Chapter 2

The de Rham cohomology

2. 1 Differential operators 2. 1 . 1 Definitions ∞ For a Riemann surface X we denote by AX the sheaf of differential ( C ) func- tions with values in C. Let U ⊂ C be an open set. We have the following differential operators: 1 1 ( 2. 1 . 1 . 1 ) ∂x , ∂y , ∂z = ( ∂x − i∂y ) , ∂¯z = ( ∂x + i∂y ) . 2 2

We compute easily ∂z ( z) = 1 , ∂¯z ( z) = 0, and

( 2. 1 . 1 . 2) ∂¯z ( f) = ∂z ( f¯) , ∂z ( f) = ∂¯z ( f¯) , where f¯ = z¯ ◦ f, i. e. f¯ = f1 − if2 if f = f1 + if2 . A function f is holomorphic iff ∂¯z ( f) = 0.

2. 1 . 2 Global point of view The differential operators 2. 1 . 1 . 1 depend on the coordinates, and therefore they do not glue to global objects on a Riemann surface. In order to get a global derivation d the values of d have to be differential forms ( which transform dually to the derivations) : 1 ( 2. 1 . 2. 1 ) d : AX −→ AX ; d( f) = ∂x ( f) dx + ∂y ( f) dy = ∂z ( f) dz + ∂¯z ( f) dz¯. Here, dz = dx + idy, dz¯ = dx − idy, and we write A1 for the 1 -forms. In 2. 1 . 2. 1 , f is supposed to be a local section, i. e. f ∈ AX ( V) , and V is isomorphic to an open set of C. The result d( f) doesn’ t depend on the choice of the isomorphism, and commutes with restriction: d( f) | U = d( f| U ) for any open U ⊂ V. For any open set V ⊂ X and f ∈ AX ( V) we can find an open cover V = ∪ i Vi with 1 Vi ⊂ C. The local sections d( f| Vi ) glue to give an element d( f) ∈ A ( V) . Therefore, d is a map of sheaves.

1 3 1 4 CHAPTER 2. THE DE RHAM COHOMOLOGY

2. 2 Poincare’ s Lemma We can go one step further and differentiate 1 -forms:

1 2 d : AX −→ AX ; d( fx dx + fy dy) = ( ∂x ( fy ) − ∂y ( fx ) ) dx ∧ dy.

Equivalently,

d( f1 , 0 dz + f0, 1 dz¯) = ( ∂z ( f0, 1 ) − ∂¯z ( f1 , 0 ) ) dz ∧ dz¯.

Again, this is a map of sheaves. The composition d ◦ d = 0 is trivial, and thus

d 1 d 2 AX −→ AX −→ AX

is a complex. Since d( f) = 0 for a function f implies that f is locally constant we obtain a complex: d 1 d 2 C −→ AX −→ AX −→ AX , where C is the constant sheaf. Theorem 2. 2. 1 ( Poincare’ s Lemma) . Let X ⊂ C be a star-shaped open set then the complex d 1 d 2 C −→ AX ( X) −→ AX ( X) −→ AX ( X) , is acyclic. If X is any Riemann surface then the complex of sheaves

d 1 d 2 AX −→ AX −→ AX ,

is a resolution of C.

1 Proof. First, let ω ∈ AX ( X) , ω = ωx dx + ωy dy, with d( ω) = 0 ⇔ ∂x ωy − ∂y ωx = 0. There is a point 0 ∈ X, such that for every point ( x, y) ∈ X the line from 0 to ( x, y) is contained in X. We paramatrize this line with γ( x, y) : [ 0, 1 ] −→ X, t �→ ( tx, ty) , and define � � 1 ( 2. 2. 1 . 1 ) f( x, y) : = ω = ( xωx + yωy ) dt. γ( x , y ) 0

An easy computation shows d( f) = ω. 2 For ω ∈ AX ( X) , ω = ωxy dx ∧ dy, we see that � � � � � � 1 1 f = − tyωxy ( tx, ty) dt dx + txωxy ( tx, ty) dt dy 0 0

solves the equation d( f) = ω. The second statement can be checked locally. For a point x ∈ X we need to show that the complex

d 1 d 2 C −→ AX, x −→ AX, x −→ AX, x 2. 2. POINCARE’ S LEMMA 1 5

for the stalks at x is acyclic. Now, i i AX, x = lim AX ( U) , x∈ U ⊂ X, U ∼= ∆ where the inductive limit is taken over all open neighbourhoods of x that are isomorphic to an open disc. Thus we are done. Definition 2. 2. 2. The cohomology groups of the complex d 1 d 2 AX ( X) −→ AX ( X) −→ AX ( X) are called the de Rham cohomology of X: 0 1 HdR ( X) =∼ ker( d : A( X) −→ A ( X) ) = { locally constant functions on X} = Homsets ( π0 ( X) , C) , 1 1 2 HdR ( X) =∼ ker( d : A ( X) −→ A ( X) ) /dA( X) = { closed 1 -forms} /{ exact 1 -forms} , 2 2 1 HdR ( X) =∼ A ( X) /dA ( X) .

If X is a compact Riemann�surface then there is natural map 2 : HdR ( X) −→ C, X � � 2 � where a class ω ∈ HdR ( X) maps to X ω ∈ C, and ω is a representative of ω ( by the Stokes-Theorem this doesn’ t depend on the choice of ω. ) Theorem 2. 2. 3. Let X be a simply connected Riemann surface then 1 HdR ( X) = 0. 1 Proof. Fix ω ∈ AX ( X) with d( ω) = 0, and a point 0 ∈ X. For every point x ∈ X we choose a path γ from 0 to x an�d claim that the integral f( x) = ω γ is independent of the choice of γ. Indeed, let γ� be another path. By assumption π1 ( X) = 0, and therefore there is a homotopy H : [ 0, 1 ] × [ 0, 1 ] −→ X such that � H| 0× [ 0, 1 ] = γ, H| 1 × [ 0, 1 ] = γ , H| [ 0, 1 ] × 0 = 0, H| [ 0, 1 ] × 1 = x.

The Stokes-The�orem implies � � d( H∗ ω) = H∗ ω − H∗ ω [ 0, 1 ] × [ 0, 1 ] �0× [ 0, 1 ] � 1 × [ 0, 1 ] = ω − ω. γ γ� Now d( H∗ ω) = H∗ ( d( ω) ) = 0 proves the claim. The equation d( f) = ω may be checked locally, and locally f is defined as in 2. 2. 1 . 1 . 1 6 CHAPTER 2. THE DE RHAM COHOMOLOGY

2. 2. 4 Comparison of de Rham and Cˇ ech cohomology Corollary 2. 2. 5. Let X be a Riemann surface. There is a natural isomorphism

1 1 H ( X, C) =∼ HdR ( X) .

Proof. Let U = ( Ui ) i∈ I be an open covering of X with Ui =∼ ∆ for very i. We form the d − δ double complex:

� � 1 � 2 C A( X) A ( X) A ( X)

� � � � ∗ � ∗ � ∗ 1 � ∗ 2 C ( U, C) C ( U, A( X) ) C ( U, A ( X) ) C ( U, A ( X) )

where C∗ ( U, ?) are the Cˇ ech complexes and the horizontal arrows are induced by d. By Proposition 1 . 3. 3 the second, third and fourth column is acyclic. Lemma 1 . 1 . 8 gives a natural map

i i ( 2. 2. 5. 1 ) Hˇ ( U, C) −→ H ( A( X) −→ A1 ( X) −→ A2 ( X) )

for all i ≥ 0. By Poincare’ s Lemma the row

0 � 0 � 0 1 � 0 2 C ( U, C) C ( U, A( X) ) C ( U, A ( X) ) C ( U, A ( X) )

is acyclic, and by Lemma 1 . 1 . 8 the map 2. 2. 5. 1 is an isomorphism for i = 1 . Since any open covering ( Vj ) j can be refined by an open covering U = ( Ui ) i∈ I with Ui =∼ ∆ open discs, we can form the inductive limit

H1 ( X, C) = lim Hˇ 1 ( U, C) U = ( Ui ) i ∈ I

by taking only these coverings into account. This proves the claim. Corollary 2. 2. 6. For any simply connected Riemann surface X we have

H1 ( X, C) = 0 = H1 ( X, Z) .

Proof. For H1 ( X, C) = 0 ⇔ H1 ( X, Z) = 0: Let U be some covering and a ∈ 1 1 1 Z ( U, Z) ⊂ Z ( U, C) . There is ( ci ) i∈ I ∈ C ( U, C) with ai 0 , i 1 = ci 1 − ci 0 . The √ ∗ ∗ ∗ elements bi = exp( 2π − 1 ci ) ∈ C ( Ui ) glue to b ∈ C ( X) = C . Choose c ∈ C √ 1 with exp( 2π − 1 c) = b then ( ci − c) i∈ I ∈ C ( U, Z) and δ( ( ci − c) i ) = a. Remark 2. 2. 7. The statement of 2. 2. 5 also holds if X is a compact manifold: let Ap be the p-forms, then p p p Hˇ ( X, C) = H ( X, C) =∼ HdR ( X) ( 2. 2. 7. 1 ) = { closed p-forms} /{ exact p-forms} .

For the proof we need to know that there are good open coverings U = ( Ui ) i∈ I , i. e. every finite intersection of open sets in U is either empty or contractible. 2. 3. H1 AND THE FUNDAMENTAL GROUP 1 7

Exercise 2. 2. 8. Let C∗ = C\{ 0} resp. ∆ ∗ = ∆\{ 0} be the punctured complex plane resp. unit disc. Show that H1 ( C∗ , Z) = H1 ( ∆ ∗ , Z) = Z. Find a generator 1 ∗ 1 ∗ for the de Rham cohomology HdR ( C ) resp. HdR ( ∆ ) . Exercise 2. 2. 9. For a torus T = C/Λ we get

A1 ( T) = { ω ∈ A1 ( C) ; λ∗ ( ω) = ω ∀λ ∈ Λ} ,

where λ : C −→ C is translation by λ. By using the Poincare Lemma for C show that

0 1 2 HdR ( T) = C, HdR ( T) = Cdx ⊕ Cdy, HdR ( T) = Cdx ∧ dy.

2. 3 H1 and the fundamental group Let X be topological space. Definition 2. 3. 1 . A Z-torsor is a (´etale) covering p : Y −→ X with Z operation

Z × Y −→ Y

over X ( Z with discrete topology) , such that Z acts freely and transitively on the fibres of p. The most important example of a Z-torsor is the following is X = S1 , Y = R, then p = y �→ exp( 2πiy) defines a Z-torsor. A morphism of Z-torsors is a continuous map of coverings Y1 −→ Y2 which is compatible with the Z action. It is automatically an isomorphism. The trivial Z-torsor is X × Z and we get

Aut( X × Z) = Z( X) = Homtop( X, Z) .

Obviously, a Z-torsor p : Y −→ X is trivial if and only if there is a section X −→ Y of p. In particular if X is simply connected then every Z-torsor is trivial. Proposition 2. 3. 2. Let X be a manifold. There is a natural bijection:

H1 ( X, Z) −→ { Z-torsors} /isomorphisms.

Proof. Let U = ( Ui ) i∈ I be a covering with Ui simply connected for every i ∈ I. 1 1 We know that H ( X, Z) = Hˇ ( U, Z) from Corollary 2. 2. 6. Let a = ( a( i 0 , i 1 ) ) ∈ 1 Z ( U, Z) , we set Yi = Ui × Z and define maps φi, j :

φ i j i − 1 � j − 1 Y | pr1 ( Ui ∩ Uj ) Y | pr1 ( Ui ∩ Uj )

= = � � ( id, + a ( i , j ) ) � ( Ui ∩ Uj ) × Z ( Ui ∩ Uj ) × Z. 1 8 CHAPTER 2. THE DE RHAM COHOMOLOGY

It is easy to see that a( i, k ) = a( i, j) + a( j, k ) implies φi, k = φj, k ◦ φi, j , and therefore 0 the ( Yi ) i∈ I glue to a Z-torsor which we denote by Y( a) . For b ∈ C ( U, Z) we get an isomorphism ∼= Y( a) −→ Y( a + δ( b) ) ,

( id, + b i ) which is given by Yi = Ui × Z −−−−−→ Ui × Z = Yi on the open sets Yi . Therefore

H1 ( X, Z) −→ { Z-torsors} /iso, [ a] �→ Y( a) ,

is well-defined. Now, suppose that p : Y −→ X is a Z-torsor. Since Ui is simply connected we see that there is an isomorphism s i : ( Ui × Z) −→ Y | Ui ( this map is unique up to translation with a section in Z( Ui ) ) . On the intersection Ui 0 ∩ Ui 1 we − 1 get s i 1 ◦ s i 0 ∈ Aut( ( Ui 0 ∩ Ui 1 ) × Z) = Z( Ui 0 ∩ Ui 1 ) . This defines an element − 1 ˇ 1 a( Y) : = ( s i 1 ◦ s i 0 ) ( i 0 , i 1 ) ∈ H ( U, Z) . It is easy to check that this gives a well- defined map: { Z-torsors} /iso −→ H1 ( X, Z) , Y �→ a( Y) .

Proposition 2. 3. 3. Let X be a connected manifold. There is a natural bijection

{ Z-torsors} /iso −→ Homgro ups ( π1 ( X) , Z) .

Proof. Let p : Y −→ X be a Z torsor. Choose points x ∈ X and y ∈ p− 1 ( x) . For a closed path γ starting and ending at x there is unique lift γ˜ to Y with γ˜ ( 0) = y. There is a unique aγ ∈ Z with y + a = γ˜ ( 1 ) . The integer aγ does only depend on the homotopy class of γ. We claim that a doesn’ t depend on the choice of y. Indeed, the path γ defines a continuous map S1 −→ X with 1 1 �→ x. Let Yγ = Y × X S be the fibre product, which is a Z-torsor over 1 1 1 S . Let U1 = S \{ − 1 } and U− 1 = S \{ 1 } . Choose sections s 1 , s − 1 of Yγ 1 over U1 and U− 1 . On U1 ∩ U− 1 = S \{ − 1 , 1 } we can compare this sections s 1 − s − 1 ∈ Z( U1 ∩ U− 1 ) = Z( 1 �→ − 1 ) × Z( − 1 �→ 1 ) . If ( a1 �→ − 1 , a1 �→ − 1 ) is the image of s 1 − s − 1 then

( 2. 3. 3. 1 ) aγ = a1 �→ − 1 − a1 �→ − 1 ,

independent of the choice of y. We define Y �→ ( φY : [ γ] �→ aγ ) . � If Y =∼ Y as Z-torsors, then φY = φY � , which follows from 2. 3. 3. 1 . This gives a map

( 2. 3. 3. 2) { Z-torsors} /iso −→ Homgroups ( π1 ( X, x) , Z) .

For two points x, x� there is in general no canonical isomorphism

π( X, x) −→ π( X, x� ) . 2. 3. H1 AND THE FUNDAMENTAL GROUP 1 9

However, choosing a homotopy class of paths from x to x� provides an isomor- phism which is independent up to conjugation with π( X, x) . In other words, for � two paths γ1 , γ2 from x to x we get isomorphism

− 1 − 1 η1 : [ γ] �→ [ γ1 ◦ γ ◦ γ1 ] , η2 : [ γ] �→ [ γ2 ◦ γ ◦ γ2 ] ,

− 1 − 1 with η2 ◦ η1 = [ γ] �→ [ ( γ2 γ1 ) ◦ γ ◦ ( γ2 γ1 ) ] . Since conjugation induces the ab identity on π1 ( X) we can write

ab Homgroups ( π1 ( X, x) , Z) = Homabelian groups ( π1 ( X) , Z) .

It is easy to see locally that the map 2. 3. 3. 2 does not depend on x. Since X is connected this also holds globally. Let us construct the inverse map. For a map φ ∈ Hom( π1 ( X, x) , Z) and π : X˜ −→ X the universal covering we get a space

X˜ × π1 ( X, x) Z = ( X × Z) /π1 ( X, x)

where π1 ( X, x) acts via g · ( x˜ , n) = ( gx˜ , − φ( g) + n) . It is easy to see that pr1 Y( φ) : X˜ × π1 ( X ) Z −−→ X is a Z-torsor over X. We claim that φY( φ) = φ for every φ ∈ Hom( π1 ( X, x) , Z) . Indeed, fix x˜ ∈ X˜ with π( x˜ ) = x and let [ γ] ∈ π1 ( X, x) . There is a unique lift γ˜ : [ 0, 1 ] −→ X˜ of γ to X˜ with γ˜ ( 0) = x˜ . The path

[ 0, 1 ] −→ Y( φ) = X˜ × π1 ( X ) Z, t �→ ( γ˜ ( t) , 0)

is a lift of γ to Y( φ) , and we compute

( γ˜ ( 1 ) , 0) = ( [ γ] · x˜ , 0) = ( x˜ , φ( [ γ] ) ) = φ( [ γ] ) + ( x˜ , 0) ,

which proves the claim. Now, let us prove that Yφ Y =∼ Y. The fibre product X˜ × X Y is a Z-torsor over ∼= X˜ and since X˜ is simply connected we can choose a section s : X˜ × Z −→ X˜ × X Y. Consider the map ∼= pr2 τ : X˜ × Z −→ X˜ × X Y −−→ Y,

we claim that it factors through X˜ × π1 ( X ) Z where π1 ( X) acts via φY on Z,

thus it induces an isomorphism of Z-torsors X˜ × π1 ( X ) Z =∼ Y. For g ∈ π1 ( X) and x˜ ∈ X˜ choose γ : [ 0, 1 ] −→ X˜ with γ( 0) = x˜ and γ( 1 ) = g · x˜ . Note that [ π( γ) ] = g. The path τ( γ, 0) is a lift of π( γ) and therefore

τ( γ( 1 ) , 0) = φY ( g) + τ( γ( 0) , 0) .

We conclude that τ( gx˜ , 0) = φY ( g) + τ( x˜ , 0) = τ( x˜ , φY ( g) ) . Theorem 2. 3. 4. There is a natural isomorphism of abelian groups

1 ab H ( X, Z) =∼ HomZ ( π( X) , Z) . 20 CHAPTER 2. THE DE RHAM COHOMOLOGY

Proof. We compose the bijections from Proposition 2. 3. 2 and 2. 3. 3. We have to prove that this is a map of abelian groups, i. e. for any a, b ∈ Z1 ( U, Z) and any closed path γ we claim

( 2. 3. 4. 1 ) φY( a+ b) ( γ) = φY( a) ( γ) + φY( b) ( γ) .

The path γ defines a continous map γ : S1 −→ X. There is an obvious restriction map γ∗ : H1 ( X, Z) −→ H1 ( S1 , Z) , which is a morphism of abelian groups. For a Z-torsor p : Y −→ X over X we 1 1 may take the fibre product Yγ : = Y × X S to get a Z-torsor over S . It is easy to see that

∗ 1 Y( γ ( a) ) =∼ Y( a) γ , for all a ∈ H ( X, Z) . ( 2. 3. 4. 2) 1 φY( a) ( γ) = φY( a) γ ( S ) .

In order to prove 2. 3. 4. 1 we may thus reduce to X = S1 and the path γ = id : 1 1 1 1 S −→ S . For a ∈ H ( X, Z) = Hˇ ( U, Z) with U = ( U1 , U− 1 ) as in Proposition 1 2. 3. 3 we may choose a representative a = ( a1 �→ − 1 , a− 1 �→ 1 ) ∈ Z ( U, Z) , then

φY( a) ( γ) = a1 �→ − 1 − a− 1 �→ 1 , which is additive in a.

Exercise 2. 3. 5. Explicitly construct all Z-torsors of the punctured unit disc and of a torus C/Λ.

2. 4 G-torsors

Definition 2. 4. 1 . A sheaf of groups on a topological space X is a sheaf G together with

1 . a global section 1 ∈ G( X) ,

2. two maps of sheaves · : G × G −→ G, ( ) − 1 : G −→ G, such that on every open U ⊂ X we get a group G( U) .

Remark 2. 4. 2. A sheaf of abelian groups is a sheaf of groups. If G is a topological group then U �→ Homcontinuous ( U, G) is a sheaf of groups on X. If X is a differentiable manifold then GLn ( AX ) is a sheaf of groups. If X is a Riemann surface then GLn ( OX ) is a sheaf of groups. Let G be a sheaf of groups on X. 2. 5. VECTOR BUNDLE 21

Definition 2. 4. 3. A G-torsor F is a sheaf on X together with an action of G, i. e. a morphism of sheaves G × F −→ F, which induces a G( U) -action on F( U) for every open U ⊂ X. And for every point x ∈ X there is a neighbourhood U and a section s ∈ F( U) such that

· s G | U −→ F | U ; g �→ g · s

is an isomorphism of sheaves on U. An isomorphism of G-torsors F =∼ H is a morphism of sheaves which is compatible with the G operation. In order to see that this definition of a Z-torsor corresponds to the one given in 2. 3. 1 , let p : Y −→ X be a Z-torsor in the sense of 2. 3. 1 , then

U �→ { s ∈ Homcontinuous ( U, Y) ; p ◦ s = idU }

defines a Z-torsor in the sense of 2. 4. 3. For a sheaf of ( maybe non-abelian) groups we have to define the Cˇ ech coho- mology more carefully. Let U = ( Ui ) i∈ I be an open cover,

1 2 Z ( U, G) : = { ( gi 0 , i 1 ) ( i 0 , i 1 ) ∈ I ; gi 0 , i 1 ∈ G( Ui 0 ∩ Ui 1 ) , gij · gjk = gik } .

1 Two elements a, b ∈ Z ( U, G) are called equivalent if there is ( ci ) i∈ I , ci ∈ G( Ui ) , − 1 1 1 such that aij = ci bij cj . Now, we define Hˇ ( U, G) to be the quotient of Z ( U, G) for this equivalence relation, and

H1 ( X, G) = lim Hˇ 1 ( U, G) . U open covering

Proposition 2. 4. 4. There is a natural bijection

H1 ( X, G) −→ { G-torsors} /isomorphism.

Proof. Mutatis mutandis as in the proof of 2. 3. 2. Exercise!

2. 5 Vector bundle Definition 2. 5. 1 . A holomorphic resp. differentiable vector bundle of rank r is a sheaf E of OX -modules resp. AX -modules such that for every x ∈ X there ⊕ r s U is a neighbourhood U and an isomorphism OU −−→ E| U of OU -modules resp. ⊕ r s U isomorphism AU −−→ E| U of AU -modules. � Example 2. 5. 2. Let D = x∈ X D( x) · x be a divisor. We define a sheaf O( D) by O( D) ( U) = { f ∈ MX ( U) ; ordx ( f) ≥ − D( x) ∀x ∈ U} .

This is an OX -module because ordx ( gf) ≥ ordx ( f) for all sections g ∈ OX . We claim that O( D) is a line bundle, i. e. a vector bundle of rank = 1 . Indeed, 22 CHAPTER 2. THE DE RHAM COHOMOLOGY

if x ∈ X then there is a neighbourhood U of x such that D( y) = 0 for all y ∈ U\{ x} , because there are only finitely many points y ∈ X with D( y) �= 0. We may choose U =∼ ∆ with holomorphic coordinate z. Set n : = ordx ( f) , then

− n s U : OU −→ O( D) | U ; f �→ f · z

is an isomorphism of OU -modules.

The sections s U ( ei ) , i = 1 , . . . , r, are called a local frame. A morphism of vector bundles φ : E1 −→ E2 is a morphism of OX resp. AX -modules. Locally, we can write ∃ ! A U ⊕ r1 � ⊕ r2 OU OU

s 1 , U s 2 , U � � φ U � E1 | U E2| U

with AU ∈ M( r2 × r1 , OX ( U) ) resp. AU ∈ M( r2 × r1 , AX ( U) ) . The matrix AU is defined by

�r2 �r2 AU ( ej ) = aij ei , φ( s 1 , U ( ej ) ) = aij s 2, U ( ei ) . i= 1 i= 1

Proposition 2. 5. 3. There is a natural bijective map

1 H ( X, GLr ( OX ) ) −→ { vector bundles of rank r} /isomorphism.

Proof. Let E be a holomorphic vector bundle. There is an open covering U = ( Ui ) i∈ I of X together with isomorphisms

⊕ r s Ui : OUi −→ E| U

for all i. The map

− 1 ⊕ r ⊕ r s Ui ◦ s Uj | Ui ∩ Uj : OUi ∩ Uj −→ OUi ∩ Uj

− 1 defines an invertible matrix Aij ∈ GLr ( OX ( Ui ∩ Uj ) ) . Since s Ui ◦ s Uj ◦ s Uj ◦ s Uk = − 1 ˇ 1 s Ui ◦ s Uk we get Aij Ajk = Aik , thus A = ( Aij ) ij ∈ H ( U, GLr ( OX ) ) . It is not difficult to see that A is independent of the choices for s Ui . The image of A in 1 Hˇ ( X, GLr ( OX ) ) is independent of the choice of the cover U. 1 1 Let g ∈ Hˇ ( X, GLr ( OX ) ) , then g is the image of ( gij ) ij ∈ Z ( U, GLr ( OX ) ) for some open covering U = ( Ui ) i∈ I . We construct a vector bundle E from the data gi j Ei = OUi , Ej | Ui ∩ Uj −−→ Ei | Ui ∩ Uj .

Since gij gjk = gik the Ei glue to a vector bundle E, which is up to isomorphism independent of the choice of the representative ( gij ) ij of g. 2. 5. VECTOR BUNDLE 23

The same statement holds for AX and differentiable vector bundle. For line bundles we get

∗ GL1 ( OX ) = OX = { sheaf of non-vanishing holomorphic function} ,

which is an abelian group.

Definition 2. 5. 4. If E, F are OX -modules then E ⊗ O X F is the sheafication of

U �→ E( U) ⊗ O X ( U ) F( U) .

The sheaf E ⊗ O X F has an OX -module structure induced by

OX ( U) × ( E( U) ⊗ O X ( U ) F( U) ) −→ E( U) ⊗ O X ( U ) F( U) , ( s, e ⊗ f) �→ se ⊗ f.

1 ∗ Proposition 2. 5. 5. 1 . The group structure on H ( X, OX ) corresponds to the tensor product ⊗ O X of line bundles. 2. The map

1 ∗ { Divisors on X} −→ H ( X, OX ) , D �→ O( D) ,

is a morphism of abelian groups.

3. If O( D) =∼ O( D � ) then there is a global meromorphic function f with � � D − D = ordx ( f) · x. x∈ X

4. Let X = P1 , then

� � � ∀D , D deg( D) = deg( D ) ⇒ O( D) =∼ O( D ) .

Proof. Exercise! Exercise 2. 5. 6. Let X be a Riemann surface. 1 . Show that the map of sheaves

∗ ∗ exp : OX −→ OX resp. exp : AX −→ AX

is surjective and find the kernel.

· 2πi 2. The morphism of sheaves Z −−→ OX induces a map

1 1 H ( X, Z) −→ H ( X, OX ) .

Show that this map is injective. Hint: Corollary 2. 2. 6. ∗ ∗ 3. If X is simply connected then OX ( X) −→ OX ( X) resp. AX ( X) −→ AX ( X) is surjective. 24 CHAPTER 2. THE DE RHAM COHOMOLOGY

4. In the following we assume that every open covering U of X has a refine- ment V such that Vj0 ∩ Vj1 is simply connected for all j0 , j1 . Construct a map of groups 1 ∗ 2 1 ∗ 2 ( 2. 5. 6. 1 ) H ( X, OX ) −→ Hˇ ( X, Z) resp. H ( X, AX ) −→ Hˇ ( X, Z) . 1 1 ∗ Hint: Observe that exp : C ( V, OX ) −→ C ( V, OX ) is surjective, and apply δ to a lifting. 5. Show that the kernel of the map ( 2. 5. 6. 1 ) is 1 1 H ( X, OX ) /H ( X, Z) , resp. 0.

2. 6 Topology of Riemann surfaces 2. 6. 1 First Betti number 1 1 We know from Corollary 2. 2. 5 that dim HdR ( X) = dim H ( X, C) is a topolog- 1 ical invariant. It is called the first Betti number b1 ( X) = dim H ( X, C) . For compact Riemann surfaces it is clear that b1 ( X) is a finite number.

2. 6. 2 Orientation All Riemann surfaces have a natural orientation. Indeed, if U ⊂ C, V ⊂ C are open sets and f : U −→ V a holomorphic function then � � ∗ ∗ − 1 − 1 f ( dx ∧ dy) = f dz ∧ dz¯ = ∂z ( f) ∂z ( f) dz ∧ dz¯ = ∂z ( f) ∂z ( f) dx ∧ dy. 2i 2i

Now, ∂z ( f) ∂z ( f) ( u) > 0 for all points u ∈ U where f is a local isomorphism.

2. 6. 3 Topology Let us recall the topological theory of compact, orientable surfaces. We denote by T = S1 × S1 the one torus; topologically there is only torus. We know that every compact, orientable surface X is homeomorphic to the g-fold connected sum of the torus:

X =∼ �T#T#��. . . #T�. g times

( Picture! ) The integer g ≥ 0 is called the genus of X, and it is an invariant for homeomorphisms. In the case g = 0 we understand X =∼ S2 . The genus g surface can be triangulated as follows. Take a regular polygon with 4g edges. Let us list the edges anti-clockwise e1 e2 . . . e4g , and consider every edge with the orientation which is pointing anti-clockwise ( the source of ei is the target of ei− 1 ) . We identify − 1 − 1 ( 2. 6. 3. 1 ) e3+ 4( i− 1 ) ∼ e1 + 4( i− 1 ) , e4i ∼ e2+ 4( i− 1 ) . 2. 6. TOPOLOGY OF RIEMANN SURFACES 25

− 1 Here, ( ) means that we reverse the orientation ( i. e. the source of e3+ 4( i− 1 ) is identified with the target of e1 + 4( i− 1 ) and vice versa) . The resulting surface X is closed and compact, the orientation of the plane descends to an orientation of X. The 4g vertices of the polygon map to a single point p ∈ X. The fundamental group π1 ( X, p) is generated by the paths e1 , e2 , e5 , e6 , . . . , e4g− 3 , e4g− 2 . The only relation is the obvious one

e1 e2 e3 e4 . . . e4g = 1

( use 2. 6. 3. 1 ) . In particular we see that Homgroups ( π1 ( X, p) , Z) is a free abelian group of rank 2g. Since

Homgroups ( π1 ( X, p) , C) = Homgroups ( π1 ( X, p) , Z) ⊗ C we conclude that the first Betti number is even and b1 ( X) /2 is the genus of the surface. ∗ ∗ ∗ ∗ ∗ ∗ 1 Let e1 , e2 , e5 , e6 , . . . , e4g− 3 , e4g− 2 ∈ Hsing ( X, Z) be the Poincare-dual base to ∗ ∗ e1 , e2 , e5 , e6 , . . . , e4g− 3 , e4g− 2 . We write ai = e4i− 3 , bi = e4i− 2 , for i = 1 , . . . , g. The intersection pairing

1 1 Hsing ( X, Z) × Hsing ( X, Z) −→ Z, is skew-symmetric, and we get

( ai , bj ) = δij , for all i, j, ( ai , aj ) = 0 = ( bi , bj ) , for all i, j.

Remark 2. 6. 4. Considered as Riemann surfaces we have to take the complex structure into account, i. e. isomorphism have to be holomorphic. The broad picture as follows. The Riemann sphere has genus 0, it is the only Riemann surface with this genus. The tori form a two dimensional family. The Riemann surfaces of genus g > 1 form a 6g − 6 dimensional family.