversely, suppose we are given a real (1, 1)-form. This means, is a (1, 1)-form and for all , () = () Dene an inner product via H(v, w) = (v iw). We have H(w, v) = (w iv) = (iv w) = (iv w) = (iv w) = (v iw) = H(v, w). (Note we could also set H(v, w) = (v iw).) Consequently, H(v, w) will be an inner product provided H(v, v) > 0 i v = 0. So, we need (v iv) = i(v v) > 0, for all v = 0. Therefore, we say is positive denite i i(v v) > 0, for all v = 0. Thus, = (1/2) (ds2 ) recaptures all of ds2 . You check (DX) that is positive denite i in local coordinates i hkl (z)dzk dz l , = 2
k,l
1 ( ), 2
(reality condition).
where (hkl ) is a Hermitian positive denite matrix. Example 1. Let X = Cn , with ds2 = (a) (ds2 ) =
n 2 k=1 (dxk n k=1
dzk dz k . As usual, if zk = xk + iyk , we have
2 + dyk ), the ordinary Euclidean metric. n k=1
(b) = (1/2) (ds2 ) = (i/2)
dzk dz k , a positive denite (1, 1)-form.
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CHAPTER 2. COHOMOLOGY OF (MOSTLY) CONSTANT SHEAVES AND HODGE THEORY
Remark: Assume that f : Y X is a complex analytic map and that we have an Hermitian metric on X. 1,0 1,0 Then, Df : TY TX maps TY,y to TX,f (y) , for all y Y . We dene an inner product on Y via , yk yl =
y
Df
, Df yk yl
.
f (y)
We get a Hermitian symmetric form on Y . If we assume that Df is everywhere an injection, then our Hermitian metric, ds2 , on X induces one on Y ; in particular, this holds if Y X.
D D Assume Df is injective everywhere. We have the dual map, f : TX TY , i.e., f : Pick U small enough in Y so that 1,0
X
1,0
Y.
(1) TY (2) TX
U is trivial f (U ) is trivial. f (U ) and f (m+1 ) = = f (n ) = 0, where
(3) We have a local coframe, 1 , . . . , n , on TX m = dim(Y ) and n = dim(X). Then, f X = f i 2
n
k k
k=1
=
i 2
m
f (k ) f (k ) = Y .
k=1
Hence, the (1, 1)-form of the induced metric on Y (from X) is the pullback of the (1, 1)-form of the metric on X. Consequently (Example 1), on an ane variety, we get an induced metric and an induced form computable from the embedding in some CN . Example 2: Fubini-Study Metric on Pn . Let be the canonical projection, : Cn+1 {0} Pn , let z0 , . . . , zn be coordinates on Cn+1 and let (Z0 : : Zn ) be homogeneous coordinates on Pn . For a small open , U , pick some holomorphic section, F : U Cn+1 {0}, of (so that F = idU ). For any p U , consider
n
F (p) Pick U small enough so that log F
2
2
=
j=0
Fj (p)Fj (p) = 0.
is dened. Now, set F = i log F 2
2
.
We need to show that this denition does not depend on the choice of the holomorphic section, F . So, let S be another holomorphic section of over U . As S = F = id on U , we have (S0 (p) : : Sn (p)) = (F0 (p) : : Fn (p)), so, there is a holomorphic function, , on U , so that (p)S(p) = F (p), We have F so we get log F
2 2
for all p U ,
for all p U .
2
= F F = SS = S
,
2
= log + log + log F
.
2.3. HODGE I, ANALYTIC PRELIMINARIES Consequently,
87
i (log + log ) + S = S , 2 since is holomorphic, is anti-holomorphic, (holo) = 0, = and (anti-holo) = 0. Clearly, our F are (1, 1)-forms. Now, cover Pn by opens, as above; pick any section on each such open, use a partition of unity and get a global (1, 1)-form on Pn which is C . We still need to check positivity, but since the unitary group, U (n + 1), acts transitively on Cn+1 , we see that PU(n) acts transitively on Pn and our form is invariant. Therefore, it is enough to check positivity at one point, say (1 : 0 : : 0). This point lies in the open Z0 = 0. Lift Z0 to Cn+1 {0} via F = F : (Z0 : : Zn ) (1, z1 , . . . , zn ), Thus, F
2
where
zj =
Zj . Z0
=1+
n
n k=1 zk z k ,
and we get
n k=1 zk dz k n + k=1 zk z k
log 1 +
k=1
zk z k
=
1
=
n k=1
dzk dz k
1+
n k=1 zk z k
n k=1 2
z k dzk
n l=1 zl dz l
.
1+ When we evaluate the above at (1 : 0 : : 0), we get F (1 : 0 : : 0) =
n k=1
n k=1 zk z k
dzk dz k and so dzk dz k ,
i 2
n
k=1
which is positive. Therefore, we get a Hermitian metric on Pn , this is the Fubini-Study metric. As a consequence, every projective manifold inherits an Hermitian metric from the Fubini-Study metric. From now on, assume that X is compact manifold (or each object has compact support). Look at the p,q bundles and choose once and for all an Hermitian metric on X and let be the associated positive (1, 1)-form. So, locally in a coframe, n i k k . = 2
k=1
At each z, a basis for
p,q z
is just {I J }, where I = {i1 < < ip }, J = {j1 < < jq } and I J = i1 ip j1 jq .
We can dene an orthonormal basis of z set I J
p,q
p,q
if we decree that the I J are pairwise orthogonal, and we = (I J , I J ) = 2p+q .
2
This gives z a C -varying Hermitian inner product. To understand where 2p+q comes from, look at C. Then, near z, we have = dz, = dz, so dz dz = (dx + idy) (dx idy) = i(dx dy + dx dy) = 2i dx dy. Therefore, dz dz = 2 and dz dz Let us write
p,q 2
= 4 = 21+1 (here, p = 1 and q = 1).
p,q
(X) for the set of global C -sections, C (X, = i 2
n 1,1
). Locally, on an open, U , we have
k k
k=1
(U )
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CHAPTER 2. COHOMOLOGY OF (MOSTLY) CONSTANT SHEAVES AND HODGE THEORY
and so, we deduce that n = i 2
n
n! (1)( 2 ) 1 n 1 n .
n
n i We call (z) = n (z)/n! = Cn 1 n 1 n the volume form and Cn = ( 2 )n (1)( 2 ) the twisting p,q constant. We can check that is a real, positive form, so we can integrate w.r.t. to it. For , (X), set (, )z (z) C. (, ) =
X
This makes
p,q
(X) a complex (innite-dimensional) inner-product space. We have
p,q1 p,q
:
(X)
(X)
p,q
p,q and say (as in the nite dimensional case) is a closed operator (i.e., B is closed in (X)). Pick some p,q p,q Z , i.e., with () = 0. All the cocyles representing the class of (an element of H ) form the p,q p,q p,q translates + B (X). This translate is a closed and convex subset of (X).
Does there exist a smallest (in the norm weve just dened) cocycle in this cohomology classif so, how to nd it? Now, we can ask if has an adjoint. If so, call it and then, : ( (), ) = (, ()), Then, Hodge observed the Proposition 2.16 The cocycle, , is of smallest norm in its cohomology class i () = 0. Proof . (). Compute +
2 p,q
(X)
p,q1
(X) and
for all , .
= ( + , + ) =
2
+
2
+ 2 (, ).
But, (, ) = ( (), ) = 0, by hypothesis, so +
2
=
2
+
2
,
p,q which shows the minimality of in + B and the uniqneness of such a .
(). We know that +
2
, for all our s. Make f (t) = ( + t, + t).
2
The function f (t) has a global minimum at t = 0 and by calculus, f (t) (, + t) + ( + t, ) that is, (, ) = 0. But, i is another element of
p,q1 t=0
t=0 =
0. We get
= 0,
X. So, let
g(t) = ( + it, + it). Repeating the above argument, we get (, ) = 0. Consequently, we have (, ) = 0, for all . Since ( (), ) = (, ()), we conclude that ( (), ) = 0, for all , so () = 0, as required. If the reasoning can be justied, then
2.3. HODGE I, ANALYTIC PRELIMINARIES
p,q (1) In each cohomology class of H , there is a unique (minimal) representative.
89
(2)
p,q H (X) = p,q
X
(a) = 0 (b) = 0
.
p,q We know from previous work that H (X) H q (X, p ). = X
Making . First, we make the Hodge operator:
p,q np,nq
: by pure algebra. We want
X
X
((z), (z))z (z) = (z) (z) for all . We need to dene on basis elements, = I J . We want (I J ,
K,L
K,L K L ) Cn 1 n 1 n = I J
|M |=np |N |=nq
aM,N M N ,
where |I| = |K| = p and |J| = |L| = q. The left hand side is equal to 2p+q I,J Cn 1 n 1 n and the right hand side is equal to aM,N I J M N = aI 0 ,J 0 I J I 0 J 0 ,
|M |=np |N |=nq
where I 0 = {1, . . . , n} I and J 0 = {1, . . . , n} J. The right hand side has 1 n 1 n in scrambled order. Consider the permutation
0 0 (1, 2, . . . , n; 1, 2, . . . , n) (i1 , . . . , ip , j1 , . . . , jq , i0 , . . . , i0 , j1 , . . . , jnq ). 1 np
If we write sgnI,J for the sign of this permutation, we get
n aI 0 ,J 0 = 2p+qn in (1)( 2 ) I,J sgnI,J .
Therefore, =
K,L
n K,L K L = 2p+qn in (1)( 2 )
sgnK,L K,L K 0 L0 .
|K 0 |=np |L0 |=nq
Now, set
p,q
= , where : X
np,nq
X
np,nq+1
X
p,q1
X.
I claim that is the formal adjoint, , we seek. Consider (, ) =
X
(, )z (z) =
X
,
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CHAPTER 2. COHOMOLOGY OF (MOSTLY) CONSTANT SHEAVES AND HODGE THEORY
p,q1
where
(X) and
p,q
(X). Now, ( ) = + (1)p+q (), so we get ().
X
( ) = (, ) + (1)p+q
X
Also,
p,q1
(X)
np,nq
(X), i.e.,
n,n1
(X). But, d = + , so
d( ) = ( ) + ( ) = ( ), and we deduce that ( ) =
X X
d( ) =
X
= 0,
if either X is compact (in which case X = ), or the forms have compact support (and hence, vanish on X). So, we have (, ) = (1)p+q
X
().
Check (DX): For
np,nq
p,q
(X), we have = (1)p+q .
np,nq+1
As (X), we have () (X), and so, () = (1)2npq+1 () = (1)p+q1 (). We conclude that (, ) =
X
() ( ())
=
X
= Therefore, = , as contended.
(, ()).
Now, we dene the Hodge Laplacian, or Laplace-Beltrami operator , = + : You check (DX) that Claim: is formally self-adjoint.
p,q p,q
, by:
(X)
(X).
() = 0 i both = 0 and = 0. () = 0 and compute (, ()). We get (, ()) = = (, ) + (, )
2
First, assume
= ( , ) + ( , ) = (, () + = Therefore, if
2
+ 2. ().
() = 0, then = 0 and = 0. The converse is obvious by denition of () = 0, where
Consequently, our minimality is equivalent to
is a second-order dierential operator.
To understand better what the operator does, consider the special case where X = Cn (use compactly 0,0 (X) = C0 . Pick f C0 , then again, supported gadgets), with the standard inner product, and (f ) C0 and on those f , we have f = 0. Consequently, (f ) = f =
n
j=1
f dz j . z j
2.3. HODGE I, ANALYTIC PRELIMINARIES We also have
n
91
j=1
f dz j z j
=
n 21n in (1)( 2 )
n
=
n 21n in (1)( 2 )
j=1 n
f z j
dz1 dzn dz {j}0 sgn,{j}
j=1
f sgn,{j} dz1 dzn dz {j}0 . zj
Taking of the above expression, we get
n 21n in (1)( 2 )
n
k,j=1
2f sgn,{j} dz k dz j dz1 dzn dz {j}0 z k zj
n = 21n in (1)( 2 ) (1)n
n
j=1
2f sgn,{j} dz1 dzn dz j dz {j}0 . zj z j
Taking of the above, we get 2i2n (1)( 2 ) (1)n
j=1
n
n
2f zj z j
n
= 2
j=1
2f . zj z j
But, 4 2 f 2f 2f = 2 + y 2 , zj z j xj j and this implies that on
0,0
(X),
(f ) up to a constant (1/2) is just the usual Laplacian. on
p,q
Write Hp,q (X) for the kernel of
(X), the space of harmonic forms. Here is Hodges theorem.
Theorem 2.17 (Hodge, (1941)) Let X be a complex manifold and assume that X is compact. Then, (1) The space Hp,q (X) is nite-dimensional. (2) There exist a projection, (Hodge decomposition)
H: p,q
(X) Hp,q (X), so that we have the orthogonal decomposition
p,q1 p,q+1
p,q
(X) = H
p,q
(X)
(X)
(X). , and it is is uniquely
(3) There exists a parametrix (= pseudo-inverse), G, (Greens operator) for determined by (a) id = H + G = H + G , and
(b) G = G, G = G and G Hp,q (X) = 0. Remarks: (1) If a decomposition ` la Hodge exists, it must be an orthogonal decomposition. Say a p,q1 p,q+1 (X) and (X), then (, ) = (0 , 0 ) = ( 0 , 0 ) = 0, and so,
p,q1
(X)
p,q+1
(X). Observe that we can write the Hodge decomposition as
p,q p,q
(X) = Hp,q (X)
(X).
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CHAPTER 2. COHOMOLOGY OF (MOSTLY) CONSTANT SHEAVES AND HODGE THEORY
p,q
For, if
(X), then = ( 0 ) + (0 ), and this implies
p,q p,q1
(X)
(X) +
p,q+1
(X).
However, the right hand side is an orthogonal decomposition and it follows that
p,q p,q1
Hp,q (X) + For perpendicularity, as
(X) = Hp,q (X) +
(X)
p,q+1
p,q
(X) =
(X).
is self-adjoint, for Hp,q (X), we have (, ()) = ( (), ) = 0,
since
() = 0. () = has a solution, given . Namely, by (3a), = H() + (G()). (),
(2) We can give a n.a.s.c. that
If H() = 0, then = (G()) and we can take = G(). Conversely, orhogonality implies that if = then H() = 0. Therefore, H() is the obstruction to solving () = . How many solutions does () = have?
The solutions of () = are in one-to-one correspondence with 0 + Hp,q (X), where 0 is a solution and if we take 0 Ker H, then 0 is unique, given by G(). (3) Previous arguments, once made correct, give us the isomorphisms
p,q Hp,q (X) H H q (X, p ). = = X
Therefore, H q (X, p ) is a nite-dimensional vector space, for X a compact, complex manifold. X
For the proof of Hodges theorem, we need some of the theory of distributions. At rst, restrict to C0 (U ) n (smooth functions of compact support) on some open, U C . One wants to understand the dual space, (C0 (U ))D . Consider g L2 (U ), then for any C0 (U ), we set
g () =
U
gd.
(Here, is the Lebesgue measure on Cn .) So, we have g C0 (U )D . Say g () = 0, for all . Take E, a measurable subset of U of nite measure with E compact. Then, as E is L2 , the function E is L2 -approximable by C0 (U )-functions. So, there is some C0 (U ) so that
E As E = E + , we get gd =
E U
2
< .
E gd =
U
(E )gd +
U
gd =
U
(E )gd
(by hypothesis...