χ(CPn , E — L—m ) = dim H 0 (CPn , E — L—m ).

i

A similar result holds true for any projective algebraic manifold X ’

CPn and any holomorphic vector bundle E over X but one has to replace

L by i— 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 ∞

rather than a holomorphic vector bundle F over a complex manifold X.

If the rank of this bundle is r then one can de¬ne so-called Chern classes

c0 (F ), c1 (F ), ..., cr (F ) of this bundle such that c0 (F ) = 1 and cj (F ) ∈

π

H 2j (X, Z). Let us construct an auxiliary bundle P(F ) ’ X with ¬ber

equal to the projective space CPr’1 obtained via the projectivization

of the bundle F . The ¬ber π ’1 (x) over x ∈ 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}/C— as a complex manfold, where the action of C— by

dilations in each ¬ber comes from the de¬nition of a vector bundle. Of

course π is a locally trivial ¬bration: if U ‚ X is an open set such that

U — Cr then P(F )|U U — CPr’1 . One has the natural algebra

F|U

homomorphism

π — : H — (X, Z) ’ H — (P(F ), Z). (2.4.1)

A point of P(F ) is a pair (x, l), where x ∈ X and l is a line in the

¬ber Fx . Let us construct a tautological line bundle E ’ P(F ) that is

a complex subbundle of the pullback vector bundle π — F . The ¬ber of

E at a point (x, l) is the line l: E(x,l) = l ‚ (π — (F ))(x,l) = Fx . Note that

48 CHAPTER 2. COHOMOLOGY OF VECTOR BUNDLES

the restriction of E to a ¬ber π ’1 (x) = P(Fx )’CPr’1 identi¬es with

˜

the tautological line bundle over projective space. Let L stand for the

dual line bundle: L = E — . 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 ξ ∈ H 2 (P(F ), Z) this class for the bundle

L. The class ξ is actually the ¬rst 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 — (P(X), Z) is a free module over H — (X, Z) with basis 1, ξ, ξ 2 , ..., ξ r’1 .

The module structure is de¬ned by

a · b = π — (a) ∪ b , a ∈ H — (X, Z) , b ∈ H — (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

de¬ned classes c1 (F ), ..., cr (F ) ∈ H — (X, Z) such that cj ∈ H 2j (X, Z)

and

ξ r + c1 (F )ξ r’1 + · · · + cr’1 (F )ξ + cr (F ) = 0. This is how we de¬ne

the Chern classes. There are several possible de¬nitions of them: via

obstruction theory, via the curvature of a connection, and so on and all

those de¬nitions re¬‚ect di¬erent sides of the same object.

Proof of Lemma. If F is a trivial bundle then we know that that

H — (CPr’1 ) is torsion free and the K¨nneth formula produces

u

H — (P(F ), Z) = H — (X, Z) — H — (CPr’1 , Z) = H — (X) — 1, ξ, ξ 2 , ..., ξ r’1 .

Next we assume that X can be covered by a ¬nite number of open

sets: X = U1 ∪ ... ∪ 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 ∪ V , where V = U2 ∪ ... ∪ Un . One has P(F ) =

(π ’1 (U1 )) ∪ (π ’1 (V )) and further

δ

’ •r’1 H p’1’2j (U1 © V ) ’ •r’1 H p’2j (X) ’ •r’1 H p’2j (U1 ) • H p’2j (V ) ’

j=0 j=1 j=1

“ψ

δ

’ H p’1 (π ’1 (U1 © V )) ’ H p (P(F )) ’ H p (π ’1 (U1 )) • H p (π ’1 (V ) ’

where ψ(a0 , ..., ar’1 ) = r’1 π — (aj ) ∪ ξ j . It then follows from the ¬ve

j=0

lemma that ψ 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 ¬ne, but in general when it is not possible to cover X by

a ¬nite 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 — (X) = lim← H — (Um ) and H — (P(F )) = lim← H — (π ’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 • G) = r cj (F ) ∪ cr’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 ) ∈

even

H (X, Z). Using this notation the Whitney sum formula becomes

c(F • G) = c(F ) ∪ 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 ¬nite-

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 ¬brations on X:

CPr’1 CPs’1 CPs+r’1

’ P (F ) ’ P (G) ’ P (F • G)

“ “ “

X X X

Also we have the classes ξF ∈ H 2 (P(F )) and ξG ∈ H 2 (P(G)). Let us

look at CPr+s’1 = P(Cr • Cs ) and de¬ne two open sets U, V ‚ CPr+s’1

such that U consists of lines not belonging to that particular Cs and

V the same with Cr . We get natural maps U ’ CPr’1 and V ’

CPs’1 . Now if E2 is the tautological line bundle over CPr+s’1 and E1 is

the tautological line bundle over CPr’1 then (E2 )|U identi¬es 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 • G). Here

U = P(F • G) \ P(G) and V = P(F • G) \ P(F ). We have projection

maps p1 : U ’ P(F ) and p2 : V ’ P(G). These projection maps are

compatible with the tautological line bundles: the restriction to U of

the tautological line bundle EF •G identi¬es with p— EF . This implies

1

— 2

that p1 ξF is the image of ξF •G in H (U ).

Returning to the Whitney formula we see that we need to prove the

identity

r’j s’k

( cj (F )ξF •G )( ck (F )ξF •G ) = 0, (2.4.2)

j k

r+s’l

because by de¬nition l cl (F •G)ξF •G = 0. Let us notice the following

LEMMA 2.4.2 Let Y = U ∪ V be the union of two open sets and let

a, b ∈ H — (Y ) be such that a has zero image in H — (U ) and b has zero

image in H — (V ) then a ∪ b = 0.

Now the Whitney formula follows because the ¬rst cohomology class

r’j

in 2.4.2 has zero image in H — (U ) since by de¬nition j cj (F )ξF = 0

r’j

and its pull-back to U under p1 : U ’ P(F ) is equal to j cj (F )ξF •G .

Analogously the second term has zero image in H — (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 ¬rst Chern class c1 (L) is the one de¬ned

in the end of Section 2.

3). Compatibility with pullbacks: if one has a map f : Y ’ X and

a complex vector bundle F over X then cj (f — F ) = f — cj (F ) ∈ H 2j (Y ).

4). The Whitney sum formula: c(F • G) = c(F ) ∪ 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 δj be a set of cohmology classes satisfying the properties

1).-4). Over P(F ) we have an exact sequence of vector bundles

0 ’ E ’ π — F ’ Q ’ 0.

2.4. CHERN CLASSES OF COMPLEX VECTOR BUNDLES 51

We may write π — (F ) = E•Q because any exact sequence of C ∞ bundles

is necessarily split. As usual we denote the total class δ(F ) = j δj (F ),

then by the Whitney formula δ(Q) = δ(π — F )δ(E)’1 . But E is the

tautological line bundle so by property 2). one has δ(E) = 1’c1 (E — ) =

1 ’ ξ. It follows that

δ(Q) = (1 + δ1 (π — F ) + δ2 (π — F ) + · · ·)(1 + ξ + ξ 2 + · · ·) =

= (1 + π — δ1 (F ) + π — δ2 (F ) + · · ·)(1 + ξ + ξ 2 + · · ·).

using property 3). Thus we obtain δr (Q) = ξ r + j δj (F )ξ r’j as desired,

because by 1). we have δr (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

φ : Y ’ X such that

φ—

1). The map H (X, Z) ’ H — (Y, Z) is injective.

—

2). φ— (F ) = L1 • · · · • 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 π — (F ) E•Q and since Q has rank r’1 there exists

by the inductive assumption a space Y and a map ψ : Y ’ P(F ) such

that one has an injection ψ — : H — (P(F ), Z) ’ H — (Y, Z) and ψ — (π — (F ))

is a direct sum of line bundles. Apparently the space Y and the map

φψ : Y ’ X are what we are looking for.

We notice that the map Y ’ X just constructed is a ¬bration. The

¬ber of Y over x ∈ X is the manifold of complete ¬‚ags in the vector

space Fx , i.e. a typical point in the ¬ber is a sequence of subspaces

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

Let σj (x1 , ..., xr ) be the elementary symmetric polynomial of homoge-

neous degree j in variables (x1 , ..., xr ) so that

σj (x1 , ..., xr ) = (1 + xi ),

j i

then we have

φ— cj (F ) = σj (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 — G in terms of the Chern classes of

each of them. It is a fact that there exist universal polynomials Qj with

integer coe¬cients such that cj (F —G) = Qj (c1 (F ), ..., cr (F ), c1 (G), ..., cs (G)).

For example when j = 1 then c1 (F — G) = rc1 (G) + sc1 (F ). To see

this we apply the splitting principle and ¬nd a space Y and a map

φ : Y ’ X such that the map H — (X, Z) ’ H — (Y, Z) is injective and

both φ— (F ) and φ— (G) split into sums of line bundles:

φ— (F ) = L1 • · · · • Lr , φ— (G) = L1 • · · · • Ls .

Further we have

φ— cj (F ) = σj (c1 (L1 ), ..., c1 (Lr )),

φ— cj (G) = σj (c1 (L1 ), ..., c1 (Ls )) , and

φ— (F — G) = •k,l Lk — Ll .

So we obtain φ— cj (F — G) = σj (c1 (Lk ) + c1 (Ll ) - the polynomial in rs

variables. It is well-known that σj (c1 (Lk ) + c1 (Ll )) can be expressed in

terms of σi (c1 (Lk )) and σi (c1 (Ll )) . So there are some polynomials Qj

such that polynomials sj (yk + zl ) = Qj (σ— (yk ), σ— (zl )). For example,

σ1 (yk zl ) = (yk + zl ) = s yk + r zl = sσ1 (yk ) + rσ1 (zl ),

1¤k¤r

1¤l¤s

thus c1 (F — G) = rc1 (G) + sc1 (F ).

2.5 Construction of the Chern character

Let F ’ X be a complex vector bundle over a CW complex X. We

will construct an element ch(F ) of H —— (X) = i≥0 H i (X, Q) called the

Chern character of F which has the following properties:

1). ch(F • G) = ch(F ) + ch(G)

2). ch(F — 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 φ : Y ’ X such that φ— (F ) = L1 • · · · • Lr . Then we

should have φ— ch(F ) = ec1 (L1 ) +· · ·+ec1 (Lr ) . On the other hand we have

φ— (cj (F )) = σj (c1 (L1 ) + · · · + c1 (Lr )). We notice that if Pi (x1 , ..., xr ) =

xi +· · ·+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 coe¬cients in σ1 , ..., σi : Pi = Ni (σ1 , ..., σi ). For instance,

2 3

P1 = σ1 , P2 = σ1 ’ 2s2 , P3 = σ1 ’ 3σ2 σ1 + 3σ3 and so on. Due to the

representation of φ— (ch(F )) as a series

1 1

φ— (ch(F )) = r + P1 (c1 (Lj )) + P2 (c1 (Lj )) + P3 (c1 (Lj )) + · · ·

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 ))+· · · .

2 3!

Now assume that we have a formal power series f (x) = 1 + a1 x +

x

a2 x2 + · · · (for instance, f (x) = 1’e’x ). Then following Hirzebruch