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j > 0 one has H j (CPn , E вЉ— LвЉ—m ) = 0. In particular,

П‡(CPn , E вЉ— LвЉ—m ) = dim H 0 (CPn , E вЉ— LвЉ—m ).
i
A similar result holds true for any projective algebraic manifold X в†’
CPn and any holomorphic vector bundle E over X but one has to replace
L by iв€— L in the above theorem.

2.4 Chern classes of complex vector bun-
dles
We shall go back to make a better leap and consider a complex C в€ћ
rather than a holomorphic vector bundle F over a complex manifold X.
If the rank of this bundle is r then one can deп¬Ѓne so-called Chern classes
c0 (F ), c1 (F ), ..., cr (F ) of this bundle such that c0 (F ) = 1 and cj (F ) в€€
ПЂ
H 2j (X, Z). Let us construct an auxiliary bundle P(F ) в†’ X with п¬Ѓber
equal to the projective space CPrв€’1 obtained via the projectivization
of the bundle F . The п¬Ѓber ПЂ в€’1 (x) over x в€€ X is the projective space
P(Fx ) of complex lines in Fx . One can see that P(F ) is the same as
F \ { zero-section}/Cв€— as a complex manfold, where the action of Cв€— by
dilations in each п¬Ѓber comes from the deп¬Ѓnition of a vector bundle. Of
course ПЂ is a locally trivial п¬Ѓbration: if U вЉ‚ X is an open set such that
U Г— Cr then P(F )|U U Г— CPrв€’1 . One has the natural algebra
F|U
homomorphism
ПЂ в€— : H в€— (X, Z) в†’ H в€— (P(F ), Z). (2.4.1)
A point of P(F ) is a pair (x, l), where x в€€ X and l is a line in the
п¬Ѓber Fx . Let us construct a tautological line bundle E в†’ P(F ) that is
a complex subbundle of the pullback vector bundle ПЂ в€— F . The п¬Ѓber of
E at a point (x, l) is the line l: E(x,l) = l вЉ‚ (ПЂ в€— (F ))(x,l) = Fx . Note that
48 CHAPTER 2. COHOMOLOGY OF VECTOR BUNDLES

the restriction of E to a п¬Ѓber ПЂ в€’1 (x) = P(Fx )в†’CPrв€’1 identiп¬Ѓes with
Лњ
the tautological line bundle over projective space. Let L stand for the
dual line bundle: L = E в€— . Earlier we have constructed a cohomology
class in H 2 (X, Z) for any line bundle over X using transition cocycles
for this bundle. We denote by Оѕ в€€ H 2 (P(F ), Z) this class for the bundle
L. The class Оѕ is actually the п¬Ѓrst Chern class c1 (L). For our purposes
we need the following
LEMMA 2.4.1 Assume that X is a CW complex (for instance a man-
ifold) then H в€— (P(X), Z) is a free module over H в€— (X, Z) with basis 1, Оѕ, Оѕ 2 , ..., Оѕ rв€’1 .
The module structure is deп¬Ѓned by

a В· b = ПЂ в€— (a) в€Є b , a в€€ H в€— (X, Z) , b в€€ H в€— (P(F ), Z).

As one of the consequences we see that the natural map 2.4.1 is an
injection. As another consequence we notice that there exist uniquely
deп¬Ѓned classes c1 (F ), ..., cr (F ) в€€ H в€— (X, Z) such that cj в€€ H 2j (X, Z)
and
Оѕ r + c1 (F )Оѕ rв€’1 + В· В· В· + crв€’1 (F )Оѕ + cr (F ) = 0. This is how we deп¬Ѓne
the Chern classes. There are several possible deп¬Ѓnitions of them: via
obstruction theory, via the curvature of a connection, and so on and all
those deп¬Ѓnitions reп¬‚ect diп¬Ђerent sides of the same object.
Proof of Lemma. If F is a trivial bundle then we know that that
H в€— (CPrв€’1 ) is torsion free and the KВЁnneth formula produces
u

H в€— (P(F ), Z) = H в€— (X, Z) вЉ— H в€— (CPrв€’1 , Z) = H в€— (X) вЉ— 1, Оѕ, Оѕ 2 , ..., Оѕ rв€’1 .

Next we assume that X can be covered by a п¬Ѓnite number of open
sets: X = U1 в€Є ... в€Є Un such that F|Ui is trivial. We would like to
prove the lemma induction on n. We write down the Mayer-Vietoris
sequence for X = U1 в€Є V , where V = U2 в€Є ... в€Є Un . One has P(F ) =
(ПЂ в€’1 (U1 )) в€Є (ПЂ в€’1 (V )) and further
Оґ
в†’ вЉ•rв€’1 H pв€’1в€’2j (U1 в€© V ) в†’ вЉ•rв€’1 H pв€’2j (X) в†’ вЉ•rв€’1 H pв€’2j (U1 ) вЉ• H pв€’2j (V ) в†’
j=0 j=1 j=1
в†“П€
Оґ
в†’ H pв€’1 (ПЂ в€’1 (U1 в€© V )) в†’ H p (P(F )) в†’ H p (ПЂ в€’1 (U1 )) вЉ• H p (ПЂ в€’1 (V ) в†’

where П€(a0 , ..., arв€’1 ) = rв€’1 ПЂ в€— (aj ) в€Є Оѕ j . It then follows from the п¬Ѓve
j=0
lemma that П€ is an isomorphism.
2.4. CHERN CLASSES OF COMPLEX VECTOR BUNDLES 49

When X is an algebraic manifold and F is an algebraic vector bundle
this proof works п¬Ѓne, but in general when it is not possible to cover X by
a п¬Ѓnite number of open sets satisfying the above property one can cover
X by an increasing family of open sets Um with compact closures, write
H в€— (X) = limв†ђ H в€— (Um ) and H в€— (P(F )) = limв†ђ H в€— (ПЂ в€’1 (Um )) as inverse
limits and apply the above procedure to each set Um as they satisfy the
above property.

Next we shall prove the Whitney sum formula that tells us that
whenever we have two complex vector bundles F and G on X then
cr (F вЉ• G) = r cj (F ) в€Є crв€’j (G). A more convenient way to write it
j=0
rank(F )
down is by introducing the total Chern class c(F ) = 1+ j=1 cj (F ) в€€
even
H (X, Z). Using this notation the Whitney sum formula becomes

c(F вЉ• G) = c(F ) в€Є c(G).

We notice that although the cohomology ring is only graded commu-
tative all the characteristic classes are in even degrees and that is the
reason why we can switch multiples when at least one of them con-
sists entirely of Chern classes. Also we mention an important property
that the total Chern class is invertible for X a manifold (or a п¬Ѓnite-
dimensional CW-complex) as it is equal to 1 + {nilpotent part}. Now
let us pass to the proof of the Whitney formula. Let F and G be two
complex vector bundles on X of ranks r and s respectively. We have
three п¬Ѓbrations on X:

CPrв€’1 CPsв€’1 CPs+rв€’1
в†’ P (F ) в†’ P (G) в†’ P (F вЉ• G)
в†“ в†“ в†“
X X X

Also we have the classes ОѕF в€€ H 2 (P(F )) and ОѕG в€€ H 2 (P(G)). Let us
look at CPr+sв€’1 = P(Cr вЉ• Cs ) and deп¬Ѓne two open sets U, V вЉ‚ CPr+sв€’1
such that U consists of lines not belonging to that particular Cs and
V the same with Cr . We get natural maps U в†’ CPrв€’1 and V в†’
CPsв€’1 . Now if E2 is the tautological line bundle over CPr+sв€’1 and E1 is
the tautological line bundle over CPrв€’1 then (E2 )|U identiп¬Ѓes with the
pullback of E1 to U via the map mentioned earlier in this paragraph.
50 CHAPTER 2. COHOMOLOGY OF VECTOR BUNDLES

We have similar constructions of open sets U, V in P(F вЉ• G). Here
U = P(F вЉ• G) \ P(G) and V = P(F вЉ• G) \ P(F ). We have projection
maps p1 : U в†’ P(F ) and p2 : V в†’ P(G). These projection maps are
compatible with the tautological line bundles: the restriction to U of
the tautological line bundle EF вЉ•G identiп¬Ѓes with pв€— EF . This implies
1
в€— 2
that p1 ОѕF is the image of ОѕF вЉ•G in H (U ).
Returning to the Whitney formula we see that we need to prove the
identity
rв€’j sв€’k
( cj (F )ОѕF вЉ•G )( ck (F )ОѕF вЉ•G ) = 0, (2.4.2)
j k
r+sв€’l
because by deп¬Ѓnition l cl (F вЉ•G)ОѕF вЉ•G = 0. Let us notice the following
LEMMA 2.4.2 Let Y = U в€Є V be the union of two open sets and let
a, b в€€ H в€— (Y ) be such that a has zero image in H в€— (U ) and b has zero
image in H в€— (V ) then a в€Є b = 0.
Now the Whitney formula follows because the п¬Ѓrst cohomology class
rв€’j
in 2.4.2 has zero image in H в€— (U ) since by deп¬Ѓnition j cj (F )ОѕF = 0
rв€’j
and its pull-back to U under p1 : U в†’ P(F ) is equal to j cj (F )ОѕF вЉ•G .
Analogously the second term has zero image in H в€— (V ).
Next we list the set of properties of the Chern classes of complex
vector bundles:
1). cn (F ) = 0 for n > rank(F ), c0 (F ) = 1
2). For a line bundle L the п¬Ѓrst Chern class c1 (L) is the one deп¬Ѓned
in the end of Section 2.
3). Compatibility with pullbacks: if one has a map f : Y в†’ X and
a complex vector bundle F over X then cj (f в€— F ) = f в€— cj (F ) в€€ H 2j (Y ).
4). The Whitney sum formula: c(F вЉ• G) = c(F ) в€Є c(G).
Those four properties actually characterize the Chern classes as we shall
show now.
THEOREM 2.4.3 For X a CW complex there exists the unique as-
signment of cj (F ) to any vector bundle F over X satisfying the prop-
erties 1).-4). above.
Proof. Let Оґj be a set of cohmology classes satisfying the properties
1).-4). Over P(F ) we have an exact sequence of vector bundles
0 в†’ E в†’ ПЂ в€— F в†’ Q в†’ 0.
2.4. CHERN CLASSES OF COMPLEX VECTOR BUNDLES 51

We may write ПЂ в€— (F ) = EвЉ•Q because any exact sequence of C в€ћ bundles
is necessarily split. As usual we denote the total class Оґ(F ) = j Оґj (F ),
then by the Whitney formula Оґ(Q) = Оґ(ПЂ в€— F )Оґ(E)в€’1 . But E is the
tautological line bundle so by property 2). one has Оґ(E) = 1в€’c1 (E в€— ) =
1 в€’ Оѕ. It follows that

Оґ(Q) = (1 + Оґ1 (ПЂ в€— F ) + Оґ2 (ПЂ в€— F ) + В· В· В·)(1 + Оѕ + Оѕ 2 + В· В· В·) =

= (1 + ПЂ в€— Оґ1 (F ) + ПЂ в€— Оґ2 (F ) + В· В· В·)(1 + Оѕ + Оѕ 2 + В· В· В·).
using property 3). Thus we obtain Оґr (Q) = Оѕ r + j Оґj (F )Оѕ rв€’j as desired,
because by 1). we have Оґr (Q) = 0, where r is the rank of F .

PROPOSITION 2.4.4 (Splitting principle). Let F be a complex
vector bundle of the rank r over X. Then there exists a continuous map
П† : Y в†’ X such that
П†в€—
1). The map H (X, Z) в†’ H в€— (Y, Z) is injective.
в€—

2). П†в€— (F ) = L1 вЉ• В· В· В· вЉ• Lr splits into a sum of line bundles.

Proof. We will work out an induction on r. As before we notice that
over P(F ) we have ПЂ в€— (F ) EвЉ•Q and since Q has rank rв€’1 there exists
by the inductive assumption a space Y and a map П€ : Y в†’ P(F ) such
that one has an injection П€ в€— : H в€— (P(F ), Z) в†’ H в€— (Y, Z) and П€ в€— (ПЂ в€— (F ))
is a direct sum of line bundles. Apparently the space Y and the map
П†П€ : Y в†’ X are what we are looking for.

We notice that the map Y в†’ X just constructed is a п¬Ѓbration. The
п¬Ѓber of Y over x в€€ X is the manifold of complete п¬‚ags in the vector
space Fx , i.e. a typical point in the п¬Ѓber is a sequence of subspaces

0 вЉ‚ V1 вЉ‚ V2 вЉ‚ В· В· В· вЉ‚ Vrв€’1 вЉ‚ Vr = Fx , rank(Vi ) = i.

Let Пѓj (x1 , ..., xr ) be the elementary symmetric polynomial of homoge-
neous degree j in variables (x1 , ..., xr ) so that

Пѓj (x1 , ..., xr ) = (1 + xi ),
j i

then we have
П†в€— cj (F ) = Пѓj (c1 (L1 ), ..., c1 (Lr )).
52 CHAPTER 2. COHOMOLOGY OF VECTOR BUNDLES

Now we would like to have an expression for the Chern classes of the
tensor product of two bundles F вЉ— G in terms of the Chern classes of
each of them. It is a fact that there exist universal polynomials Qj with
integer coeп¬ѓcients such that cj (F вЉ—G) = Qj (c1 (F ), ..., cr (F ), c1 (G), ..., cs (G)).
For example when j = 1 then c1 (F вЉ— G) = rc1 (G) + sc1 (F ). To see
this we apply the splitting principle and п¬Ѓnd a space Y and a map
П† : Y в†’ X such that the map H в€— (X, Z) в†’ H в€— (Y, Z) is injective and
both П†в€— (F ) and П†в€— (G) split into sums of line bundles:

П†в€— (F ) = L1 вЉ• В· В· В· вЉ• Lr , П†в€— (G) = L1 вЉ• В· В· В· вЉ• Ls .

Further we have

П†в€— cj (F ) = Пѓj (c1 (L1 ), ..., c1 (Lr )),

П†в€— cj (G) = Пѓj (c1 (L1 ), ..., c1 (Ls )) , and
П†в€— (F вЉ— G) = вЉ•k,l Lk вЉ— Ll .
So we obtain П†в€— cj (F вЉ— G) = Пѓj (c1 (Lk ) + c1 (Ll ) - the polynomial in rs
variables. It is well-known that Пѓj (c1 (Lk ) + c1 (Ll )) can be expressed in
terms of Пѓi (c1 (Lk )) and Пѓi (c1 (Ll )) . So there are some polynomials Qj
such that polynomials sj (yk + zl ) = Qj (Пѓв€— (yk ), Пѓв€— (zl )). For example,

Пѓ1 (yk zl ) = (yk + zl ) = s yk + r zl = sПѓ1 (yk ) + rПѓ1 (zl ),
1в‰¤kв‰¤r
1в‰¤lв‰¤s

thus c1 (F вЉ— G) = rc1 (G) + sc1 (F ).

2.5 Construction of the Chern character
Let F в†’ X be a complex vector bundle over a CW complex X. We
will construct an element ch(F ) of H в€—в€— (X) = iв‰Ґ0 H i (X, Q) called the
Chern character of F which has the following properties:
1). ch(F вЉ• G) = ch(F ) + ch(G)
2). ch(F вЉ— G) = ch(F )ch(G)
3). ch(L) = exp(c1 (L)) for a line bundle L
4). the usual compatibility with pullbacks
2.6. RIEMANN-ROCH-HIRZEBRUCH THEOREM 53

We will construct the Chern character using the splitting principle. Let
us have a map П† : Y в†’ X such that П†в€— (F ) = L1 вЉ• В· В· В· вЉ• Lr . Then we
should have П†в€— ch(F ) = ec1 (L1 ) +В· В· В·+ec1 (Lr ) . On the other hand we have
П†в€— (cj (F )) = Пѓj (c1 (L1 ) + В· В· В· + c1 (Lr )). We notice that if Pi (x1 , ..., xr ) =
xi +В· В· В·+xi is the so-called power-sum symmetric polynomial, then by a
1 r
theorem of Sir Isaac Newton it is possible to express Pi as a polynomial
with integer coeп¬ѓcients in Пѓ1 , ..., Пѓi : Pi = Ni (Пѓ1 , ..., Пѓi ). For instance,
2 3
P1 = Пѓ1 , P2 = Пѓ1 в€’ 2s2 , P3 = Пѓ1 в€’ 3Пѓ2 Пѓ1 + 3Пѓ3 and so on. Due to the
representation of П†в€— (ch(F )) as a series

1 1
П†в€— (ch(F )) = r + P1 (c1 (Lj )) + P2 (c1 (Lj )) + P3 (c1 (Lj )) + В· В· В·
2 3!
we get the following expression of the Chern character of F in terms of
its Chern classes:
1 1
ch(F ) = r+N1 (c1 (F ))+ N2 (c1 (F ), c2 (F ))+ N3 (c1 (F ), c2 (F ), c3 (F ))+В· В· В· .
2 3!
Now assume that we have a formal power series f (x) = 1 + a1 x +
x
a2 x2 + В· В· В· (for instance, f (x) = 1в€’eв€’x ). Then following Hirzebruch
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