is clearly compact.

The next case to consider is n = 2. Here we have CP2 = U0 ∪U1 ∪U2 .

The subset U0 identi¬es with C2 via ψ0 [z0 : z1 : z2 ] = (z1 /z0 , z2 /z0 ).

C2 is provided by ψ1 [z0 : z1 : z2 ] = ( z0 , z2 ). We

z

Analogously, U1 1 z1

have:

ψ0 2

C2 \ (u = 0) (u, v) ∈ C

U0,1 ’

ψ1 φ0,1

(±, β) ∈ C2

C2 \ (± = 0)

We notice that

1v

φ0,1 (u, v) = ( , )

uu

is a holomorphic map. Its inverse

1β

φ’1 (±, β) = ( , )

0,1

±±

10 CHAPTER 1. HOLOMORPHIC VECTOR BUNDLES

is a holomorphic map too. We conclude that φ0,1 is biholomorphic.

Another way to deduce biholomorphicity is to consider the di¬erential

map dφ0,1 (u, v) : C2 ’ C2 . We have

‚β

‚± 1 v

’ u2 ’ u2

‚u ‚v

dφ0,1 = = .

1

‚β

‚±

0 u

‚v ‚u

1

The determinant of dφ0,1 (u, v) is equal to ’ u3 and is never zero. Since

φ0,1 is holomorphic and bijective, then it is automatically a biholomor-

phic map.

We have a well-known

Fact. Any holomorphic function on a compact connected manifold M

is constant.

Proof. Let f : M ’ C be a holomorphic function. Since M is compact,

|f | attains its maximum value at some point. The maximum principle

implies that f is constant in some neighbourhood of this point. Now

the principle of analytical continuation tells us that f is locally constant

on M .

Exercise. Another consequence of the maximum principle is that a

compact connected complex Lie group is always abelian.

1.3 Holomorphic line bundles

Now we are ready to consider the ¬rst example of a non-trivial line

bundle E on CPn , which is called the tautological line bundle. The bun-

dle E is de¬ned as a subbundle of the trivial bundle Cn+1 — CPn . More

precisely, E = {(v, l) ∈ Cn+1 — CPn ; v ∈ l}. We can show that this line

bundle satis¬es the local triviality condition by ¬nding a non-vanishing

section over each open set of some covering. Let us de¬ne the section σi

over Ui ‚ CPn by σi [z0 : ... : zn ] = ((z0 /zi , ..., zi’1 /zi , 1, zi+1 /zi , ..., zn /zi ),

line through (z0 , ..., zn )).

It turns out that the line bundle E has no non-zero global sections.

LEMMA 1.3.1 “hol (E) = {0}.

Proof. We have the inclusion “hol (E) ‚ “hol (Cn+1 — CPn ). The latter

space is the same as the space of (n+1)-tuples (f1 , ..., fn+1 ) of functions

1.3. HOLOMORPHIC LINE BUNDLES 11

on CPn . But each fi is a constant function by the above lemma. Now

we take any v ∈ Cn+1 \ {0}. For n > 0, which is the case, there always

exists a line l not containing v. So, v is not a section of E.

Similarly, one can prove that the space “(E —k ) has dimension 0,

where E —k = E — · · · — E for k > 0. When k = 0, E —0 is by de¬nition

k

the trivial line bundle 1CPn .

On the contrary, the tensor powers of dual bundle L = E — have

many non-trivial sections.

PROPOSITION 1.3.2 “hol (L—k ) = { degree k homogeneous polyno-

mials on Cn+1 } =: S k (Cn+1 )

Before proving this assertion, we establish some auxiliary results. First,

we notice that the space of holomorphic sections of L—k is the same as

the space of holomorphic functions f on the total space of E such

that f (»x) = »k f (x) for any x ∈ E and » ∈ C— . Verbally, such a

function is homogeneous of degree k with respect to the C— -action. To

see this, we start with k = 1. Consider a holomorphic section σ of

L. We have L = E — = Hom(E, C — CPn ), where C — CPn = 1CPn is

the trivial line bundle. So, σ is a bundle homomorphism, which gives

rise to a holomorphic function f : E ’ C as f (x) = σp(x) (x). And we

have f (»x) = »f (x). Now we treat the general case k ≥ 1. Similarly,

we have L—k = Hom(E —k , C — CPn ). A section σ of L—k is a bundle

homomorphism E —k ’ C — CPn . Now we de¬ne a function f on E by

f (x) = σ(x—k ). And we have f (»x) = »k f (x).

Let us take a closer look at the geometry of E (we always assume

n ≥ 1). In addition to the projection p : E ’ CPn we have another

projection q : E ’ Cn+1 , q(x, l) = x:

q

CPn = q ’1 (0) Cn+1

’ E ’

Id|| p ∪ ∪ .

Id

CPn Cn+1 \ {0} ’ Cn+1 \ {0}

When x = 0, the preimage q ’1 (x) is just the line C.x. E is the so-called

blow-up of Cn+1 at the point 0. So, we see that the maps Hol(Cn+1 ) ’

Hol(E) ’ Hol(Cn+1 \ {0}) are injective. But in complex analysis the

singularities in codimension ≥ 2 are removable and we prove a special

case of this assertion

12 CHAPTER 1. HOLOMORPHIC VECTOR BUNDLES

THEOREM 1.3.3 (Hartog). Any holomorphic function on Cn \

{0}, n ≥ 2 is extendable to a holomorphic function on Cn .

Proof. It is enough to consider n = 2. We have a holomorphic func-

tion f (z1 , z2 ) de¬ned in R = {|z1 |, |z2 | ¤ 1}. For z1 ¬xed we con-

struct a new function of one variable g(z) = f (z1 , z) in the region

|z| ¤ 1, z = 0. It has a Laurent series like g(z) = n∈Z an z n , where

1

an = an (z1 ) = 2π√’1 C z ’n’1 f (z1 , z)dz. The series converges uniformly

on the annulus µ < z < 1. We have that whenever z1 = 0, g(z) is holo-

morphic in |z| ¤ 1, which implies that an (z1 ) = 0, whenever n < 0 and

z1 = 0. Just by the continuity, an (z1 ) = 0 in the disk |z1 | ¤ 1 when

n

n < 0. Whence, f (z1 , z2 ) = n≥0 an (z1 )z2 converges absolutely in R.

The above theorem shows that we have the same space of holo-

morphic functions on Cn+1 , Cn+1 \ {0}, and E. Now, “hol (L—k ) = {

holomorphic homogeneous of degree k functions on E} = { homoge-

neous of degree k holomorphic functions on Cn+1 } = { homogeneous

polynomials of degree k on Cn+1 }. This concludes the proof of the

proposition.

If one wants to measure the growth of the dimension of “hol (L—k ),

one can form the power series

dim “hol (L—k )tk .

F (t) =

k≥0

Such a series is called the Hilbert-Samuel series or Poincar´ series. In

e

1

dimension 1 (n = 0) we have F (t) = 1 + t + t2 + · · · = 1’t . Looking at

the relations S(Cn+1 ) = •k≥0 S k (Cn+1 ) and S(Cl • Cm ) = S(Cl ) —S(Cm )

one sees that in our situation

1 (n + 1)(n + 2) · · · (n + k)

, and dim S k (Cn+1 ) =

F (t) = .

(1 ’ t)n+1 k!

In our further study we need the notion of a pull-back of a vector

bundle. For a smooth map f : Y ’ X consider the diagram

V ← V —X Y

p“ “q

f

X← Y

1.3. HOLOMORPHIC LINE BUNDLES 13

The manifold V —X Y ‚ V — Y is the ¬ber product and is de¬ned as

V —X Y = {(v, y) : p(v) = f (y)}. So, the ¬bers of the map q do not

change and are the same as those of p. In fact, V —X Y is again a vector

bundle, which is called the pull-back of V via f and usually is denoted

by f — (V ).

An example of a pull-back bundle is provided by a restriction of a

vector bundle V on X to the submanifold Y ‚ X. The notation for this

restriction which we are going to use is V|Y . For instance, the tangent

±

bundle to Y is the subbundle of T X|Y : T Y ’ T X|Y . The bundle map

± has a constant rank equal to the dimension of Y . The quotient bundle

is called the normal bundle NY ’X = T X|Y /T Y . We may summarize

the results by so-called exact sequence of vector bundles:

0 ’ T Y ’ T X|Y ’ NY ’X ’ 0

Example. The inclusion Cn ’ Cn+1 induces an inclusion CPn’1 ’

CPn . The normal bundle here is just a line bundle and we will see that

it is isomorphic to L.

We brie¬‚y mention here an important property of the line bundle

L called the universality property. It means the following. Given a

complex line bundle F over a smooth compact manifold X, there exists

an integer n and a smooth map f : X ’ CPn such that F f — L.

Let us stress here two signi¬cant di¬erences between smooth and

complex setups. The ¬rst one consists of the fact that given an arbitrary

vector bundle E over a smooth manifold X, for any point x ∈ X, there

always exists a section σ of X such that σ(x) = 0. On the contrary, for

a holomorphic vector bundle over a complex manifold it happens quite

often that the bundle does not have non-zero sections at all. The second

important distinction is that an exact sequence of C ∞ vector bundles

over a smooth manifold is always split. Though, for the holomorphic

bundles we will see myriads of examples when exact sequences are not

split.

Returning to the example of CPn we shall give a geometric picture

of its holomorphic tangent bundle T (CPn ). We have a principal ¬ber

14 CHAPTER 1. HOLOMORPHIC VECTOR BUNDLES

bundle with structure group C— :

C— ’ Cn+1 \ {0}

q“

CPn

The ¬ber q ’1 [z0 : z1 : · · · : zn ] is the C— -orbit of (z0 , z1 , ..., zn ). Over

Cn+1 \ {0} we have the holomorphic vector ¬elds ‚/‚z0 , ..., ‚/‚zn . If

we make a dilation (z0 , ..., zn ) ’ (»z0 , ..., »zn ), each ‚/‚zj will be mul-

tiplied by »’1 . We explain the following

Basic fact. The vector ¬elds ‚/‚z0 , ..., ‚/‚zn give holomorphic sec-

tions of T (CPn ) — E.

So, we have (n + 1) holomorphic sections of the vector bundle T (CPn ) —

L— . In fact, this bundle is generated everywhere by these sections, so

that we have a surjective bundle morphism: Cn+1 — CPn ’ T (CPn ) —

L— . If we tensor this with L, we get another surjective bundle mor-

phism: Cn+1 — L ’ T (CPn ). The bundle Cn+1 — L is isomorphic to

L • L • · · · • L.

n+1

We take a point z ∈ CPn and consider the di¬erential map dq :

Cn+1 ’ T[z] CPn . It is surjective and its kernel is the tangent space to the

line C.z spanned by the holomorphic Euler vector ¬eld Eu = n zi ‚zi . ‚

i=0

It is homogeneous of degree zero and Eu(zi ) = zi . Hence we obtain an

exact sequence of holomorphic vector bundles over CPn :

0 ’ 1CPn ’ Cn+1 — L ’ T CPn ’ 0. (1.3.1)

This is an example of an exact sequence which is not split. To see it,

notice that any bundle homomorphism Cn+1 —L ’ 1CPn is zero, because

E does not have non-trivial sections and hence is not surjective.

Next, we remark that the group GL(n + 1, C) acts transitively on

both Cn+1 \ {0} and CPn . Also it acts on L. The exact sequence

(?.?) is equivariant under the action of GL(n + 1, C). Notice, that

its center C— ‚ GL(n + 1, C) acts trivially on CPn and all the above

bundles. So, the whole business is equivariant under the quotient group

P GL(n + 1, C).

Another very interesting example of a line bundle is provided by the

canonical line bundle C. On the projective space it is de¬ned as C =

1.4. DIVISORS ON RIEMANN SURFACES 15

§n T — (CPn ) - the maximum exterior power of the holomorphic cotangent

bundle. Locally, a section of C looks like ω = f (z)dz1 § · · · § dzn , where

f (z) is locally a holomorphic function. We can understand C better

if we notice that C = (§n T (CPn ))— and use the fact, that whenever

we have an exact sequence of vector bundles in the form 0 ’ F ’

G ’ H ’ 0, the isomorphism §max G §max F — §max H takes place.

Applying this to exact sequence (?.?), we get C = (§n+1 (Cn+1 — L))— =

(L—n+1 )— E —n+1 .

It is true in general that all the line bundles over CPn are isomorphic

to L—k , k ∈ Z. (Here we adopt the notation L—’1 = L— .) As one notices,

isomorphism classes of line bundles over any manifold form an abelian

group with respect to the tensor product. In our case, this group is

isomorphic to Z. In general, it is called the Picard group P ic(X) = {

the group of isomorphism classes of line bundles on X}.

We notice that there exists no non-zero holomorphic n-form on CPn .

This may be easily checked for n = 1. As before, CP1 = U0 ∪ U1

with local coordinates z and w on U0 and U1 respectively. On the

intersection U0 ©U1 one has z = 1/w and dz = ’dw/w2 . Let f (z)dz be a

holomorphic form on U0 . We have a Taylor expansion f (z) = n≥0 an z n

and f (z)dz = ’ n≥0 an w’n’2 dw. So, it is not holomorphic. The form

dz has a pole of order 2 at z = ∞. In general, a holomorphic section

of C has a pole of order n + 1 on a hyperplane X ‚ CPn .

1.4 Divisors on Riemann surfaces

We take a compact connected Riemann surface X and a holomorphic

line bundle L on it. We may associate an integer number to L which

is called the degree of L. The construction of deg(L) is as follows.

A line bundle L always has a non-zero meromorphic section s. It

means that s is a holomorphic section everywhere except for a ¬nite set

of points, and in a neighbourhood of any point p ∈ X it is equal to z m .σ,