Introduction To Algebraic Topology and Algebraic Geometry: International School For Advanced Studies Trieste
Introduction To Algebraic Topology and Algebraic Geometry: International School For Advanced Studies Trieste
Introduction To Algebraic Topology and Algebraic Geometry: International School For Advanced Studies Trieste
U. Bruzzo
INTRODUCTION TO
ALGEBRAIC GEOMETRY
Preface
These notes assemble the contents of the introductory courses I have been giving at
SISSA since 1995/96. Originally the course was intended as introduction to (complex)
algebraic geometry for students with an education in theoretical physics, to help them to
master the basic algebraic geometric tools necessary for doing research in algebraically
integrable systems and in the geometry of quantum field theory and string theory. This
motivation still transpires from the chapters in the second part of these notes.
The first part on the contrary is a brief but rather systematic introduction to two
topics, singular homology (Chapter 2) and sheaf theory, including their cohomology
(Chapter 3). Chapter 1 assembles some basics fact in homological algebra and develops
the first rudiments of de Rham cohomology, with the aim of providing an example to
the various abstract constructions.
Chapter 5 is an introduction to spectral sequences, a rather intricate but very pow-
erful computation tool. The examples provided here are from sheaf theory but this
computational techniques is also very useful in algebraic topology.
I thank all my colleagues and students, in Trieste and Genova and other locations,
who have helped me to clarify some issues related to these notes, or have pointed out
mistakes. In this connection special thanks are due to Fabio Pioli. Most of Chapter 3 is
an adaptation of material taken from [2]. I thank my friends and collaborators Claudio
Bartocci and Daniel Hernández Ruipérez for granting permission to use that material.
I thank Lothar Göttsche for useful suggestions and for pointing out an error and the
students of the 2002/2003 course for their interest and constant feedback.
Chapter 7. Divisors 93
1. Divisors on Riemann surfaces 93
2. Divisors on higher-dimensional manifolds 100
3. Linear systems 101
4. The adjunction formula 103
Bibliography 131
Part 1
Algebraic Topology
CHAPTER 1
Introductory material
The aim of the first part of these notes is to introduce the student to the basics of
algebraic topology, especially the singular homology of topological spaces. The future
developments we have in mind are the applications to algebraic geometry, but also
students interested in modern theoretical physics may find here useful material (e.g.,
the theory of spectral sequences).
As its name suggests, the basic idea in algebraic topology is to translate problems
in topology into algebraic ones, hopefully easier to deal with.
In this chapter we give some very basic notions in homological algebra and then
introduce the fundamental group of a topological space. De Rham cohomology is in-
troduced as a first example of a cohomology theory, and is homotopic invariance is
proved.
Example 1.1. Set R = Z, the ring of integers (recall that Z-modules are just abelian
groups), and consider the sequence
i exp
(1.1) 0 → Z −−→ C −−→ C∗ → 1
where i is the inclusion of the integers into the complex numbers C, while C∗ = C − {0}
is the multiplicative group of nonzero complex numbers. The morphism exp is defined
as exp(z) = e2πiz . The reader may check that this sequence is exact.
The elements of the spaces M , Z(M, d) ≡ ker d and B(M, d) ≡ Im d are called
cochains, cocycles and coboundaries of (M, d), respectively. The condition d2 = 0 implies
that B(M, d) ⊂ Z(M, d), and the R-module
H(M, d) = Z(M, d)/B(M, d)
is called the cohomology group of the differential module (M, d). We shall often write
Z(M ), B(M ) and H(M ), omitting the differential d when there is no risk of confusion.
Let (M, d) and (M 0 , d0 ) be differential R-modules.
n ∈ Z.
For a complex M • the cocycle and coboundary modules and the cohomology group
split as direct sums of terms Z n (M • ) = ker dn , B n (M • ) = Im dn−1 and H n (M • ) =
Z n (M • )/B n (M • ) respectively. The groups H n (M • ) are called the cohomology groups
of the complex M • .
fn
Mn −−−−→ Nn
dy
yd .
fn+1
M n+1 −−−−→ N n+1
i p
Proposition 1.7. Let 0 → N • −−→ M • −−→ P • → 0 be an exact sequence of com-
plexes of R-modules. There exist connecting morphisms δn : H n (P • ) → H n+1 (N • ) and
a long exact sequence of cohomology
... / M k−1 d / Mk d /
M k+1 / ...
w w
K www K www
ww f ww
g
{ww
{ww
Proof. One could use the previous definitions and results to yield a proof, but it
is easier to note that if m ∈ M k is a cocycle (so that dm = 0), then
d(K(m)) = m − K(dm) = m
Remark 1.14. More generally, one can state that if a homotopy K : M k → M k−1
exists for k ≥ k0 , then H k (M, d) = 0 for k ≥ k0 . In the case of complexes bounded
below zero (i.e., M = ⊕k∈N M k ) often a homotopy is defined only for k ≥ 1, and it
2. DE RHAM COHOMOLOGY 7
may happen that H 0 (M, d) 6= 0. Examples of such situations will be given later in this
chapter.
2. De Rham cohomology
Ωk (X) = 0 for k > n and k < 0, the groups HdR k (X) vanish for k > n and k < 0.
0 1
Moreover, since ker[d : Ω (X) → Ω (X)] is formed by the locally constant functions on
X, we have HdR 0 (X) = RC , where C is the number of connected components of X.
Proof. We define a linear operator K : Ωk (Rn ) → Ωk−1 (Rn ) by letting, for any
k-form ω ∈ Ωk (Rn ), k ≥ 1, and all x ∈ Rn ,
Z 1
(Kω)(x) = k k−1
t ωi1 i2 ...ik (tx) dt xi1 dxi2 ∧ · · · ∧ dxik .
0
One easily shows that dK + Kd = Id; this means that K is a homotopy of the de Rham
complex of Rn defined for k ≥ 1, so that, according to Proposition 1.13 and Remark
1.14, all cohomology groups vanish in positive degree. Explicitly, if ω is closed, we have
ω = dKω, so that ω is exact.
Exercise 1.2. Realize the circle S 1 as the unit circle in R2 . Show that the in-
tegration of 1-forms on S 1 yields an isomorphism HdR 1 (S 1 ) ' R. This argument can
p]1 : HdR
• •
(X) → HdR (X × R), •
s] : HdR •
(X × R) → HdR (X×)
are isomorphisms.
Exercise 1.5. By a similar argument one proves that for all k > 0
k (X × S 1 ) ' H k (X) ⊕ H k−1 (X).
HdR
dR dR
Now we give an example of a long cohomology exact sequence within de Rham’s the-
ory. Let X be a differentiable manifold, and Y a closed submanifold. Let rk : Ωk (X) →
1In intrinsic notation this means that
Ωk (Y ) be the restriction morphism; this is surjective. Since the exterior differential com-
mutes with the restriction, after letting Ωk (X, Y ) = ker rk a differential d0 : Ωk (X, Y ) →
Ωk+1 (X, Y ) is defined. We have therefore an exact sequence of differential modules, in
a such a way that the diagram
rk−1
0 / Ωk−1 (X, Y ) / Ωk−1 (X) / Ωk−1 (Y ) /0
d0 d d
rk
0 / Ωk (X, Y ) / Ωk (X) / Ωk (Y ) /0
commutes. The complex (Ω• (X, Y ), d0 ) is called the relative de Rham complex, 3 and its
k (X, Y ) are called the relative de Rham cohomology groups.
cohomology groups by HdR
One has a long cohomology exact sequence
0 0 0 1 δ
0 → HdR (X, Y ) → HdR (X) → HdR (Y ) → HdR (X, Y )
1 1 2 δ
→ HdR (X) → HdR (Y ) → HdR (X, Y ) → . . .
Exercise 1.6. 1. Prove that the space ker d0 : Ωk (X, Y ) → Ωk+1 (X, Y ) is for all
k ≥ 0 the kernel of rk restricted to Z k (X), i.e., is the space of closed k-forms on X
which vanish on Y . As a consequence HdR0 (X, Y ) = 0 whenever X and Y are connected.
2. Let n = dim X and dim Y ≤ n − 1. Prove that HdRn (X, Y ) → H n (X) surjects,
dR
and that HdRk (X, Y ) = 0 for k ≥ n + 1. Make an example where dim X = dim Y and
Example 1.7. Given the standard embedding of S 1 into R2 , we compute the relative
• (R2 , S 1 ). We have the long exact sequence
cohomology HdR
0 δ
0 → HdR (R2 , S 1 ) → HdR
0
(R2 ) → HdR
0
(S 1 ) → HdR
1
(R2 , S 1 )
1 δ
→ HdR (R2 ) → HdR
1
(S 1 ) → HdR
2
(R2 , S 1 ) → HdR
2
(R2 ) → 0 .
As in the previous exercise, we have HdRk (R2 , S 1 ) = 0 for k ≥ 3. Since H 0 (R2 ) ' R,
dR
1 (R2 ) = H 2 (R2 ) = 0, H 0 (S 1 ) ' H 1 (S 1 ) ' R, we obtain the exact sequences
HdR dR dR dR
0 r
0 → HdR (R2 , S 1 ) → R → R → HdR
1
(R2 , S 1 ) → 0
2
0 → R → HdR (R2 , S 1 ) → 0
where the morphism r is an isomorphism. Therefore from the first sequence we get
0 (R2 , S 1 ) = 0 (as we already noticed) and H 1 (R2 , S 1 ) = 0. From the second we
HdR dR
obtain HdR2 (R2 , S 1 ) ' R.
From this example we may abstract the fact that whenever X and Y are connected,
0 (X, Y ) = 0.
then HdR
3Sometimes this term is used for another cohomology complex, cf. [3].
10 1. INTRODUCTORY MATERIAL
Exercise 1.1. Use the Mayer-Vietoris sequence (1.4) to compute the de Rham
cohomology of the circle S 1 .
Example 1.2. We use the Mayer-Vietoris sequence (1.4) to compute the de Rham
cohomology of the sphere S 2 (as a matter of fact we already know the 0th and 2nd
group, but not the first). Using suitable stereographic projections, we can assume that
U and V are diffeomorphic to R2 , while U ∩ V ' S 1 × R. Since S 1 × R is homotopic to
S 1 , it has the same de Rham cohomology. Hence the sequence (1.4) becomes
0
0 → HdR (S 2 ) → R ⊕ R → R → HdR
1
(S 2 ) → 0
2
0 → R → HdR (S 2 ) → 0.
4The Mayer-Vietoris sequence foreshadows the Čech cohomology we shall study in Chapter 3.
4. HOMOTOPY THEORY 11
From the first sequence, since HdR 0 (S 2 ) ' R, the map H 0 (S 2 ) → R ⊕ R is injective,
dR
and then we get HdR1 (S 2 ) = 0; from the second sequence, H 2 (S 2 ) ' R.
dR
k (S n ) ' R for k = 0, n,
Exercise 1.3. Use induction to show that if n ≥ 3, then HdR
k (S n ) = 0 otherwise.
HdR
If the two paths have the same end points (i.e. γ1 (0) = γ2 (0) = x1 , γ1 (1) = γ2 (1) = x2 ),
we may introduce the stronger notion of homotopy with fixed end points by requiring
additionally that Γ(0, s) = x1 , Γ(1, s) = x2 for all s ∈ I.
Let us fix a base point x0 ∈ X. A loop based at x0 is a path such that γ(0) = γ(1) =
x0 . Let us denote L(x0 ) th set of loops based at x0 . One can define a composition
between elements of L(x0 ) by letting
(
γ1 (2t), 0 ≤ t ≤ 21
(γ2 · γ1 )(t) =
γ2 (2t − 1), 21 ≤ t ≤ 1.
This does not make L(x0 ) into a group, since the composition is not associative (com-
posing in a different order yields different parametrizations).
Proof. Let c be such a path, and let γ1 ∈ L(x1 ). We define γ2 ∈ L(x2 ) by letting
c(1 − 3t), 0 ≤ t ≤ 31
γ2 (t) = γ1 (3t − 1), 13 ≤ t ≤ 32
2
c(3t − 2), ≤ t ≤ 1.
3
4.2. The fundamental group. Again with reference with a base point x0 , we
consider in L(x0 ) an equivalence relation by decreeing that γ1 ∼ γ2 if there is a homotopy
with fixed end points between γ1 and γ2 . The composition law in Lx0 descends to a
group structure in the quotient
π1 (X, x0 ) = L(x0 )/ ∼ .
π1 (X, x0 ) is the fundamental group of X with base point x0 ; in general it is nonabelian,
as we shall see in examples. As a consequence of Proposition 1.1, if x1 , x2 ∈ X and
there is a path connecting x1 with x2 , then π1 (X, x1 ) ' π1 (X, x2 ). In particular, if
X is pathwise connected the fundamental group π1 (X, x0 ) is independent of x0 up to
isomorphism; in this situation, one uses the notation π1 (X).
The simplest example of a simply connected space is the one-point space {∗}.
Since the definition of the fundamental group involves the choice of a base point, to
describe the behaviour of the fundamental group we need to introduce a notion of map
which takes the base point into account. Thus, we say that a pointed space (X, x0 ) is a
pair formed by a topological space X with a chosen point x0 . A map of pointed spaces
f : (X, x0 ) → (Y, y0 ) is a continuous map f : X → Y such that f (x0 ) = y0 . It is easy
to show that a map of pointed spaces induces a group homomorphism f∗ : π(X, x0 ) →
π1 (Y, y0 ).
Definition 1.3. One says that two topological spaces X, Y are homotopically equiv-
alent if there are continuous maps f : X → Y , g : Y → X such that g ◦ f is homotopic
to idX , and f ◦ g is homotopic to idY . The map f , g are said to be homotopical equiv-
alences,.
Example 1.4. For any manifold X, take Y = X × R, f (x) = (x, 0), g the projection
onto X. Then F : X × I → X, F (x, t) = x is a homotopy between g ◦ f and idX , while
G : X × R × I → X × R, G(x, s, t) = (x, st) is a homotopy between f ◦ g and idY . So X
and X × R are homotopically equivalent. The reader should be able to concoct many
similar examples.
Given two pointed spaces (X, x0 ), (Y, y0 ), we say they are homotopically equivalent
if there exist maps of pointed spaces f : (X, x0 ) → (Y, y0 ), g : (Y, y0 ) → (X, x0 ) that
make the topological spaces X, Y homotopically equivalent.
4. HOMOTOPY THEORY 13
Proof. Let g : (Y, y0 ) → (X, x0 ) be a map that realizes the homotopical equiva-
lence, and denote by F a homotopy between g ◦ f and idX . Let γ be a loop based at
x0 . Then g ◦ f ◦ γ is again a loop based at x0 , and the map
Γ : I × I → X, Γ(s, t) = F (γ(s), t)
is a homotopy between γ and g ◦ f ◦ γ, so that γ = g ◦ f ◦ γ in π1 (X, x0 ). Hence,
g∗ ◦ f∗ = idπ1 (X,x0 ) . In the same way one proves that f∗ ◦ g∗ = idπ1 (Y,y0 ) , so that the
claim follows.
Corollary 1.6. If two pathwise connected spaces X and Y are homotopic, then
their fundamental groups are isomorphic.
4.5. The van Kampen theorem. The computation of the fundamental group
of a topological space is often unsuspectedly complicated. An important tool for such
computations is the van Kampen theorem, which we state without proof. This theorem
allows one, under some conditions, to compute the fundamental group of an union U ∪V
if one knows the fundamental groups of U , V and U ∩ V . As a prerequisite we need
the notion of amalgamated product of two groups. Let G, G1 , G2 be groups, with fixed
morphisms f1 : G → G1 , f2 : G → G2 . Let F the free group generated by G1 q G2 and
denote by · the product in this group.6 Let R be the normal subgroup generated by
elements of the form7
(xy) · y −1 · x−1 with x, y both in G1 or G2
Exercise 1.11. (1) Prove that if G1 = G2 = {e}, and G is any group, then
G1 ∗G G2 = {e}.
(2) Let G be the group with three generators a, b, c, satisfying the relation ab = cba.
Let Z → G be the homomorphism induced by 1 7→ c. Prove that G ∗Z G is
isomorphic to a group with four generators m, n, p, q, satisfying the relation
m n m−1 n−1 p q p−1 q −1 = e.
Suppose now that a pathwise connected space X is the union of two pathwise con-
nected open subsets U , V , and that U ∩ V is pathwise connected. There are morphisms
π1 (U ∩ V ) → π1 (U ), π1 (U ∩ V ) → π1 (V ) induced by the inclusions.
This is a simplified form of van Kampen’s theorem, for a full statement see [7].
Example 1.13. We compute the fundamental group of a figure 8. Think of the figure
8 as the union of two circles X in R2 which touch in one pount. Let p1 , p2 be points
in the two respective circles, different from the common point, and take U = X − {p1 },
V = X − {p2 }. Then π1 (U ) ' π1 (V ) ' Z, while U ∩ V is simply connected. It follows
that π1 (X) is a free group with two generators. The two generators do not commute;
this can also be checked “experimentally” if you think of winding a string along the
6F is the group whose elements are words x1 x . . . x or the empty word, where the letters x are
1 2 n i
or G2 , when this makes sense. The second relation kind of “glues” G1 and G2 along the images of G.
4. HOMOTOPY THEORY 15
figure 8 in a proper way... More generally, the fundamental group of the corolla with n
petals (n copies of S 1 all touching in a single point) is a free group with n generators.
Exercise 1.14. Prove that for any n ≥ 2 the sphere S n is simply connected. Deduce
that for n ≥ 3, Rn minus a point is simply connected.
4.6. Other ways to compute fundamental groups. Again, we state some re-
sults without proof.
Exercise 1.18. Prove that, given two pointed topological spaces (X, x0 ), (Y, y0 ),
then
π1 (X × Y, (x0 , y0 )) ' π1 (X, x0 ) × π1 (Y, y0 ).
This gives us another way to compute the fundamental group of the n-dimensional
torus T n (once we know π1 (S 1 )).
Exercise 1.19. Prove that the manifolds S 3 and S 2 × S 1 are not homeomorphic.
Exercise 1.20. Let X be the space obtained by removing a line from R2 , and a
circle linking the line. Prove that π1 (X) ' Z ⊕ Z. Prove the stronger result that X is
homotopic to the 2-torus.
CHAPTER 2
1. Singular homology
1.1. Definitions. The basic blocks of singular homology are the continuous maps
from standard subspaces of Euclidean spaces to the topological space one considers. We
shall denote by P0 , P1 , . . . , Pn the points in Rn
The convex hull of these points is denoted by ∆n and is called the standard n-simplex.
Alternatively, one can describe ∆k as the set of points in Rn such that
n
X
xi ≥ 0, i = 1, . . . , n, xi ≤ 1.
i=1
also be denoted < P0 , . . . , Pn >, and the face Fni of ∆n is the singular (n − 1)-simplex
< P0 , . . . , P̂i , . . . , Pn >, where the hat denotes omission.
Choose now a commutative unital ring R. We denote by Sk (X, R) the free group
generated over R by the singular k-simplexes in X. So an element in Sk (X, R) is a
“formal” finite linear combination (called a singular chain)
X
σ= aj σj
j
with aj ∈ R, and the σj are singular k-simplexes. Thus, Sk (X, R) is an R-module, and,
via the inclusion Z → R given by the identity in R, an abelian group. For k ≥ 1 we
define a morphism ∂ : Sk (X, R) → Sk−1 (X, R) by letting
k
X
∂σ = (−1)i σ ◦ Fki
i=0
Proposition 2.2. ∂ 2 = 0.
Resumming the first sum by letting i = j, j = i − 1 the last two terms cancel.
So (S• (X, R), ∂) is a (homology) graded differential module. Its homology groups
Hk (X, R) are the singular homology groups of X with coefficients in R. We shall use
the following notation and terminology:
Zk (X, R) = ker ∂ : Sk (X, R) → Sk−1 (X, R) (the module of k-cycles);
Bk (X, R) = Im ∂ : Sk+1 (X, R) → Sk (X, R) (the module of k-boundaries);
therefore, Hk (X, R) = Zk (X, R)/Bk (X, R). Notice that Z0 (X, R) ≡ S0 (X, R).
Proof. Any singular k-simplex must map ∆k inside a pathwise connected com-
ponents (if two points of ∆k would map to points lying in different components, that
would yield path connecting the two points).
if γj is a path joining x0 to xj .
This means that B0 (X, R) is the kernel of the surjective map Z0 (X, R) = S0 (X, R) →
P P
R given by j aj xj 7→ j aj , so that H0 (X, R) = Z0 (X, R)/B0 (X, R) ' R.
This implies that f induces a morphism Hk (X, R) → Hk (Y, R), that we denote f[ . It
is also easy to check that, if g : Y → W is another continous map, then Sk (g ◦ f ) =
Sk (g) ◦ Sk (f ), and (g ◦ f )[ = g[ ◦ f[ .
It should be by now clear that this yields as an immediate consequence the homotopic
invariance of the singular homology.
Corollary 2.6. If two topological spaces are homotopically equivalent, their singu-
lar homologies are isomorphic.
To prove Proposition 2.5 we build, for every k ≥ 0 and any topological space X, a
morphism (called the prism operator ) P : Sk (X) → Sk+1 (X × I) (here I denotes again
the unit closed interval in R). We define the morphism P in two steps.
Step 1. The first step consists in definining a singular (k + 1)-chain πk+1 in the
topological space ∆k × I by subdiving the polyhedron ∆k × I ⊂ Rk+1 (a “prysm”
20 2. HOMOLOGY THEORY
B0 B1
A0 A1
Figure 1. The prism π2 over ∆1
over the standard symplesx ∆k ) into a number of singular (k + 1)-simplexes, and sum-
ming them with suitable signs. The polyhedron ∆k × I ⊂ Rk+1 has 2(k + 1) vertices
A0 , . . . , Ak , B0 , . . . , Bk , given by Ai = (Pi , 0), Bi = (Pi , 1). We define
k
X
πk+1 = (−1)i < A0 , . . . , Ai , Bi , . . . , Bk > .
i=0
Sk (X)
P / Sk+1 (X × I)
Sk (f ) Sk+1 (f ×id)
Sk (Y )
P / Sk+1 (Y × I)
commutes.
The relevant property of the prism operator is proved in the next Lemma.
1. SINGULAR HOMOLOGY 21
Lemma 2.8. Let λ0 , λi : X → X × I be the maps λ0 (x) = (x, 0), λ1 (x) = (x, 1).
Then
All terms with i = j cancel with the exception of < B0 , . . . , Bk > − < A0 , . . . Ak >. So
we have
Since
X
P (< P0 , . . . , P̂j , . . . , Pk >) = (−1)i < A0 , . . . , Ai , Bi , . . . , B̂j , . . . , Bk >
i<j
X
− (−1)i < A0 , . . . , Âj , . . . , Ai , Bi , . . . , Bk >
i>j
we obtain the equation (2.2) (note that exchanging the indices i, j changes the sign).
22 2. HOMOLOGY THEORY
We must now prove that if equation (2.2) holds when both sides are applied to δk ,
then it holds in general. One has indeed
so that
Equation (2.2) states that P is a hotomopy (in the sense of homological algebra)
between the maps λ0 and λ1 , so that one has (λ1 )[ = (λ2 )[ in homology.
Proof of Proposition 2.5. Let F be a hotomopy between the maps f and g. Then,
f = F ◦ λ0 , g = F ◦ λ1 , so that
f[ = (F ◦ λ0 )[ = F[ ◦ (λ0 )[ = F[ ◦ (λ1 )[ = (F ◦ λ1 )[ = g[ .
1.4. Relation between the first fundamental group and homology. A loop
γ in X may be regarded as a closed singular 1-simplex. If we fix a point x0 ∈ X, we
have a set-theoretic map χ : L(x0 ) → S1 (X, Z). The following result tells us that χ
descends to a group homomorphism χ : π1 (X, x0 ) → H1 (X, Z).
Proposition 2.10. If two loops γ1 , γ2 are homotopic, then they are homologous
as singular 1-simplexes. Moreover, given two loops at x0 , γ1 , γ2 , then χ(γ2 ◦ γ1 ) =
χ(γ1 ) + χ(γ2 ) in H1 (X, Z).
Proof. Choose a homotopy with fixed endpoints between γ1 and γ2 , i.e., a map
Γ : I × I → X such that
Define the loops γ3 (t) = Γ(1, t), γ4 (t) = Γ(0, t), γ5 (t) = Γ(t, t). Both loops γ3 and
γ4 are actually the constant loop at x0 . Consider the points P0 , P1 , P2 , Q = (1, 1) in
R2 , and define the singular 2-simplex
P2 γ2 Q
>
γ4 ∧ ∧ γ3
γ5
>
P0 γ1 P1
Figure 2
This proves that χ(γ1 ) and χ(γ2 ) are homologous. To prove the second claim we need
to define a singular 2-simplex σ such that
∂σ = γ1 + γ2 − γ2 · γ1 .
Consider the point T = (0, 21 ) in the standard 2-simplex ∆2 and the segment Σ
joining T with P1 (cf. Figure 3). If Q ∈ ∆2 lies on or below Σ, consider the line joining
P0 with Q, parametrize it with a parameter t such that t = 0 in P0 and t = 1 in the
intersection of the line with Σ, and set σ(Q) = γ1 (t). Analogously, if Q lies above or
on Σ, consider the line joining P2 with Q, parametrize it with a parameter t such that
t = 1 in P2 and t = 0 in the intersection of the line with Σ, and set σ(Q) = γ2 (t). This
defines a singular 2-simplex σ : ∆2 →X, and one has
We recall from basic group theory the notion of commutator subgroup. Let G be
any group, and let C(G) be the subgroup generated by elements of the form ghg −1 h−1 ,
g, h ∈ G. The subgroup C(G) is obviously normal in G; the quotient group G/C(G) is
abelian. We call it the abelianization of G. It turns out that the first homology group
of a space with integer coefficients is the abelianization of the fundamental group.
P2
@
A
A@
γ2 ∧ A @
A•Q@
A @ γ
T H A @ 2
HHA @
HA
γ1 ∧ H
H
@
@
H
•
HH@
Q Σ H@
> H
H
@
P0 γ1 P1
Figure 3
P
Proof. Let c = j aj σj be a 1-cycle. So we have
X
0 = ∂c = ai (σj (1) − σj (0)).
j
In this linear combination of points with coefficients in Z some of the points may coin-
cide; the sum of the coefficients corresponding to the same point must vanish. Choose a
base point x0 ∈ X and for every j choose a path αj from x0 to σj (0) and a path βj from
x0 to σj (1), in such a way that they depend on the endpoints and not on the indexing
(e.g, if σj (0) = σk (0), choose αj = αk ). Then we have
X
aj (βj − αj ) = 0.
j
then,
h i
a
χ( Πj γj j ) = [c]
so that χ is surjective.
To prove the second claim we need to show that the commutator subgroup of
π1 (X, x0 ) coincides with ker χ. We first notice that since H1 (X, Z) is abelian, the
commutator subgroup is necessarily contained in ker χ. To prove the opposite inclusion,
P
let γ be a loop that in homology is a 1-boundary, i.e., γ = ∂ j aj σj . So we may write
P2
α2j
γ0j
γ1j
α0j
P1 x0
γ2j P0
α1j
Figure 4
are homotopic to the constant loop at x0 (since the image of a singular 2-simplex is
contractible). As a consequence one has the equality in π1 (X, x0 )
Πj [βj ]aj = e.
This implies that the image of Πj [βj ]aj in π1 (X, x0 )/C(π1 (X, x0 )) is the identity. On the
a
other hand from (2.3) we see that γ coincides, up to reordering of terms, with Πj βj j , so
that the image of the class of γ in π1 (X, x0 )/C(π1 (X, x0 )) is the identity as well. This
means that γ lies in the commutator subgroup.
Exercise 2.13. Compute H1 (X, Z) when X is: 1. the corolla with n petals, 2. Rn
minus a point, 3. the circle S 1 , 4. the torus T 2 , 5. a punctured torus, 6. a Riemann
surface of genus g.
2. Relative homology
The relative homology is more conveniently defined in a slightly different way, which
makes clearer its geometrical meaning. It will be useful to consider the following diagram
qk
Zk (X) / Z 0 (X, A)
k
qk
Sk (A) / Sk (X) / Sk (X)/Sk (A) /0
∂ ∂ ∂
0 / Bk−1 (A) / Bk−1 (X) / B 0 (X, A) /0
qk−1 k−1
Let
Zk (X, A) = {c ∈ Sk (X) | ∂c ∈ Sk−1 (A)}
Thus, Zk (X, A) is formed by the chains whose boundary is in A, and Bk (A) by the
chains that are boundaries up to chains in A.
Lemma 2.2. Zk (X, A) is the pre-image of Zk0 (X, A) under the quotient homomor-
phism qk ; that is, an element c ∈ Sk (X) is in Zk (X, A) if and only if qk (c) ∈ Zk0 (X, A).
Proof. If qk (c) ∈ Zk0 (X, A) then 0 = ∂ ◦ qk (c) = qk−1 ◦ ∂(c) so that c ∈ Zk (X, A).
If c ∈ Zk (X, A) then qk−1 ◦ ∂(c) = 0 so that qk (c) ∈ Zk0 (X, A).
Lemma 2.3. c ∈ Sk (X) is in Bk (X, A) if and only if qk (c) ∈ Bk0 (X, A).
Proposition 2.4. For all k ≥ 0, Hk (X, A) ' Zk (X, A)/Bk (X, A).
2. RELATIVE HOMOLOGY 27
Proof. What we should do is to prove the commutativity and the exactness of the
rows of the diagram
qk
0 / Sk (A) / Bk (X, A) / B 0 (X, A) /0
k
∼
qk
0 / Sk (A) / Zk (X, A) / Z 0 (X, A) /0
k
Commutativity is obvious. For the exactness of the first row, it is obvious that Sk (A) ⊂
Bk (X, A) and that qk (c) = 0 if c ∈ Sk (A). On the other hand if c ∈ Bk (X, A) we have
c = ∂b + c0 with b ∈ Sk+1 (X) and c0 ∈ Sk (A), so that qk (c) = 0 implies 0 = qk ◦ ∂b =
∂ ◦ qk+1 (b), which in turn implies c ∈ Sk (A). To prove the surjectivity of qk , just notice
that by definition an element in Bk0 (X, A) may be represented as ∂b with b ∈ Sk+1 (X).
As for the second row, we have Sk (A) ⊂ Zk (X, A) from the definition of Zk (X, A).
If c ∈ Sk (A) then qk (c) = 0. If c ∈ Zk (X, A) and qk (c) = 0 then c ∈ Sk (A) by the
definition of Zk0 (X, A). Moreover qk is surjective by Lemma 2.2.
2.2. Main properties of relative homology. We list here the main properties
of the cohomology groups Hk (X, A). If a proof is not given the reader should provide
one by her/himself.
• If A is empty, Hk (X, A) ' Hk (X).
• The relative cohomology groups are functorial in the following sense. Given topo-
logical spaces X, Y with subsets A ⊂ X, B ⊂ Y , a continous map of pairs is a con-
tinuous map f : X → Y such that f (A) ⊂ B. Such a map induces in natural way a
morphisms of R-modules f[ : H• (X, A) → H• (Y, B). If we consider the inclusion of pairs
(X, ∅) ,→ (X, A) we obtain a morphism H• (X) →• H(X, A).
• The inclusion map i : A ,→ X induces a morphism H• (A) → H• (X) and the
composition H• (A) → H• (X) → H• (X, A) vanishes (since Zk (A) ⊂ Bk (X, A)).
• If X = ∪j Xj is a union of pathwise connected components, then Hk (X, A) '
⊕j Hk (Xj , Aj ) where Aj = A ∩ Xj .
Proof.
Zk (X, A) = {c ∈ Sk (X) | ∂c ∈ Sk−1 (A)} = Zk (X) when k > 0
Bk (X, A) = {c ∈ Sk (X) | c = ∂b + c0 with b ∈ Sk+1 (X), c0 ∈ Sk (A)}
= Bk (X) when k > 0.
2.3. The long exact sequence of relative homology. By definition the relative
homology of X with respect to A is the homology of the quotient complex S• (X)/S• (A).
By Proposition 1.7, adapted to homology by reversing the arrows, one obtains a long
exact cohomology sequence
The Mayer-Vietoris sequence (in its simplest form, that we are going to consider
here) allows one to compute the homology of a union X = U ∪ V from the knowledge
of the homology of U , V and U ∩ V . This is quite similar to what happens in de Rham
cohomology, but in the case of homology there is a subtlety. Let us denote A = U ∩ V .
One would think that there is an exact sequence
i p
0 → Sk (A) → Sk (U ) ⊕ Sk (V ) → Sk (X) → 0
where i is the morphism induced by the inclusions A ,→ U , A ,→ V , and p is given by
p(σ1 , σ2 ) = σ1 − σ2 (again using the inclusions U ,→ X, V ,→ X). However, it is not
possible to prove that p is surjective (if σ is a singular k-simplex whose image is not
contained in U or V , it is not in general possible to write it as a difference of standard
k-simplexes in U , V ). The trick to circumvent this difficulty consists in replacing S• (X)
with a different complex that however has the same homology.
Let U = {Uα } be an open cover of X.
P
Definition 2.1. A singular k-chain σ = j aj σj is U-small if every singular k-
simplex σj maps into an open set Uα ∈ U for some α. Moreover we define S•U (X) as
the subcomplex of S• (X) formed by U-small chains.1
The homology differential ∂ restricts to S•U (X), so that one has a homology H•U (X).
1Again, we understand the choice of a coefficient ring R.
3. THE MAYER-VIETORIS SEQUENCE 29
E0
HH
B HH
HH
E1
Figure 5. The join B(< E0 , E1 >)
To prove this isomorphism we shall build a homotopy between the complexes S•U (X)
and S• (X). This will be done in several steps.
Given a singular k-simplex < Q0 , . . . , Qk > in Rn and a point B ∈ Rn we consider
the singular simplex < B, Q0 , . . . , Qk >, called the join of B with < Q0 , . . . , Qk >. This
operator B is then extended to singular chains in Rn by linearity. The following Lemma
is easily proved.
Next we define operators Σ : Sk (X) → Sk (X) and T : Sk (X) → Sk+1 (X). The
operator Σ is called the subdivision operator and its effect is that of subdividing a
singular simplex into a linear combination of “smaller” simplexes. The operators Σ
and T , analogously to what we did for the prism operator, will be defined for X = ∆k
(the space consisting of the standard k-simplex) and for the “identity” singular simplex
δk : ∆k → ∆k , and then extended by functoriality. This should be done for all k. One
defines
Σ(δ0 ) = δ0 , T (δ0 ) = 0.
and then extends recursively to positive k:
Example 2.4. For k = 1 one gets Σ(δ1 ) =< B1 P1 > − < B1 P0 >; for k = 2, the
action of Σ splits ∆2 into smaller simplexes as shown in Figure 6.
30 2. HOMOLOGY THEORY
P2
@
A
A@
A @
A @
A @ M
M1 H A @ 0
HHA @
HAH @
B2A HH @
A HH@
AA H@H
H
@
P0 M2 P1
Figure 6. The subdivision operator Σ splits ∆2 into the chain
< B2 , M0 , P2 > − < B2 , M0 , P1 > − < B2 , M1 , P2 > + < B2 , M1 , P0 >
+ < B2 , M2 , P1 > − < B2 , M2 , P0 >
The definition of Σ and T for every topological space and every singular k-simplex
σ in X is
Σ(σ) = Sk (σ)(Σ(δk )), T (σ) = Sk+1 (σ)(T (δk )).
∂ ◦ Σ = Σ ◦ ∂, ∂ ◦ T + T ◦ ∂ = Id −Σ.
Proof. These identities are proved by direct computation (it is enough to consider
the case X = ∆k ).
The first identity tells us that Σ is a morphism of differential complexes, and the
second that T is a homotopy between Σ and Id, so that the morphism Σ[ induced in
homology by Σ is an isomorphism.
The diameter of a singular k-simplex σ in Rn is the maximum of the lengths of
the segments contained in σ. The proof of the following Lemma is an elementary
computation.
there is an r > 0 such that Σr (δk ) is a linear combination of simplexes whose diameter
is less than . But as Σr (σ) = Sk (σ)(Σr (δk )) we are done.
This completes the proof of Proposition 2.2. We may now prove the exactness of
the Mayer-Vietoris sequence in the following sense. If X = U ∪ V (union of two open
subsets), let U = {U, V } and A = U ∩ V .
? U @@
`U ~~~ @@ jU
@@
~~ @@
~~
A@ X
@ @@ ~~>
@@ ~~
@ ~~
`V ~~ jV
V
Defining i(σ) = (`U ◦ σ, −`V ◦ σ) and p(σ1 , σ2 ) = jU ◦ σ1 + jV ◦ σ2 , the exactness of the
Mayer-Vietoris sequence is easily proved.
The morphisms i and p commute with the homology operator ∂, so that one obtains
a long homology exact sequence involving the homologies H• (A), H• (V ) ⊕ H• (V ) and
H•U (X). But in view of Proposition 2.2 we may replace H•U (X) with the homology
H• (X), so that we obtain the exact sequence
· · · → H2 (A) → H2 (U ) ⊕ H2 (V ) → H2 (X)
→ H1 (A) → H1 (U ) ⊕ H1 (V ) → H1 (X)
→ H0 (A) → H0 (U ) ⊕ H0 (V ) → H0 (X) → 0
Exercise 2.9. Prove that for any ring R the homology of the sphere S n with
coefficients in R, n ≥ 2, is
(
R for k = 0 and k = n
Hk (S n , R) =
0 for 0 < k < n and k > n .
Exercise 2.10. Show that the relative homology of S 2 mod S 1 with coefficients in
Z is concentrated in degree 2, and H2 (S 2 , S 1 ) ' Z ⊕ Z.
4. Excision
Theorem 2.4. If the closure U of U lies in the interior int(A) of A, then U can be
excised.
P
Proof. We consider the cover U = {X − U , int(A)} of X. Let c = j aj σj ∈
Zk (X, A), so that ∂c ∈ Sk−1 (A). In view of Proposition 2.2 we may assume that c is U-
small. If we cancel from σ those singular simplexes σj taking values in int(A), the class
[c] ∈ Hk (X, A) is unchanged. Therefore, after the removal, we can regard c as a relative
cycle in X −U mod A−U ; this implies that the morphism Hk (X −U, A−U ) → Hk (X, A)
is surjective.
To prove that it is injective, let [c] ∈ Hk (X − U, A − U ) be such that, regarding c as
a cycle in X mod A, it is a boundary, i.e., c ∈ Bk (X, A). This means that
c = ∂b + c0 with b ∈ Sk+1 (X), c0 ∈ Sk (A) .
We apply the operator Σr to both sides of this inequality, and split Σr (b) into b1 + b2 ,
where b1 maps into X − Ū and b2 into int(A). We have
Σr (c) − ∂b1 = Σr (c0 ) + ∂b2 .
4. EXCISION 33
The chain in the left side is in X − U while the chain in the right side is in A; therefore,
both chains are in (X − U ) ∩ A = A − U . Now we have
with Σr (c0 )+∂b2 ∈ Sk (A−U ) and ∂b1 ∈ Sk+1 (X −U ) so that Σr (c) ∈ Bk (X −U, A−U ),
which implies [c] = 0 (in Hk (X − U, A − U )).
Exercise 2.5. Let B an open band around the equator of S 2 , and x0 ∈ B. Compute
the relative homology H• (S 2 − x0 , B − x0 ; Z).
∂ ] : S0 (X, R) → R
X X
aj σj 7→ aj .
j j
H0] (X, R) = ker ∂ ] /B0 (X, R) , Hk] (X, R) = Hk (X, R) for k > 0 .
Exercise 2.6. Prove that there is a long exact sequence for the augmented relative
homology modules.
Exercise 2.7. Let B n be the closed unit ball in Rn+1 , S n its boundary, and let En±
be the two closed (northern, southern) emispheres in S n .
1. Use the long exact sequence for the augmented relative homology modules to
prove that Hk] (S n ) ' Hk] (S n , En− ) and Hk−1 ]
(S n−1 ) ' Hk] (B n , S n−1 ). So we have
Hk] (B n , S n−1 ) = 0 for k < n, Hn] (B n , S n−1 ) ' R
2. Use excision to show that Hk] (S n , En− ) ' Hk] (B n , S n−1 ).
3. Deduce that Hk] (S n ) ' Hk−1
]
(S n−1 ).
Exercise 2.8. Let S n be the sphere realized as the unit sphere in Rn+1 , and let
r : S n → S n → S n be the reflection
r(x0 , x1 , . . . , xn ) = (−x0 , x1 , . . . , xn ).
34 2. HOMOLOGY THEORY
r[ r[
∼ / H ] (S n−1 )
Hn (S n ) n−1
Exercise 2.9. 1. The rotation group O(n + 1) acts on S n . Show that for any
M ∈ O(n + 1) the induced morphism M[ : Hn (S n ) → Hn (S n ) is the multiplication by
det M = ±1.
2. Let a : S n → S n be the antipodal map, a(x) = −x. Show that a[ : Hn (S n ) →
Hn (S n ) is the multiplication by (−1)n+1 .
Example 2.10. We show that the inclusion map (En+ , S n−1 ) → (S n , En− ) is an
excision. (Here we are excising the open southern emisphere, i.e., with reference to the
general theory, X = S n , U = the open southern emisphere, A = En− .)
The hypotheses of Theorem 2.4 are not satisfied. However it is enough to consider
the subspace
V = x ∈ S n | x0 > − 21 .
the three maps Id, w, a are homotopic. But as a consequence of Exercise 2.9, n must
be odd.
CHAPTER 3
1This rather naive terminology can be made more precise by saying that a presheaf on X is a
contravariant functor from the category OX of open subsets of X to the category of Abelian groups.
OX is defined as the category whose objects are the open subsets of X while the morphisms are the
inclusions of open sets.
37
38 3. SHEAVES AND THEIR COHOMOLOGY
Px = lim P(U )
−→
U
Remark 3.4. We recall here the notion of direct limit. A directed set I is a partially
ordered set such that for each pair of elements i, j ∈ I there is a third element k such
that i < k and j < k. If I is a directed set, a directed system of Abelian groups is
a family {Gi }i∈I of Abelian groups, such that for all i < j there is a group morphism
` `
fij : Gi → Gj , with fii = id and fij ◦ fjk = fik . On the set G = i∈I Gi , where
denotes disjoint union, we put the following equivalence relation: g ∼ h, with g ∈ Gi
and h ∈ Gj , if there exists a k ∈ I such that fik (g) = fjk (h). The direct limit l of the
system {Gi }i∈I , denoted l = limi∈I Gi , is the quotient G/ ∼. Heuristically, two elements
−→
in G represent the same element in the direct limit if they are ‘eventually equal.’ From
this definition one naturally obtains the existence of canonical morphisms Gi → l. The
following discussion should make this notion clearer; for more detail, the reader may
consult [13].
S1) If two sections s ∈ F(U ), s̄ ∈ F(U ) coincide when restricted to any Ui , s|Ui =
s̄|Ui , they are equal, s = s̄.
S2) Given sections si ∈ F(Ui ) which coincide on the intersections, si |Ui ∩Uj =
sj |Ui ∩Uj for every i, j, there exists a section s ∈ F(U ) whose restriction to
each Ui equals si , i.e. s|Ui = si .
Thus, roughly speaking, sheaves are presheaves defined by local conditions. The
stalk of a sheaf is defined as in the case of a presheaf.
1. PRESHEAVES AND SHEAVES 39
Example 3.6. If F is a sheaf, and Fx = {0} for all x ∈ X, then F is the zero sheaf,
F(U ) = {0} for all open sets U ⊂ X. Indeed, if s ∈ F(U ), since sx = 0 for all x ∈ U ,
there is for each x ∈ U an open neighbourhood Ux such that s|Ux = 0. The first sheaf
axiom then implies s = 0. This is not true for a presheaf, cf. Example 3.15 below.
Example 3.8. Let G be an Abelian group. Defining P(U ) ≡ G for every open
subset U and taking the identity maps as restriction morphisms, we obtain a presheaf,
called the constant presheaf G̃X . All stalks (G̃X )x of G̃X are isomorphic to the group
G. The presheaf G̃X is not a sheaf: if V1 and V2 are disjoint open subsets of X, and
U = V1 ∪ V2 , the sections g1 ∈ G̃X (V1 ) = G, g2 ∈ G̃X (V2 ) = G, with g1 6= g2 , satisfy the
hypothesis of the second sheaf axiom S2) (since V1 ∩ V2 = ∅ there is nothing to satisfy),
but there is no section g ∈ G̃X (U ) = G which restricts to g1 on V1 and to g2 on V2 .
Example 3.10. In the same way one can define the following sheaves:
∞ of differentiable functions on a differentiable manifold X.
The sheaf CX
The sheaves ΩpX of differential p-forms, and all the sheaves of tensor fields on a
differentiable manifold X.
The sheaf of holomorphic functions on a complex manifold and the sheaves of holo-
morphic p-forms on it.
The sheaves of forms of type (p, q) on a complex manifold X.
Example 3.11. Let X be a differentiable manifold, and let d : Ω•X → Ω•X be the
p
exterior differential. We can define the presheaves ZX of closed differential p-forms, and
40 3. SHEAVES AND THEIR COHOMOLOGY
p
BX of exact p-differential forms,
p
ZX (U ) = {ω ∈ ΩpX (U ) | dω = 0},
p
BX (U ) = {ω ∈ ΩpX (U ) | ω = dτ for some τ ∈ Ωp−1
X (U )}.
p
ZX is a sheaf, since the condition of being closed is local: a differential form is closed if
p
and only if it is closed in a neighbourhood of each point of X. On the contrary, BX is
2 1
not a sheaf. In fact, if X = R , the presheaf BX of exact differential 1-forms does not
fulfill the second sheaf axiom: consider the form
xdy − ydx
ω=
x2 + y 2
defined on the open subset U = X − {(0, 0)}. Since ω is closed on U , there is an
open cover {Ui } of U by open subsets where ω is an exact form, ω|Ui ∈ BX 1 (U ) (this is
i
Poincaré’s lemma). But ω is not an exact form on U because its integral along the unit
circle is different from 0.
∞ −−→ Z 1 → 0 d
This means that, while the sequence of sheaf morphisms 0 → R → CX X
d
∞ (U ) −−→ Z 1 (U ) may fail to be surjective.
is exact (Poincaré lemma), the morphism CX X
Example 3.12. Let X be a complex manifold, Z the constant sheaf with stalk the
integers, O the sheaf of holomorphic functions on X, and O∗ the sheaf of nowhere
vanishing holomorphic functions. In analogy with the exact sequence (1.1) we may
consider the sequence
exp
(3.1) 0 → Z → O −−→ O∗ → 1
This is an exact sequence of sheaves, in particular exp : C → C∗ is surjective as a map
of sheaves, since in a neighbourhood of every point, an inverse my be found by applying
the logarithm function. However, since the latter is multi-valued, surjectivity fails on
non-simply connected open sets. See Example 3.11.
1.1. Étalé space. We wish now to describe how, given a presheaf, one can natu-
rally associate with it a sheaf having the same stalks. As a first step we consider the case
of a constant presheaf G̃X on a topological space X, where G is an Abelian group. We
can define another presheaf GX on X by putting GX (U ) = {locally constant functions
f : U → G}, 2 where G̃X (U ) = G is included as the constant functions. It is clear that
(GX )x = Gx = G at each point x ∈ X and that GX is a sheaf, called the constant sheaf
with stalk G. Notice that the functions f : U → G are the sections of the projection
`
π : x∈X Gx → X and the locally constant functions correspond to those sections which
locally coincide with the sections produced by the elements of G.
Now, let P be an arbitrary presheaf on X. Consider the disjoint union of the stalks
`
P = x∈X Px and the natural projection π : P → X. The sections s ∈ P(U ) of the
2A function is locally constant on U if it is constant on any connected component of U .
1. PRESHEAVES AND SHEAVES 41
Definition 3.13. The set P, endowed with the topology whose base of open subsets
consists of the sets s(U ) for U open in X and s ∈ P(U ), is called the étalé space of the
presheaf P.
P \ is called the sheaf associated with the presheaf P. In general, the morphism
φ : P → P \ is neither injective nor surjective: for instance, the morphism between the
constant presheaf G̃X and its associated sheaf GX is injective but nor surjective.
Example 3.15. As a second example we study the sheaf associated with the presheaf
k
BX of exact k-forms on a differentiable manifold X. For any open set U we have an
exact sequence of Abelian groups (actually of R-vector spaces)
k k k
0 → BX (U ) → ZX (U ) → HX (U ) → 0
where HXk is the presheaf that with any open set U associates its k-th de Rham coho-
such that i < j. By the definition of direct limit we see that, given a directed family of Abelian groups
{Gi }i∈I , if {Gj }j∈J is the subfamily indexed by J, then
lim Gi ' lim Gj ;
−→ −→
i∈I j∈J
that is, direct limits can be taken over cofinal subsets of the index set.
42 3. SHEAVES AND THEIR COHOMOLOGY
If F and G are sheaves, then the presheaves F ⊕ G and Hom(F, G) are sheaves.
On the contrary, the tensor product of F and G previously defined may not be a sheaf.
Indeed one defines the tensor product of the sheaves F and G as the sheaf associated
with the presheaf U → F(U ) ⊗ G(U ).
It should be noticed that in general Hom(F, G)(U ) 6' Hom(F(U ), G(U )) and
Hom(F, G)x 6' Hom(Fx , Gx ).
1.2. Direct and inverse images of presheaves and sheaves. Here we study
the behaviour of presheaves and sheaves under change of base space. Let f : X → Y be
a continuous map.
Let P be a presheaf on Y .
The inverse image sheaf of a sheaf F on Y is the sheaf f −1 F associated with the inverse
image presheaf of F.
of sheaves on X, is also exact (that is, the inverse image functor for sheaves of Abelian
groups is exact).
The étalé space f −1 F of the inverse image sheaf is the fibred product 5 Y ×X F. It
follows easily that the inverse image of the constant sheaf GX on X with stalk G is the
constant sheaf GY with stalk G, f −1 GX = GY .
2. Cohomology of sheaves
Here a caret denotes omission of the index. For instance, if p = 0 we have α = {αi } and
(3.2) (δα)ik = αk|Ui ∩Uk − αi|Ui ∩Uk .
It is an easy exercise to check that δ 2 = 0. Thus we obtain a cohomology theory. We
denote the corresponding cohomology groups by Ȟ k (U, P).
Example 3.2. We consider an open cover U of the circle S 1 formed by three sets
which intersect only pairwise. We compute the Čech cohomology of U with coefficients
in the constant sheaf R. We have C 0 (U, R) = C 1 (U, R) = R ⊕ R ⊕ R, C k (U, R) =
0 for k > 1 because there are no triple intersections. The only nonzero differential
d0 : C 0 (U, R) → C 1 (U, R) is given by
d0 (x0 , x1 , x2 ) = (x1 − x2 , x2 − x0 , x0 − x1 ).
Hence
Ȟ 0 (U, R) = ker d0 ' R
It is possible to define Čech cohomology groups depending only on the pair (X, F),
and not on a cover, by letting
The direct limit is taken over a cofinal subset of the directed set of all covers of X (the
order is of course the refinement of covers: a cover V = {Vj }j∈J is a refinement of U if
there is a map f : I → J such that Vf (i) ⊂ Ui for every i ∈ I). The order must be fixed
at the outset, since a cover may be regarded as a refinement of another in many ways.
As different cofinal families give rise to the same inductive limit, the groups Ȟ k (X, F)
are well defined.
2.2. Fine sheaves. Čech cohomology is well-behaved when the base space X is
paracompact. (It is indeed the bad behaviour of Čech cohomology on non-paracompact
spaces which motivated the introduction of another cohomology theory for sheaves,
usually called sheaf cohomology; cf. [6].) In this and in the following sections we consider
some properties of Čech cohomology that hold in that case.
Definition 3.3. A sheaf of rings R on a topological space X is fine if, for any
locally finite oper cover U = {Ui }i∈I of X,6 there is a family {si }i∈I of global sections
of R such that:
P
(1) i∈I si = 1;
(2) for every i ∈ I there is a closed subset Si ⊂ Ui such that (si )x = 0 whenever
x∈
/ Si .
6We recall that an oper cover U is locally finite if every point in X has an open neighbourhood which
intersects only a finite number of elements of U. It is possible to show that whenever X is paracompact,
any open cover has a locally finite refinement [17].
2. COHOMOLOGY OF SHEAVES 45
The family {si } is called a partition of unity subordinated to the cover U. For
instance, the sheaf of continuous functions on a paracompact topological space as well
as the sheaf of smooth functions on a differentiable manifold are fine, while sheaves of
complex or real analytic functions are not.
Proof. Let U = {Ui }i∈I be a locally finite open cover of X, and let {ρi } be a
partition of unity of R subordinated to U. For any α ∈ Č q (U, M) with q > 0 we set
X X
(Kα)i0 ...iq−1 = ρj aji0 ...iq−1 − ρj ai0 ji1 ...iq−1 + . . .
j∈I j∈I
j<i0 i0 <j<i1
q
X X
= (−1)k ρj ai0 ...ik−1 jik ...iq−1 .
k=0
j∈I
ik−1 <j<ik
Example 3.6. Using this result we may recast the proof of the exactness of the
Mayer-Vietoris sequence for de Rham cohomology in a slightly different form. Given a
differentiable manifold X, let U be the open cover formed by two sets U and V . Since
Č 2 (U, Ωk ) = 0 (there are no triple intersections) we have an exact sequence
δ
0 → Ȟ 0 (U, Ωk ) → Č 0 (U, Ωk ) → Č 1 (U, Ωk ) → 0 .
which in principle is exact everywhere but at C 1 (U, Ωk ). However since the sheaves Ωk
are acyclic by Proposition 3.5, one has Ȟ 1 (U, Ωk ) = 0, which means that δ is surjective,
and the sequence is exact at that place as well. We have the identifications
2.3. Long exact sequences in Čech cohomology. We wish to show that when
X is paracompact, any exact sequence of sheaves induces a corresponding long exact
sequence in Čech cohomology.
46 3. SHEAVES AND THEIR COHOMOLOGY
(3.3) 0 → P 0 → P → P 00 → 0
Proof. For any open cover U the exact sequence (3.3) induces an exact sequence
of differential complexes
Proof. We shall need to use the following fact [5, ?]: given an open cover U =
{Ui }i∈I of a paracompact space X,7 there is an open cover V = {Vi }i∈I having the same
cardinality of U, such that V̄i ⊂ Ui .
0 → Q1 → P → P \ → Q2 → 0
with
(3.5) 0 → Q1 → P → T → 0 , 0 → T → P \ → Q2 → 0
7It is enough that X is normal, however, any paracompact space is normal [17].
2. COHOMOLOGY OF SHEAVES 47
where T is the quotient presheaf P/Q1 , i.e. the presheaf U → P(U )/Q1 (U ). By Lemma
3.8 the isomorphisms (3.4) yield Ȟ k (X, Q1 ) = Ȟ k (X, Q2 ) = 0. Then by taking the long
exact sequences of cohomology from the exact sequences (3.5) we obtain the desired
isomorphism.
Using these results we may eventually prove that on paracompact spaces one has
long exact sequences in Čech cohomology.
Proof. Let P be the quotient presheaf F/F 0 ; then P \ ' F 00 . One has an exact
sequence of presheaves
0 → F0 → F → P → 0 .
By taking the associated long exact sequence in cohomology (cf. Lemma 3.7) and using
the isomorphism Ȟ k (X, P) = Ȟ k (X, F 00 ) one obtains the required exact sequence.
Example 3.11. The long exact sequence in cohomology associated with the exact
sequence (3.1) starts with
0 → H 0 (U, Z) → H 0 (U, O) → H 0 (U, O∗ ) → H 1 (U, Z) → . . .
This shows that the obstruction to the sequence (3.1) to be exact as a sequence of
presheaves is the first cohomology group with coefficients in Z. Since X, being a mani-
fold, is paracompact and locally Euclidean , the Čech cohomology of Z coincides with the
singular cohomology (see Proposition 3.29); therefore the above mentioned obstruction
is the non-simply connectedness of U .
2.4. Abstract de Rham theorem. We describe now a very useful way of com-
puting cohomology groups; this result is sometimes called “abstract de Rham theorem.”
As a particular case it yields one form of the so-called de Rham theorem, which states
that the de Rham cohomology of a differentiable manifold and the Čech cohomology of
the constant sheaf R are isomorphic.
0 → F → L0 → Q → 0
is not exact. We shall consider its cohomology H • (L• (X), d). By the previous Lemma
we have H 0 (L• (X), d) ' H 0 (X, F).
Since the sheaves Lk are acyclic by taking the long exact sequences of cohomology we
obtain a chain of isomorphisms
Ȟ 0 (X, Qk )
Ȟ k (X, F) ' Ȟ k−1 (X, Q1 ) ' · · · ' Ȟ 1 (X, Qk−1 ) '
Im Ȟ 0 (X, Lk−1 )
By Exercise 3.7 Ȟ 0 (X, Qk ) = Qk (X) is the kernel of dk : Lk (X) → Lk+1 so that the
claim is proved.
Corollary 3.15. (de Rham theorem.) Let X be a differentiable manifold. For all
k (X) and Ȟ k (X, R) are isomorphic.
k ≥ 0 the cohomology groups HDR
(where Ω0X ≡ CX ∞ ) is exact (this is Poincaré’s lemma). Moreover the sheaves Ω• are
X
∞
modules over the fine sheaf of rings CX , hence are acyclic. The claim then follows for
the previous theorem.
2.5. Soft sheaves. For later use we also introduce and study the notion of soft
sheaf. However, we do not give the proofs of most claims, for which the reader is referred
to [2, 6, 25]. The contents of this subsection will only be used in Section 5.5.
Definition 3.17. Let F be a sheaf a on a topological space X, and let U ⊂ X be
a closed subset of X. The space F(U ) (called “the space of sections of F over U ”) is
defined as
F(U ) = lim F(V )
−→
V ⊃U
where the direct limit is taken over all open neighbourhoods V of U .
Proof. The proof of Proposition 3.2 can be easily adapted to this situation.
Corollary 3.20. The quotient of two soft sheaves on a paracompact space is soft.
Proof. If F 00 = F/F 0 is the quotient of two soft sheaves, by Proposition 3.19 the
restriction morphism F(X) → F(V ) is surjective (where V is any closed subset of X),
so that F 00 (X) → F 00 (V ) is surjective as well.
Proposition 3.21. Any soft sheaf of rings R on a paracompact space is fine.
Proof. Let S 0 (F) be the sheaf of discontinuous sections of F (i.e., the sheaf of
all sections of the sheaf space F). The sheaf S 0 (F) is obviously soft. Now we have an
exact sequence 0 → F → S 0 (F) → F1 → 0. The sheaf F1 is not soft in general, but it
may embedded into the soft sheaf S 0 (F1 ), and we have an exact sequence 0 → F1 →
S 0 (F1 ) → F2 → 0. Upon iteration we have exact sequences
i p
k
0 → Fk −−→ S k (F) −−→
k
Fk+1 → 0
where S k (F) = S 0 (Fk ). One can check that the sequence of sheaves
f0 f1
0 → F → S 0 (F) −−→ S 1 (F) −−→ . . .
(where fk = ik+1 ◦ pk ) is exact.
50 3. SHEAVES AND THEIR COHOMOLOGY
Proof. The endomorphism sheaf End(S 0 (F)) is soft, hence fine by Proposition
3.21. Since S 0 (F) is an End(S 0 (F))-module, it is acyclic.8
Note that in this way we have shown that for any sheaf F on a paracompact space
there is a canonical soft resolution.
To prove this theorem we need to construct the so-called Čech sheaf complex. For
every nonvoid intersection Ui0 ...ip let ji0 ...ip : Ui0 ...ip → X be the inclusion. For every p
define the sheaf
Y
(3.7) Cˇp (U, F) = (ji0 ...ip )∗ F|Ui0 ...ip
i0 <···<ip
(every factor (ji0 ...ip )∗ F|Ui0 ...ip is the sheaf F first restricted to Ui0 ...ip and the extended by
zero to the whole of X). The Čech differential induces sheaf morphisms δ : Cˇp (U, F) →
Cˇp+1 (U, F). From the definition, we get isomorphisms
i.e., by taking global sections of the Čech sheaf complex we get the Čech cochain group
complex. Moreover we have:
where V runs over all open covers of X. The groups Ȟ k (V, Cˇp (U, F)) are the cohomology
of the complex Č • (V, Cˇp (U, F)), which may be written as
Y
Č k (V, Cˇp (U, F)) = Cˇp (U, F)(V`0 ...`k ))
`0 <···<`k
Y Y
' F(V`0 ...`k ∩ Ui0 ...ip )
`0 <···<`k i0 <···<ip
where Vi0 ...ip is the restriction of the cover V to Ui0 ...ip . This implies the claim.
2.7. Good covers. By means of Leray’s theorem we may reduce the problem of
computing the Čech cohomology of a differentiable manifold with coefficients in the
constant sheaf R (which, via de Rham theorem, amounts to computing its de Rham
cohomology) to the computation of the cohomology of a cover with coefficients in R;
thus a problem which in principle would need the solution of differential equations on
topologically nontrivial manifolds is reduced to a simpler problem which only involves
the intersection pattern of the open sets of a cover.
Good covers exist on any differentiable manifold (cf. [19]). Due to Corollary 3.16,
good covers are acyclic for the constant sheaf R. We have therefore
52 3. SHEAVES AND THEIR COHOMOLOGY
Proposition 3.28. For any good cover U of a differentiable manifold X one has
isomorphisms
Ȟ k (U, R) ' Ȟ k (X, R) , k ≥ 0.
The cover of Example 3.2 was good, so we computed there the de Rham cohomology
of the circle S 1 .
3. Sheaf cohomology
Another kind of sheaves which can be introduced is that of flabby sheaves (also called
“flasque”). A sheaf F on a topological space X is said to be flabby if for every open
subset U ⊂ X the restriction morphism F(X) → F(U ) is surjective. It is easy to prove
that flabby sheaves are soft: if U ⊂ X is a closed subset, by definition of direct limit,
for every s ∈ F(U ) there is an open neighbourhood V of U and a section s0 ∈ F(V )
which restricts to s. Since F is flabby, s0 can be extended to the whole of X. So on a
paracompact space, flabby sheaves are acyclic, and by the abstract de Rham theorem
flabby resolution can be used to compute cohomology. We should also notice that the
canonical soft resolution S • (F) we constructed in Section 2.5 is flabby, as one can easily
check by the definition itself. We shall then call S • (F) the canonical flabby resolution
of the sheaf F (this is also called the Godement resolution of F).
One can further pursue this line and use flabby resolutions (for instance, the canon-
ical flabby resolution) to define cohomology. That is, for every sheaf F, its cohomology
is by definition the cohomology of the global sections of its canonical flabby resolution
(it then turns out that cohomology can be computed with any acyclic resolution). This
has the advantage of producing a cohomology theory (called sheaf cohomology) which
is bell-behaved (e.g., it has long exact sequences in cohomology) on every topological
space, not just on paracompact ones. In this section we briefly outline the basics of
this theory; for a more comprehensive treatment the reader may refer to [6, 4, 2], or to
[23] where a different and more general approach to sheaf cohomology (using injective
resolutions) is pursued; also the original paper by Grothendieck [9] can be fruitfully
read. It follows from our treatment that on a paracompact topological space the sheaf
3. SHEAF COHOMOLOGY 53
and Čech cohomology coincide, but in general they do not. In the next chapter we shall
establish the relation between the two cohomologies in terms of a spectral sequence
(cf. also [12], especially the exercise section, for a discussion of the comparison between
the two cohomologies).
The following two results are basic for this construction. Here X is any topological
space.
Corollary 3.4. If
0 → L0 → L1 → . . .
is an exact sequence of flabby sheaves, for every open set U ⊂ X the sequence of abelian
groups
0 → L0 (U ) → L1 (U ) → . . .
is exact.
54 3. SHEAVES AND THEIR COHOMOLOGY
Corollary 3.5. Flabby sheaves are acyclic with respect to sheaf cohomology, i.e.,
H p (X, F)
= 0 for all p > 0 if F is a flabby sheaf.
Corollary 3.6. Flabby sheaves are acyclic with respect to Čech cohomology, i.e.,
Ȟ p (U, F) = 0 for every open cover U of X and for all p > 0 if F is a flabby sheaf.
Proof. Since F is flabby, the sheaves Cˇp (U, F) defined in Eq. (3.7) are flabby as
well. By Corollary 3.4 the sequence
δ δ
Cˇ0 (U, F)(X) →
− Cˇ1 (U, F)(X) →
− ...
is exact. Since Cˇp (U, F)(X) = Č p (U, F), this implies that the Čech complex Č • (U, F)
is exact.
As a further consequence, we have the isomorphism between Čech and sheaf coho-
mology on a paracompact space.
Corollary 3.7. For any sheaf F on a paracompact space X, the Čech cohomology
Ȟ • (X, F)
and the sheaf cohomology H • (X, F) are isomorphic.
We want to show that sheaf cohomology is well behaved with respect to exact
sequences of sheaves on any topological space. Let us denote by Sh/X , K(Sh/X ) and
K(Ab) the categories of sheaves (of abelian groups) on X, of complexes of sheaves on
X, and of complexes of abelian groups, respectively. The canonical flabby resolution
allows one to define two functors:
F1 : Sh/X → K(Sh/X )
F 7→ S • (F)
F2 : Sh/X → K(Ab)
F 7→ S • (F)(X)
Proposition 3.8. The two functors F1 , F2 are exact (i.e., they map exact sequences
to exact sequences).
Proof. If
(3.9) 0 → F 0 → F → F 00 → 0
Proof. The long exact sequence of cohomology associated with the exact sequence
of complexes of abelian groups (3.10) is just the sequence (3.11).
An immediate consequence of this result is that the proof of the abstract de Rham
theorem for the Čech cohomology on a paracompact space may be applied to provide a
proof of the same theorem for sheaf cohomology on any space; thus, on any topological
space, the sheaf cohomology of a sheaf F is isomorphic to the cohomology of the complex
of global sections of a resolution of F which is acyclic for the sheaf cohomology.
CHAPTER 4
57
CHAPTER 5
Spectral sequences
Spectral sequences are a powerful tool for computing homology, cohomology and
homotopy groups. Often they allow one to trade a difficult computation for an easier
one. Examples that we shall consider are another proof of the Čech-de Rham theorem,
the Leray spectral sequence, and the Künneth theorem.
Spectral sequences are a difficult topic, basically because the theory is quite intrin-
cate and the notation is correspondingly cumbersome. Therefore we have chosen what
seems to us to be the simplest approach, due to Massey [20]. Our treatment basically
follows [3].
1. Filtered complexes
K = K0 ⊃ K1 ⊃ K2 ⊃ . . .
is a filtration of (K, d). We then say that (K, d) is filtered, and associate with it the
graded complex1
M
Gr(K) = Kp /Kp+1 , Kp = K if p ≤ 0 .
p∈Z
Note that by assumption (since every Kp+1 is a graded subgroup of Kp ) the filtration
is compatible with the grading, i.e., if we define Kpi = K i ∩ Kp , then
59
60 5. SPECTRAL SEQUENCES
δ1 2 = δ2 2 = 0 , δ1 δ2 + δ2 δ1 = 0 .
we obtain a filtration of (T, d). This satifies Tp ' T for p ≤ 0. The successive quotients
of the filtration are
M
Tp /Tp+1 = K p,q .
q∈N
For instance, the filtration in Example 5.1 is regular since Tpi = 0 for p > i, and
indeed
i−p
M
i i
Tp = T ∩ Tp = K i−j,j .
j=0
2This assumption is made here for simplicity but one could let p, q range over the integers; however
with E ' Gr(K). The differential d induces differentials in G and E, so that from (5.2)
one gets an exact triangle in cohomology (cf. Section 1.1)
(5.3) H(G)
i / H(G)
cHH v
HH vv
HH vv
H v
k HH v
v{ v j
H(E)
where k is the connecting morphism.
Let us now assume that the filtration K• has finite length, i.e., Kp = 0 for p greater
than some ` (called the length of the filtration).
Since dKp ⊂ Kp for every p, we may consider the cohomology groups H(Kp ). The
morphism i induces morphisms i : H(Kp+1 ) → H(Kp ). Define G1 to be the direct sum
of the terms on the sequence (which is not exact)
i i
0 → H(K` ) −−→ H(K`−1 ) −−→ . . .
i i ∼ ∼
−−→ H(K1 ) −−→ H(K) −−→ H(K−1 ) −−→ . . . ,
L
i.e., G1 = p∈Z H(Kp ) ' H(G). Next we define G2 as the sum of the terms of the
sequence
We come now to the construction of the spectral sequence. Recall that since dKp ⊂
L
Kp , and E = p Kp /Kp+1 , the differential d acts on E, and one has a cohomology
group H(E) wich splits into a direct sum
M
H(E) ' H(Kp /Kp+1 , d) .
p∈Z
The cohomology group H(E) fits into the exact triangle (5.3), that we rewrite as
i1
(5.5) G1 / G1
`BB |
BB |
BB ||
k1 BB ||| j1
~|
E1
where E1 = H(E). We define d1 : E1 → E1 by letting d1 = j1 ◦ k1 ; then d21 = 0 since
the triangle is exact. Let E2 = H(E1 , d1 ) and recall that G2 is the image of G1 under
i : G1 → G1 . We have morphisms
i 2 : G2 → G2 , , j2 : G2 → E2 , k2 : E2 → G2
where
(i) i2 is induced by i1 by letting i2 (i1 (x)) = i1 (i1 (x)) for x ∈ G1 ;
(ii) j2 is induced by j1 by letting j2 (i1 (x)) = [j1 (x)] for x ∈ G1 , where [ ] denotes
taking the homology class in E2 = H(E1 , d1 ).
(iii) k2 is induced by k1 by letting k2 ([e]) = i1 (k1 (e)).
Exercise 5.1. Show that the morphisms j2 and k2 are well defined, and that the
triangle
i2
(5.6) G2 / G2
`BB |
BB |
BB ||
k2 BB ||| j2
~|
E2
is exact.
We call (5.6) the derived triangle of (5.5). The procedure leading from (5.5) to the
triangle (5.6) can be iterated, and we get a sequence of exact triangles
ir
Gr / Gr
`BB |
BB |
BB ||
kr BB ||| jr
~|
Er
where each group Er is the cohomology group of the differential module (Er−1 , dr−1 ),
with dr−1 = jr−1 ◦ kr−1 .
As we have already noticed, due to the assumption that the filtration K• has finite
length `, the groups Gr stabilize when r ≥ ` + 1, and the morphisms ir : Gr → Gr
2. THE SPECTRAL SEQUENCE OF A FILTERED COMPLEX 63
become injective. Thus all morphisms kr : Er → Gr vanish in that range, which implies
dr = 0, so that the groups Er stabilize as well: Er+1 ' Er for r ≥ ` + 1. We denote by
E∞ = E`+1 the stable value.
Thus, the sequence
∞ i
0 → G∞ −−→ G∞ → E∞ → 0
is exact, which implies that E∞ is the associated graded module of the filtration (5.4)
of H(K): M
E∞ ' Fp /Fp+1 .
p≤`
So what we have seen so far in this section is that if (K, d) is a differential module
with a filtration of finite length, one can build a spectral sequence which converges to
H(K).
Remark 5.3. It may happen in special cases that the groups Er stabilize before
getting the value r = ` + 1. That happens if and only if dr = 0 for some value r = r0 .
This implies that dr = 0 also for r > r0 , and Er+1 ' Er for all r ≥ r0 . When this
happens we say that the spectral sequence degenerates at step r0 .
Theorem 5.4. Let (K, d) be a graded differential module, and K• a regular filtration.
There is a spectral sequence {(Er , dr )}, where each Er is graded, which converges to the
graded group H • (K, d).
Note that the filtration need not be of finite length: the length `(i) of the filtration
of K i is finite for every i, but may increase with i.
Proof. For every n and p we have d(Kpn ) ⊂ Kpn+1 , therefore we have cohomology
groups H n (Kp ). As a consequence, the groups Gr are graded:
M M
Gr ' Frn = ir−1 (H n (Kp ))
n∈Z n,p∈Z
and the groups Er are accordingly graded. We may construct the derived triangles as
before, but now we should pay attention to the grading: the morphisms i and j have
degree zero, but k has degree one (just check the definition: k is basically a connecting
morphism).
Fix a natural number n, and let r ≥ `(n + 1) + 1; for every p the morphisms
ir : Frn+1 → Frn+1
64 5. SPECTRAL SEQUENCES
The last statement in the proof means that for each n, F•n is a filtration of H n (K, d),
n ' n n
L
and E∞ p∈Z Fp /Fp+1 .
The terms Er of the spectral sequence are actually bigraded; for instance, since the
filtration and the degree of K are compatible, we have
q p+q
M M
Kp /Kp+1 ' Kpq /Kp+1 ' Kpp+q /Kp+1
q∈Z q∈Z
and E0 = E is bigraded by
M p,q
E0 = E0 p+q
with E0p,q = Kpp+q /Kp+1 .
p,q∈Z
Note that the total complex associated with this bidegree yields the gradation of E.
Let us go to next step. Since d : Kpp+q → Kpp+q+1 , i.e., d : E0p,q → E0p,q+1 , and
E1 = H(E, d), if we set
E1p,q = H q (E0p,• , d) ' H p+q (Kp /Kp+1 )
L p,q
we have E1 ' p,q∈Z E1 .
If we go one step further we can show that
d1 : E1p,q → E1p+1,q .
Indeed if x ∈ E1p,q ' H p+q (Kp /Kp+1 ) we write x as x = [e] where e ∈ Kpp+q /Kp+1
p+q
so
that k1 (x) = i1 (k(e)) ∈ H p+q+1 (Kp+1 ) and
d1 (x) = j1 (k1 (x)) = j1 (k(e)) ∈ H p+q+1 (Kp+1 /Kp+2 ) ' E1p+1,q .
As a result we have E2 ' p,q∈Z E2p,q with
L
dr : Erp,q → Erp+r,q−r+1
4. THE SPECTRAL SEQUENCES ASSOCIATED WITH A DOUBLE COMPLEX 65
The next two Lemmas establish the existence of the morphisms that we shall use to
introduce the so-called five-term sequence, and will anyway be useful in the following.
Proof. Since Kp ' K for p ≤ 0 we have Fpn ' H n (Kp ) = H n (K) for p ≤ 0,
p,q 0,q
hence E∞ = 0 for p < 0 and E∞ ' F0q /F1q ' H q (K)/F1q , so that there is a surjective
0,q
morphism H q (K) → E∞ .
Note now that a nonzero class in Er0,q cannot be a boundary, since then it should
come from Er−r,q+r−1 = 0. So cohomology classes are cycles. Since cohomology classes
0,q 0,q
are elements in Er+1 , we have inclusions Er+1 ⊂ Er0,q (Er+1
0,q
is the subgroup of cycles
0,q 0,q 0,q
in Er ). This yields an inclusion E∞ ⊂ Er for all r.
Combining the two arguments we obtain morphisms H q (K) → Er0,q .
Lemma 5.2. Assume that Kpn = 0 if p > n (so, in particular, the filtration is
regular). Then for every r ≥ 2 there is a morphism Erp,0 → H p (K).
Proof. The hypothesis of the Lemma implies that Erp,q = 0 for q < 0 (indeed,
Fpp+q= ir (H p+q (Kp )) for r big enough, so that Fqp+q = 0 if q < 0 since then Kpp+1 = 0).
As a consequence, for r ≥ 2 the differential dr : Erp,0 → Erp+r,1−r maps to zero, i.e., all
elements in Erp,0 are cycles, and determine cohomology classes in Er+1 p,0
. This means we
have a morphism Erp,0 → Er+1 p,0
, and composing, morphisms Erp,0 → E∞ p,0
.
p,0
Since Fpn = 0 for p > n we have E∞ ' Fpp /Fp+1
p
' Fpp so that one has an injective
p,0
morphism E∞ → H p (K). Composing we have a morphism Erp,0 → H p (K).
Proposition 5.3. (The five-term sequence). Assume that Kpn = 0 if p > n. There
is an exact sequence
d
0 → E21,0 → H 1 (K) → E20,1 −−→
2
E22,0 → H 2 (K) .
Proof. The morphisms involved in the sequence in addition to d2 have been defined
in the previous two Lemmas. We shall not prove the exactness of the sequence here, for
a proof cf. e.g. [6].
Gp,q = Tqp+q .
Notice that if we form the total complex p+q=n Gp,q we obtain the complex (5.7) back:
L
M q
M M n−p
M
p,q p+q−j,j
G ' K = K n−j,j = Gn .
p+q=n p+q=n j=0 j=0
We analyze the spectral sequence associated with these data. The first three terms
are easily described. One has
so that the differential d0 : E0p,q → E0p,q+1 coincides with δ2 : K p,q → K p,q+1 , and one
has
At next step we have d1 : E1p,q → E1p+1,q with E1p,q ' H p+q (Tp /Tp+1 ) and Tp /Tp+1 '
p,q . Hence the differential
L
q∈Z K
M M
d1 : H p+q ( K p,n ) → H p+q+1 ( K p+1,n )
n∈Z n∈Z
One should notice that by exchanging the two degrees in K (i.e., considering another
double complex 0 K such that 0 K p,q = K q,p ), we obtain another spectral sequence, that
we denote by { 0 Er , 0 dr }. Both sequences converge to the same graded group, i.e., the
cohomology of the total complex (but the corresponding filtrations are in general differ-
ent), and this often provides interesting information. For the second spectral sequence
we get
(5.10) 0
E1q,p ' H p (K •,q , δ1 )
(5.11) 0
E2q,p ' H q (0 E1p,• , δ2 ) .
4. THE SPECTRAL SEQUENCES ASSOCIATED WITH A DOUBLE COMPLEX 67
Example 5.1. A simple application of the two spectral sequences associated with a
double complex provides another proof of the Čech-de Rham theorem, i.e., the isomor-
phism H • (X, R) ' HDR• (X) for a differentiable manifold X. Let U = {U } be a good
i
cover of X, and define the double complex
K p,q = Č p (U, Ωq ) ,
i.e., K •,q is the complex of Čech cochains of U with coefficients in the sheaf of differential
q-forms. The first differential δ1 is basically the Čech differential δ, while δ2 is the
exterior differential d.3 Actually δ and d commute rather than anticommute, but this
is easily settled by defining the action of δ1 on K p,q as δ1 = (−1)q δ (this of course
leaves the spaces of boundaries and cycles unchanged). We start analyzing the spectral
sequences from the terms E1 . For the first, we have
This implies that d2 = 0, so that the spectral sequence degenerates at the second step,
p,q p,0
and E∞ = 0 for q 6= 0 and E∞ ' H p (U, R). The resulting filtration of H p (T, D) has
only one nonzero quotient, so that H p (T, D) ' H p (U, R).
Let us now consider the second spectral sequence. We have
0
E1p,q ' H q (K •,p , δ) = H q (Č • (U, Ωp ), δ) = H q (U, Ωp ) .
Again the spectral sequence degenerates at the second step, and we have H p (T, D) '
p p
HDR (X). Comparing with what we got from the first sequence, we obtain HDR (X) '
p p p
H (U, R). Taking a direct limit on good covers, we obtain H (X, R) ' HDR (X).
3Here a notational conflict arises, so that we shall denote by D the differential of the total complex
T.
68 5. SPECTRAL SEQUENCES
Remark 5.2. From this example we may get the general result that if at step r,
with r ≥ 1, we have Erp,q = 0 for q 6= 0 (or for p 6= 0) then the sequence degenerates at
step r, and Erp,0 ' H p (T, d) (or Er0,q ' H q (T, d)).
5. Some applications
E2p,q ' H p (E1•,q , δ1 ) ' H p (Č • (U, H̃q (L• )), δ) ' H p (U, Hq (L• ))
where, since X is paracompact, we have replaced the presheaves H̃• with the corre-
sponding sheaves H• (possibly replacing the cover U by a suitable refinement).
For the second spectral sequence we have
0
E1p,q ' H q (K •,p , δ1 ) ' H q (Č • (U, Lp ), δ1 ) ' H p (U, Lq )
0
E2q,p ' H p (0 E1q,• , δ2 ) ' H p (H q (U, L• ), f ) .
The canonical filtrations of a double complex always satisfy the hypothesis of Lemma
5.2. So, considering the first spectral sequence, we obtain morphisms (again taking a
direct limit)
H q (L• (X), f ) → H q (X, F) .
5. SOME APPLICATIONS 69
In general these are not isomorphisms. The same morphisms could be obtained by
breaking the exact sequence 0 → F → L• into short exact sequences, taking the asso-
ciated long exact cohomology sequences and suitably composing the morphisms, as in
the proof of the abstract de Rham theorem 3.14.
A further specialization is obtained if the resolution L• is acyclic; then the second
spectral sequence degenerates at the second step as well, and we get isomorphisms
H p (X, F) ' H p (L• (X), f ), i.e., we have another proof of the abstract de Rham theorem
3.14.
Proposition 5.2. The sheaf Rk π∗ F is isomorphic to the sheaf associated with the
presheaf P k on Y defined by P k (U ) = H k (π −1 (U ), F).
H k (L• (π −1 (U ), f ) ' H k (π −1 (U ), F) ,
Let us consider the double complex Č p (U, π∗ Lq ), where U is a locally finite open
cover of Y . The two spectral sequences we have previously studied yield at the second
term
E2p,q ' H p (U, Rq π∗ F)
0
E2p,q ' H q (H p (U, π∗ L• ), f )
70 5. SPECTRAL SEQUENCES
Since the sheaves π∗ L• are soft (hence acyclic) the second spectral sequence degenerates,
p,q
and one has 0 E∞ = 0 for p 6= 0, and
0 0,q
E∞ ' 0
E20,q ' H q (H 0 (Y, π∗ L• ), f )
' H q (L• (X), f ) ' H q (X, F) .
Again after taking a direct limit, we have:
We describe without proof the relation between the stalks of the sheaf Rk π∗ F
at points y ∈ Y and the cohomology groups H k (π −1 (y), F); here F is to be consid-
ered as restricted to π −1 , i.e., more precisely we should write H k (π −1 (y), i−1
y F) where
−1
iy : π (y) → X is the inclusion. Since
(Rk π∗ F )y = lim (Rk π∗ F)(U ) ' lim H k (π −1 (U ), F) ,
−→ −→
y∈U y∈U
while H k (π −1 (y), F) is the direct limit of the groups H k (V, F) where V ranges over all
open neighbourhoods of π −1 (y), there is a natural map
(5.13) (Rk π∗ F)y → H k (π −1 (y), F) .
This is an isomorphism under some conditions, e.g., if Y is locally compact and π is
proper (cf. [6]). This happens for instance when both X and Y are compact.
As a simple Corollary to Proposition 5.3 one obtains Leray’s theorem:
Proof. The hypothesis of the Corollary means that every y ∈ Y has a system of
neighbourhoods {U } such that H k (π −1 (U ), F) = 0 for all k > 0. This implies that
Rk π∗ F = 0 for k > 0, so that the only nonzero terms in the spectral sequence E2 are
Ep,0 p
2 ' H (Y, π∗ F). The sequence degenerates and the claim follows.
Proposition 5.6. Assume that the groups H • (Y, G) have no torsion over Z, and
that X and Y are compact Hausdorff and locally Euclidean. Then,
M
H k (X × Y, G) ' H p (X, Z) ⊗ H q (Y, G) .
p+q=k
(6.1) αx = βy , αy = −βx
are biholomorphisms.
Example 6.3. (The Riemann sphere) Consider the sphere in R3 centered at the
origin and having radius 12 , and identify the tangent planes at (0, 0, 12 ) and (0, 0, − 12 ) with
C. The stereographic projections give local complex coordinates z1 , z2 ; the transition
function z2 = 1/z1 is defined in C? = C − {0} and is biholomorphic.
75
76 6. COMPLEX MANIFOLDS AND VECTOR BUNDLES
Example 6.7. For any k < n the inclusion of Ck+1 into Cn+1 obtained by setting
to zero the last n − k coordinates in Cn+1 yields a map Pk → Pn ; the reader may check
that this realizes Pk as a submanifold of Pn .
where the vi are a basis of tangent vectors to the given k-plane. So Gk,n imbeds into
P(Λk Cn ) = PN , where N = nk − 1 (this is called the Plücker embedding. If a basis
the numbers Pi1 ...ik are the Plücker coordinates on the Grassmann variety.
2.1. Orientation. All complex manifolds are oriented. Consider for simplicity the
1-dimensional case; the jacobian matrix of a transition function z 0 = f (z) = α(x, y) +
iβ(x, y) is (by the Cauchy-Riemann conditions)
! !
αx αy αx αy
J= =
βx βy −αy αx
2.2. Forms of type (p, q). Let X be an n-dimensional complex manifold; by the
identification Cn ' R2n , and since a biholomorphic map is a C ∞ diffeomorphism, X
has an underlying structure of 2n-dimensional real manifold. Let T X be the smooth
tangent bundle (i.e. the collection of all ordinary tangent spaces to X). If (z 1 , . . . , z n ) is
a set of local complex coordinates around a point x ∈ X, then the complexified tangent
space Tx X ⊗R C admits the basis
∂ ∂ ∂ ∂
,..., , ,..., .
∂z 1 x ∂z n x ∂ z̄ 1 x ∂ z̄ n x
This yields a decomposition
T X ⊗ C = T 0 X ⊕ T 00 X
which is intrinsic because X has a complex structure, so that the transition functions
are holomorphic and do not mix the vectors ∂z∂ i with the ∂∂z̄ i . As a consequence one has
a decomposition
M
Λi T ∗ X ⊗ C = Ωp,q X where Ωp,q X = Λp (T 0 X)∗ ⊗ Λq (T 00 X)∗ .
p+q=i
The elements in Ωp,q X are called differential forms of type (p, q), and can locally be
written as
η = ηi1 ...ip ,j1 ...jq (z, z̄) dz i1 ∧ · · · ∧ dz ip ∧ dz̄ j1 ∧ · · · ∧ dz̄ jq .
4. HOLOMORPHIC VECTOR BUNDLES 79
The compositions
Ω p+1,q X
o7
∂ oooo
ooo
ooo
Ωp,q X
d / Λp+q+1 T ∗ X
OOO
OOO
OOO
∂¯ OO'
Ωp,q+1 X
define differential operators ∂, ∂¯ such that
∂ 2 = ∂¯2 = ∂ ∂¯ + ∂∂
¯ =0
¯ = 0).
(notice that the Cauchy-Riemann condition can be written as ∂f
3. Dolbeault cohomology
Another interesting cohomology theory one can consider is the Dolbeault cohomology
associated with a complex manifold X. Let Ωp,q denote the sheaf of forms of type (p, q)
on X. The Dolbeault (or Cauchy-Riemann) operator ∂¯ : Ωp,q → Ωp,q+1 squares to zero.
¯ is for any p ≥ 0 a cohomology complex. Its cohomology
Therefore, the pair (Ωp,• (X), ∂)
p,q
groups are denoted by H∂¯ (X), and are called the Dolbeault cohomology groups of X.
We have for this theory an analogue of the Poincaré Lemma, which is sometimes
¯
called the ∂-Poincaré Lemma (or Dolbeault or Grothendieck Lemma).
Moreover, the kernel of the morphism ∂¯ : Ωp,0 → Ωp,1 is the sheaf of holomorphic
p-forms Ωp . Therefore, the Dolbeault complex of sheaves Ωp,• is a resolution of Ωp ,
i.e. for all p = 0, . . . , n (where n = dimC X) the sheaf sequence
∂¯ ∂¯ ∂¯
0 → Ωp → Ωp,0 −−→ Ωp,1 −−→ . . . −−→ Ωp,1 → 0
∞ -modules). Then, exactly as
is exact. Moreover, the sheaves Ωp,q are fine (they are CX
one proves the de Rham theorem (Theorem 3.3.15), one obtains the Dolbeault theorem:
Exercise 6.2. Show that two trivializations are equivalent (i.e. describe isomorphic
bundles) if there exist holomorphic maps λα : Uα → Gl(n, C) such that
(6.3) 0
gαβ = λα gαβ λ−1
β
Exercise 6.3. Show that the rule that to any open subset U ⊂ X assigns the
∞ (U )-module
OX of sections of a holomorphic vector bundle E defines a sheaf E (which
actually is a sheaf of OX -modules).
(where T denotes transposition) define another vector bundle, called the dual vector
bundle to E, and denoted by E ∗ . Sections of E ∗ can be paired with (or act on) sections
of E, yielding holomorphic (smooth complex-valued) functions on (open sets of) X.
Example 6.4. The space E = X × Cn , with the projection onto the first factor, is
obviously a holomorphic vector bundle, called the trivial vector bundle of rank n. We
shall denote such a bundle by Cn (in particular, C denotes the trivial line bundle). A
holomorphic vector bundle is said to be trivial when it is isomorphic to Cn .
Every holomorphic vector bundle has an obvious structure of smooth complex vector
bundle. A holomorphic vector bundle may be trivial as a smooth bundle while not being
trivial as a holomorphic bundle. (In the next sections we shall learn some homological
techniques that can be used to handle such situations).
Example 6.6. (The tautological bundle) Let (w1 , . . . , wn+1 ) be homogeneous coor-
dinates in Pn . If to any p ∈ Pn (which is a line in Cn+1 ) we associate that line we obtain
a line bundle, the tautological line bundle L of Pn . To be more concrete, let us exhibit
a trivialization for L and the related transition functions. If {Ui } is the standard cover
of Pn , and p ∈ Ui , then wi can be used to parametrize the points in the line p. So if
p has homogeneous coordinates (w0 , . . . , wn ), we may define ψi : π −1 (Ui ) → Ui × C as
ψi (u) = (p, wi ) if p = π(u). The transition function is then gik = wi /wk . The dual
bundle H = L∗ acts on L, so that its fibre at p = π(u), u ∈ Cn+1 can be regarded as
82 6. COMPLEX MANIFOLDS AND VECTOR BUNDLES
the space of linear functionals on the line Cu ≡ Lp , i.e. as hyperplanes in Cn+1 . Hence
H is called the hyperplane bundle. Often L is denoted O(−1), and H is denoted O(1)
— the reason of this notation will be clear in Chapter 7.
In the same way one defines a tautological bundle on the Grassmann variety Gk,n ;
it has rank k.
Exercise 6.7. Show that that the elements of a basis of the vector space of global
sections of L can be identified homogeneous coordinates, so that dim H 0 (Pn , L) = n +
1. Show that the global sections of H can be identified with the linear polynomials
in the homogeneous coordinates. Hence, the global sections of H r are homogeneous
polynomials of order r in the homogeneous coordinates.
4.2. More constructions. Additional operations that one can perform on vector
bundles are again easily described in terms of transition functions.
(1) Given two vector bundles E1 and E2 , of rank r1 and r2 , their direct sum E1 ⊕ E2
is the vector bundle of rank r1 + r2 whose transition functions have the block matrix
form !
(1)
gαβ 0
(2)
0 gαβ
(2) We may also define the tensor product E1 ⊗ E2 , which has rank r1 r2 and has
(1) (2)
transition functions gαβ gαβ . This means the following: assume that E1 and E2 trivialize
over the same cover {Uα }, a condition we may always meet, and that in the given
trivializations, E1 and E2 have local bases of sections {s(α)i } and {t(α)k }. Then E1 ⊗ E2
has local bases of sections {s(α)i ⊗ t(α)k } and the corresponding transition functions are
given by
r1 X r2
(1) (2)
X
s(α)i ⊗ t(α)k = (gαβ )im (gαβ )kn s(β)m ⊗ t(β)n .
m=1 n=1
Exercise 6.8. Show that the canonical bundle of the projective space Pn is isomor-
phic to O(−n − 1).
Example 6.9. Let π : Cn+1 − {0} → Pn be the usual projection, and let (w1 , . . . ,
wn+1 ) be homogeneous coordinates in Pn . The tangent spaces to Pn are generated by
5. CHERN CLASSES 83
∂
the vectors π∗ ∂w i , and these are subject to the relation
n+1
X ∂
wi π∗ = 0.
∂wi
i=1
0 λα
gαβ = gαβ with λα ∈ O∗ (Uα ),
λβ
so that one has an identification Pic(X) ' H 1 (X, O∗ ). The long cohomology sequence
associated with the exact sequence
exp
0 → Z → O −−→ O∗ → 0
84 6. COMPLEX MANIFOLDS AND VECTOR BUNDLES
is the first Chern class1 of L. The fact that δ is a group morphism means that
c1 (L ⊗ L0 ) = c1 (L) + c1 (L0 ) .
Exercise 6.2. Show that there exist nontrivial holomorphic line bundles which are
trivial as smooth complex line bundles.
is exact, so that Pic0 (X) = H 1 (X, O)/H 1 (X, Z). If in addition dim X = 1 we have
H 2 (X, O) = 0, so that every element in H 2 (X, Z) is the first Chern class of a holomorphic
line bundle.2
From the definition of connecting morphism we can deduce an explicit formula for
a Čech cocycle representing c1 (L) with respect to the cover {Uα }:
1
{c1 (L)}αβγ = 2πi (log gαβ + log gβγ + log gγα ) .
From this one can easily prove that, if f : X → Y is a holomorphic map, and L is a line
bundle on Y , then
c1 (f ∗ L) = f ] (c1 (L)) .
1This allows us also to define the first Chern class of a vector bundle E of any rank by letting
k > n.
6. CHERN CLASSES 85
5.2. Smooth line bundles. The first Chern class can equally well be defined
for smooth complex line bundles. In this case we consider the sheaf C of complex-
valued smooth functions on a differentiable manifold X, and the subsheaf C ∗ of nowhere
vanishing functions of such type. The set of isomorphism classes of smooth complex line
bundles is identified with the cohomology group H 1 (X, C ∗ ). However now the sheaf C
is acyclic, so that the obstruction morphism δ establishes an isomorphism H 1 (X, C ∗ ) '
H 2 (X, Z). The first Chern class of a line bundle L is again defined as c1 (L) = δ([L]),
but now c1 (L) classifies the bundle (i.e. L ' L0 if and only if c1 (L) = c1 (L0 )).
In this section we define higher Chern classes for complex vector bundles of any rank.
Since the Chern classes of a vector bundle will depend only on its smooth structure, we
may consider a smooth complex vector bundle E on a differentiable manifold X. We
are already able to define the first Chern class c1 (L) of a line bundle L, and we know
that c1 (L) ∈ H 2 (X, Z). We proceed in two steps:
(1) we first define Chern classes of vector bundles that are direct sums of line bundles;
(2) and then show that by means of an operation called cohomology base change we
can always reduce the computation of Chern classes to the previous situation.
Step 1. Let σi , i = 1 . . . k, denote the symmetric function of order i in k arguments.3.
Since these functions are polynomials with integer coefficients, they can be regarded as
functions on the cohomology ring H • (X, Z). In particular, if α1 , . . . , αk are classes in
H 2 (X, Z), we have σi (α1 , . . . , αk ) ∈ H 2i (X, Z).
If E = L1 ⊕ · · · ⊕ Lk , where the Li ’s are line bundles, for i = 1...k we define the i-th
Chern class of E as
ci (E) = σi (c1 (L1 ), . . . , c1 (Lk )) ∈ H 2i (X, Z) .
3The symmetric functions are defined as
X
σi (x1 , . . . , xk ) = xj1 · · · · · xji .
1≤j1 <···<ji ≤n
σ1 (x1 , . . . , xk ) = x1 + · · · + xk
σ2 (x1 , . . . , xk ) = x1 x2 + x1 x3 + · · · + xk−1 xk
...
σk (x1 , . . . , xk ) = x1 · · · · · xk .
Definition 6.2. The i-th Chern class ci (E) of E is the unique class in H 2i (X, Z)
such that f ] (ci (E)) = ci (f ∗ E).
f ] (ci (E)) = ci (f ∗ E) .
Exercise 6.3. Prove that for any vector bundle E one has c1 (E) = c1 (det E).
7. KODAIRA-SERRE DUALITY 87
7. Kodaira-Serre duality
In this section we introduce Kodaira-Serre duality, which will be one of the main
tools in our study of algebraic curves. To start with a simple situation, let us study
the analogous result in de Rham theory. Let X be a differentiable manifold. Since the
exterior product of two closed forms is a closed form, one can define a bilinear map
i j i+j
HDR (X) ⊗ HDR (X) → HDR (X), [τ ] ⊗ [ω] → [τ ∧ ω].
As we already know, via the Čech-de Rham isomorphism this product can be identified
with the cup product. If X is compact and oriented, by composition with the map4
Z
n
∫ : HDR (X) → R, ∫ [ω] = ω
X X X
and a duality
H∂p,q ∗ n−p,n−q
¯ (X) ' H∂¯ (X).
∂¯E (f s) = f ∂¯E s + ∂f
¯ ⊗s
for s ∈ E ∞ (U ), f ∈ C ∞ (U ).
(2) Show that ∂¯E defines an exact sequence of sheaves
∂¯ ∂¯ ∂¯
(6.7) 0 → Ωp ⊗ E → Ωp,0 ⊗ E ∞ −−→
E
Ωp,1 ⊗ E ∞ −−→
E E
. . . −−→ Ωp,n ⊗ E ∞ → 0.
4This map is well defined because different representatives of [ω] differ by an exact form, whose
By combining the pairing (6.5) with the action of the sections of E ∗ on the sections
of E we obtain a nondegenerate pairing
H∂p,q n−p,n−q
¯ (X, E) ⊗ H∂¯ (X, E ∗ ) → C
and therefore a duality
H∂p,q ∗ n−p,n−q
¯ (X, E) ' H∂¯ (X, E ∗ ).
Using the isomorphism (6.8) we can express this duality in the form
H p (X, Ωq ⊗ E)∗ ' H n−p (X, Ωn−q ⊗ E ∗ ).
This is the Kodaira-Serre duality. In particular for q = 0 we get (denoting K = Ωn =
det T ∗ X, the canonical bundle of X)
H p (X, E)∗ ' H n−p (X, K ⊗ E ∗ ).
This is usually called Serre duality.
8. Connections
In this section we give the basic definitions and sketch the main properties of con-
nections. The concept of connection provides the correct notion of differential operator
to differentiate the sections of a vector bundle.
by letting
∇(ω ⊗ s) = dω ⊗ s + (−1)k ω ⊗ ∇(s).
Exercise 6.1. Prove that if gαβ denotes the transition functions of E with respect
to the chosen local basis sections (i.e., sα = gαβ sβ ), the transformation formula for the
connection 1-forms is
−1 −1
(6.9) ωα = gαβ ωβ gαβ + dgαβ gαβ .
The connection is not a tensorial morphism, but rather satifies a Leibniz rule; as a
consequence, the transformation properties of the connection 1-forms are inhomogeneous
and contain an affine term.
Exercise 6.2. Prove that if E and F are vector bundles, with connections ∇1 and
∇2 , then the rule
∇(s ⊗ t) = ∇1 (s) ⊗ t + s ⊗ ∇2 (t)
(minimal coupling) defines a connection on the bundle E ⊗ F (here s and t are sections
of E and F , respectively).
Exercise 6.3. Prove that is E is a vector bundle with a connection ∇, the rule
defines a connection on the dual bundle E ∗ (here τ , s are sections of E ∗ and E, respec-
tively, and < , > denotes the pairing between sections of E ∗ and E).
It is an easy exercise, which we leave to the reader, to check that the square of the
connection
∇2 : ΩkX ⊗ E → Ωk+2
X ⊗E
is f -linear, i.e., it satisfies the property
∇2 (f s) = f ∇2 (s)
Θ(sα ) = Θα ⊗ sα .
90 6. COMPLEX MANIFOLDS AND VECTOR BUNDLES
Exercise 6.4. Prove that the curvature 2-forms may be expressed in terms of the
connection 1-forms by the equation (Cartan’s structure equation)
(6.10) Θα = dωα − ωα ∧ ωα .
Exercise 6.5. Prove that the transformation formula for the curvature 2-forms is
−1
Θα = gαβ Θβ gαβ .
Due to the tensorial nature of the curvature morphism, the curvature 2-forms obey a
homogeneous transformation rule, without affine term.
Since we are able to induce connections on tensor products of vector bundles (and
also on direct sums, in the obvious way), and on the dual of a bundle, we can induce
connections on a variety of bundles associated to given vector bundles with connec-
tions, and thus differentiate their sections. The result of such a differentiation is called
the covariant differential of the section. In particular, given a vector bundle E with
connection ∇, we may differentiate its curvature as a section of Ω2X ⊗ End(E).
Obviously we have
Proposition 6.8. Given a hermitian bundle (E, h), there is a unique connection ∇
on E which is metric with respect to h and is compatible with the holomorphic structure
of E.
Proof. If we use holomophic local bases of sections, the connection forms are of
type (1,0). Then equation (6.12) yields
and this equations shows the uniqueness. As for the existence, one can easily check
that the connection forms as defined by equation (6.13) satisfy the condition (6.9) and
therefore define a connection on E. This is metric w.r.t. h and compatible with the
holomorphic structure of E by construction.
92 6. COMPLEX MANIFOLDS AND VECTOR BUNDLES
Example 6.9. (Chern classes and Maxwell theory) The Chern classes of a complex
vector bundle E can be calculated in terms of a connection on E via the so-called Chern-
Weil representation theorem. Let us discuss a simple situation. Let L be a complex line
bundle on a smooth 2-dimensional manifold X, endowed with a connection, and let F
be the curvature of the connection. F can be regarded as a 2-form on X. In this case
the Chern-Weil theorem states that
Z
i
(6.14) c1 (L) = F
2π X
where we regard c1 (L) as an integer number via the isomorphism H 2 (X, Z) ' Z given
by integration over X. Notice that the Chern class of F is independent of the connection
we have chosen, as it must be. Alternatively, we notice that the complex-valued form F
is closed (Bianchi identity) and therefore singles out a class [F ] in the complexified de
Rham group HDR 2 (X) ⊗ C ' H 2 (X, C); the class i [F ] is actually real, and one has
R 2π
the equality
i
c1 (L) = [F ]
2π
in HDR2 (X). If we consider a static spherically symmetric magnetic field in R3 , by
solving the Maxwell equations we find a solution which is singular at the origin. If we
do not consider the dependence from the radius the vector potential defines a connection
on a bundle L defined on an S 2 which is spanned by the angular spherical coordinates.
The fact that the Chern class of L as given by (6.14) can take only integer values is
known in physics as the quantization of the Dirac monopole.
CHAPTER 7
Divisors
Divisors are a powerful tool to study complex manifolds. We shall start with the one-
dimensional case. The notion will be later generalized to higher dimensional manifolds.
0 → O∗ → M∗ → M∗ /O∗ → 0
obviously satisfy the cocycle condition, and define a line bundle, which we denote by [D].
The line bundle [D] in independent, up to isomorphism, of the set of functions defining
D; if {fα0 } is another set, then ordpi fα = ordpi fα0 , so that the functions hα = fα /fα0 are
holomorphic and nowhere vanishing, and
0 fα0 fα hβ hβ
gαβ = 0 = = gαβ ,
fβ fβ hα hα
0
so that the transition functions gαβ define an isomorphic line bundle.
(1) (2)
If D = D(1) + D(2) then fα = fα fα by eq. (7.1), so that [D(1) + D(2) ] = [D(1) ] ⊗
[D(2) ], and one has a homomorphism Div(S) → Pic(S).
We offer now a sheaf-theoretic description of this homomorphism. Let f = {fα } ∈
H 0 (S, M∗ ); let us set fα = gα /hα , with gα ,hα ∈ O(Uα ) relatively prime. We have
(f ) = (g) − (h), with (g) and (h) effective divisors. The line bundle [(f )] has transition
functions
gα hβ fα
gαβ = = =1
gβ hα fβ
(since f is a Čech cocycle) so that [(f )] = C, i.e. [(f )] is the trivial line bundle.
Conversely, let D be a divisor such that [D] = C; then the transition functions of
[D] have the form
hα
gαβ = with hα ∈ O∗ (Uα ).
hβ
fα
Let {fα } be meromorphic functions which define D, so that one also has gαβ = fβ , and
fα fβ fβ
= gαβ = ;
hα hα hβ
fα
the quotients hα therefore determine a global nonzero meromorphic function, namely:
Proposition 7.2. The line bundle associated with a divisor D is trivial if and only
if D is the divisor of a global meromorphic function.
In view of the identifications Div(S) ' H 0 (S, M∗ /O∗ ) and Pic(S) ' H 1 (S, O∗ ) this
statement is equivalent to the exactness of the sequence
Quite evidently, D and D0 are linearly equivalent if and only if [D] ' [D0 ], so that
there is an injective group homomorphism
Proposition 7.4. A line bundle L is associated with a divisor D (i.e. L = [D] for
some D ∈ Div(S)) if and only if it has a global nontrivial meromorphic section. L is the
line bundle associated with an effective divisor if and only if it has a global nontrivial
holomorphic section.
Proof. The “if” part has already been proven. For the “only if” part, let L = [D]
with D a divisor with local equations fα = 0. Then fα = gαβ fβ , where the functions
gαβ are transition functions for L; the functions fα glue to yield a global meromorphic
section s of L. If D is effective the functions fα are holomorphic so that s is holomorphic
as well.
Corollary 7.5. The line bundle [p] trivializes over the cover {U1 , U2 }, where U1 =
S − {p} and U2 is a neighbourhood of p, biholomorphic to a disc in C.
Proof. Since [p] is effective it has a global holomorphic section which vanishes only
at p, so that [p] is trivial on U1 . Of course it is trivial on U2 as well.
For the remainder of this section we assume that S is compact. Let us define the
P
degree of a divisor D = ai pi as the integer
X
deg D = ai .
For simplicity we shall write O(D) for O([D]).
Before proving this result we need some preliminaries. We define a hermitian metric
on a line bundle L as an assignment of a hermitian scalar product in each Lp which is
C ∞ in p; thus a hermitian metric is a C ∞ section h of the line bundle L∗ ⊗ L∗ such that
each h(p) is a hermitian scalar product in Lp . In terms of a local trivialization over an
open cover {Uα } a hermitian metric is represented by nonvanishing real functions hα on
i ¯
Uα . On Uα ∩ Uβ one has hα = |gαβ |2 hβ , so that the 2-form 2π ∂∂ log hα does not depend
on α, and defines a global closed 2-form on S, which we denote by Θ.
2 (S) of c (L).
Lemma 7.8. The class of Θ is the image in HDR 1
Proof. We need the explicit form of the de Rham correspondence. One has exact
sequences
(7.2) 0 → R → C ∞ → Z 1 → 0, 0 → Z 1 → Ω1 → Z 2 → 0 .
(Here Ω1 is the sheaf of smooth real-valued 1-forms.) From the long exact cohomology
sequences of the second sequence we get
H 0 (S, Ω1 ) → H 0 (S, Z 2 ) → H 1 (S, Z 1 ) → 0
so that the connecting morphism H 0 (S, Z 2 ) → H 1 (S, Z 1 ) induces an isomorphism
2 (S) → H 1 (S, Z 1 ). Since we may write Θ = i d∂ log h a cocycle representing
HDR 2π α
the image of [Θ] in H 1 (S, Z 1 ) is {θα − θβ }, with
i
θα = 2π ∂ log hα .
Notice that
i i
θα − θβ = 2π ∂ (log hα − log hβ ) = 2π d log gαβ
so that d(θα − θβ ) = 0.
1The reader should check that the integral does not depend on the choice of the representative.
98 7. DIVISORS
If we consider now the first of the sequences (7.2) we obtain from its long cohomology
exact sequence a segment
0 → H 1 (S, Z 1 ) → H 2 (S, R) → 0
Proof of Proposition 7.7: Since c1 and deg are both group homomorphisms, it is
enough to consider the case D = [p]. Consider the open cover {U1 , U2 }, where U1 =
S − {p}, and U2 is a small patch around p. Then
Z Z Z
i
c1 (D) = Θ = 2π lim d∂ log h1
S S →0 S−B()
where B() is the disc |z| < , with z a local coordinate around p, and z(p) = 0. Since
¯ = 1 d(∂ − ∂),
∂∂ ¯ and assuming that h1|U −B() = |z|2 , which can always be arranged, we
2 2
have Z Z Z
1 1 dz
c1 (D) = 2πi lim ∂ log z z̄ = 2πi =1
S →0 ∂B() ∂B() z
having used Stokes’ theorem and the residue theorem (note a change of sign due to a
reversal of the orientation of ∂B()).
Proof. If there is a nonzero s ∈ H 0 (S, O(L)), then L = [D] with D = (s). Since
deg D < 0 by the previous Proposition, this contradicts Corollary 7.6.
1. DIVISORS ON RIEMANN SURFACES 99
D.
Proposition 7.11. The sequence
(7.3) 0 → O → O(D) → kD → 0
is exact.
The sheaf O(−D) can be regarded as the sheaf of holomorphic functions which at
pi have a zero of order at least ai . Since O(D) ' O(−D)∗ , the O(D) may be identified
with the sheaf of meromorphic functions which at pi have a pole of order at most ai .
In particular one may write
0 → O(−2p) → O → kp ⊕ Tp∗ S → 0
where Tp∗ S is considered as a skyscraper sheaf concentrated at p (indeed the quantity
f 0 (z0 ) determines an element in Tp∗ S).
If E is a holomorphic vector bundle on S, let us denote E(D) = E ⊗ [D]. Then by
tensoring the exact sequence (7.4) by O(E) we get
2Here we use the fact that tensoring all elements of an exact sequence by the sheaf of sections of a
vector bundle preserves exactness. This is quite obvious because by the local triviality of E the stalk of
O(E) at p is Opk , with k the rank of E.
100 7. DIVISORS
0 → O(E(−D)) → O(E) → ED → 0
where ED = ⊕i Ep⊕a
i
i is a skyscraper sheaf concentrated on D.
Proof. This follows from the fact that the stalk Op is a unique factorization domain
([10] page 12).3 Let us sketch the proof for hypersurfaces. In a neighbourhood of p the
hypersurface V is given by f = 0. Denoting by the same letter the germ of f in p,
since Op (where O is the sheaf of holomorphic functions on X) is a unique factorization
domain we have
f = f1 · · · · · fm ,
where the fi ’s are irreducible in Op , and are defined up to multiplication by invertible
elements in Op ; if Vi is the locus of zeroes of fi , then V = ∪i Vi . Since fi irreducible, Vi
is irreducible as well; since it is not true that fj = gfi for some g ∈ Op which vanishes
at p, we also have Vi 6⊂ Vj .
3Let us recall this notion: one says that a ring R is an integral domain if uv = 0 implies that either
f (w, z 2 , . . . , z n ) = wa h(w, z 2 , . . . , z n )
with h(p) 6= 0. The function f has the same representation in all nearby points of V ,
so that a is constant on the connected components of V , namely, it is constant on V ,
so that we may define
ordV f = a.
With this proviso all the theory previously developed applies to this situation; the
only definition which no longer applies is that of degree of a line bundle, in that c1 (L)
is still represented by a 2-form, while the quantities that can be integrated on X are
the 2n-forms if dimC X = n. Proposition 7.7 must now be reformulated as follows. Let
ai Vi be a divisor, and let Vi∗ be the smooth locus of Vi . We then have:
P
D=
Proposition 7.4. For any divisor D ∈ Div(X) and any (2n − 2)-form φ on X,
Z X Z
c1 (D) ∧ φ = ai φ.
X i Vi∗
Proof. The proof is basically the same as in Proposition 7.7 (cf. [10] page 141).
3. Linear systems
So a linear system is of the form E = {Dλ }λ∈Pm for some m. The number m is
called the dimension of E. A one-dimensional linear system is called a pencil, a two-
dimensional one a net, and a three-dimensional one a web. Since all divisors in a linear
system have the same degree, one can associate a degree to a linear system.
Definition 7.3. If E = {Dλ }λ∈Pm is a linear system, we define its base locus as
B(E) = ∩λ∈Pm Dλ .
(7.6) f (z 1 , . . . , z n ) + λg(z 1 , . . . , z n ) = 0
where f and f are local defining functions for D0 and D∞ (holomorphic because the
divisors in E are effective). f and g do not vanish simultaneously on X − B(E),
so that they do not vanish separately either. Then the above map is given by λ =
−f (z 1 , . . . , z n )/g(z 1 , . . . , z n ).
By this we mean that the set of divisors in a linear system E which have singular
points outside the base locus form a subvariety of E of dimension strictly smaller than
that of E.
∂f ∂f
(7.8) (pλ ) + λ i (pλ ) = 0, i = 1, . . . , n
∂z i ∂z
f (pλ ), g(pλ ) 6= 0.
We then have λ = −f (pλ )/g(pλ ), so that
∂f f ∂g
− =0 in pλ ,
∂z i g ∂z i
and
∂ f
(7.9) =0 in pλ .
∂z i g
Let Y be the locus in ∆ × P1 cut out by the conditions (7.7) and (7.8); Y is an analytic
variety, so the same holds true for its image V in ∆. Actually V is nothing but the locus
of all singular points of the divisors Dλ . Equation (7.9) shows that f /g is constant on
the connected components of V −B, that is, every connected component of V −B meets
only one divisor of the pencil. Since the connected components of V − B are finitely
many by Proposition 7.2, the divisors which have singularities outside B(E) are finite
in number.
The dual NV∗ , the conormal bundle to V , is the subbundle of ι∗V T ∗ X whose sections
are holomorphic 1-forms which are zero on vectors tangent to V .
We first prove the isomorphism
(7.10) NV∗ ' ι∗V [−V ].
We consider the exact sequence of vector bundles on V
0 → NV∗ → ι∗V T ∗ X → T ∗ V → 0
whence we get4
(7.11) ι∗V KX ' KV ⊗ NV∗ .
If, relative to an open cover {Uα } of X, the divisor V is locally given by functions
fα ∈ O(Uα ), the line bundle [V ] has transition functions gαβ = fα /fβ . The 1-form
dfα |V ∩Uα is a section of NV∗ |V ∩Uα , which never vanishes because V is smooth. On
Uα ∩ Uβ we have
dfα = d(gαβ fβ ) = dgαβ fβ + gαβ dfβ = gαβ dfβ
the last equality holding on V ∩ Uα ∩ Uβ . This equation shows that the 1-forms dfα
do not glue to a global section of NV∗ , but rather to a global section of the line bundle
NV∗ ⊗ ι∗V [V ], so that this bundle is trivial, and the isomorphism (7.10) holds.
By combining the formula (7.10) with the isomorphism (7.11) we obtain the adjunc-
tion formula:
(7.12) KV ' ι∗V (KX ⊗ [V ]).
Sometimes an additive notation is used, and then the adjunction formula reads
KV = KX |V + [V ]|V .
Example 7.1. Let V be the divisor cut out from P3 by the quartic equation
(7.13) w04 + w14 + w24 + w34 = 0
where the wi ’s are homogeneous coordinates in P3 . It is easily shown the V is smooth,
and it is of course compact: so it is a 2-dimensional compact complex manifold, called
the Fermat surface. By a nontrivial result, known as Lefschetz hyperplane theorem
([10] p. 156) one has H 1 (V, R) = 0, so that H 1 (V, OV ) = 0. Then the group Pic0 (V ),
which classifies the line bundles whose first Chern classes vanishes, is trivial: if a line
bundle L on V is such that c1 (L) = 0, then it is trivial, and every line bundle is fully
classified by its first Chern class. (The same happens on P3 , since H 1 (P3 , OP3 ) = 0).
4We use the fact that whenever
0→E→F →G→0
is an exact sequence of vector bundles, then det F ' det E ⊗ det G, as one can prove by using transition
functions.
4. THE ADJUNCTION FORMULA 105
We also know that KP3 = OP3 (−4H), where H is any hyperplane in P3 . Therefore
ι∗V KX ' OV (−4HV ), where HV = H ∩ V is a divisor in V .
Let us compute c1 ([V ]|V ) = ι∗V c1 ([V ]). We use the following fact: if D1 , D2 , D3 are
irreducible divisors in P3 , then we can move the divisors inside their linear equivalence
classes in such a way that they intersects at a finite number of points. This number is
computed by the integral
Z
c1 ([D1 ]) ∧ c1 ([D2 ]) ∧ c1 ([D3 ])
P3
where one considers the Chern classes c1 ([Di ]) as de Rham cohomology classes. If we
take D1 = V , D2 = D3 = H the number of intersection points is 4, because such is the
degree of the algebraic system formed by the equation (7.13) and by the equations of
two (different) hyperplanes. Since the class h, where h = c1 ([H]), generates H 2 (P3 , Z),
we have c1 ([V ]) = 4h, that is, V ∼ 4H. Then [V ]|V ' OV (4HV ).
From the adjunction formula we get KV ' C: the canonical bundle of V is trivial.
1 (V ) = 0, V is an example of a K3 surface.
Since we also have HDR
CHAPTER 8
Algebraic curves I
The main purpose of this chapter is to show that compact Riemann surfaces can be
imbedded into projective space (i.e. they are algebraic curves), and to study some of
their basic properties.
Proof. Pick up a line bundle L on S such that deg L > deg K + 2 (choose an
effective divisor D with enough points, and let L = [D]). By Serre duality we have
for any p ∈ S, since deg(K − L + 2p) < 0 (here L − 2p = L ⊗ [−2p]). Consider now the
exact sequence
dp ⊕evp
0 → O(L − 2p) → O(L) −−→ Tp∗ S ⊕ Lp → 0
so that dim |D| ≥ 1. Let N = dim |D|, and let {s0 , . . . , sN } be a basis of |D|. If U is an
open neighbourhood of p, and φ : L|U → U × C is a local trivialization of L, the quantity
(8.2) [φ ◦ s0 , . . . , φ ◦ sN ] ∈ PN
107
108 8. ALGEBRAIC CURVES I
Given any complex manifold X, one says that a line bundle L on X is very ample
if the construction (8.2) defines an imbedding of X into PH 0 (X, O(L)). A line bundle
L is said to be ample if Ln is very ample for some natural n. A sufficient condition for
a line bundle to be ample may be stated as follows (cf. [10]).
Proposition 8.3. If the first Chern class of a line bundle L on a complex manifold
can be represented by a positive 2-form, then L is ample.
While we have seen that any compact Riemann surface carries plenty of very ample
line bundles, this in general is not the case: there are indeed complex manifolds which
cannot be imbedded into any projective space.
A first consequence of the imbedding theorem expressed by Proposition 8.1 is that
any line bundle on a compact Riemann surface comes from a divisor, i.e. Div(S)/linear
equivalence ' Pic(S).
We shall now proceed to identify compact Riemann surfaces with (smooth) algebraic
curves. Given a homogeneous polynomial F on Cn+1 the zero locus of F in Pn is by
definition the projection to Pn of the zero locus of F in Cn+1 .
2This result is actually true whatever is the dimension of M , cf. [10].
110 8. ALGEBRAIC CURVES I
Exercise 8.7. Use Chow’s lemma to show that H 0 (Pn , H d ) — where H is the
hyperplane line bundle — can be identified with the space of homogeneous polynomials
of degree d on Cn+1 .
Using Chow’s lemma together with the imbedding theorem (Proposition 8.1) we
obtain
2. Riemann-Roch theorem
Proposition 8.1. (Riemann-Roch theorem) For any line bundle L on C one has
h0 (L) − h1 (L) = deg L − g + 1.
Proof. If L = C is the trivial line bundle, the result holds obviously (notice that
H 1 (C, O)∗ ' H 0 (C, K) by Serre duality). Exploiting the fact that L = [D] for some
divisor D, it is enough to prove that if the results hold for L = [D], then it also holds
for L0 = [D + p] and L00 = [D − p].
In the first case we start from the exact sequence
0 → O(D) → O(D + p) → kp → 0
3Strictly speaking an algebraic curve consists of more data than a compact Riemann surface S,
since the former requires an imbedding of S into a projective space, i.e. the choice of an ample line
bundle.
4We introduce the following notation: if E is a sheaf of O -modules, then hi (E) = dim H i (C, E).
C
3. GENERAL RESULTS 111
0 → H 0 (S, O(D)) → H 0 (S, O(D + p)) → C → H 1 (S, O(D)) → H 1 (S, O(D + p)) → 0
whence
By using the Riemann-Roch theorem and Serre duality we may compute the degree
of K, obtaining
deg K = 2g − 2.
This is called the Riemann-Hurwitz formula. It allows us to identify g with the topolog-
ical genus gtop of C regarded as a compact oriented 2-dimensional real manifold S. To
this end we may use the Gauss-Bonnet theorem, which states that the integral of the Eu-
ler class of the real tangent bundle to S is the Euler characteristic of S, χ(S) = 2−2gtop .
On the other hand the complex structure of C makes the real tangent bundle into a
complex holomorphic line bundle, isomorphic to the holomorphic tangent bundle T C,
and under this identification the Euler class corresponds to the first Chern class of T C.
Therefore we get deg K = 2gtop − 2, namely,5
g = gtop .
still true (provided we know what a line bundle on a singular curve is!) with g the arithmetic genus.
6Since two holomorphic functions of one variable which agree on a nondiscrete set are identical,
(8.4) w = zr .
The number r −1 is called the ramification index of f at q (or at p = f (q)), and p = f (q)
is said to be a branch point if r(p) > 1. The branch locus of f is the divisor in C 0
X
B0 = (r(q) − 1) · q
q∈C 0
or its image in C
X
B= (r(q) − 1) · f (q).
q∈C 0
For any p ∈ C we have
X
f ∗ (p) = r(q) · q
q∈f −1 (p)
X
deg f ∗ (p) = r(q) = n.
q∈f −1 (p)
From these formulae we may draw the following picture. If p ∈ C 0 does not lie in
the branch locus, then exactly n distinct points of C 0 are mapped to f (p), which means
that f : C 0 − B 0 → C − B is a covering map.7 It p ∈ C 0 is a branch point of ramification
index r − 1, at p exactly r sheets of the covering join together.
There is a relation between the canonical divisors of C 0 and C and the branch locus.
Let η be a meromorphic 1-form on C, which can locally be written as
g(w)
η= dw.
h(w)
From (8.4) we get
g(z r ) r r
r−1 g(z )
f ∗η = dz = rz dz
h(z r ) h(z r )
so that
ordp f ∗ η = ordf (p) η + r − 1.
This implies the relation between divisors
X
(f ∗ η) = f ∗ (η) + (r(p) − 1) · p.
p∈C 0
On the other hand the divisor (η) is just the canonical divisor of C, so that
KC 0 = f ∗ KC + B 0 .
7A (holomorphic) covering map f : X → Y , with X connected, is a map such that each p ∈ Y has a
connected neighbourhood U such that f −1 (U ) = ∪α Uα is the disjoint union of open subsets of X which
are biholomorphic to U via f .
3. GENERAL RESULTS 113
From this formula we may draw an interesting result. By taking degree we get
X
deg KC 0 = n deg KC + (r(p) − 1);
p∈C 0
3.3. The genus formula for plane curves. An algebraic curve C is said to be
plane if it can be imbedded into P2 . Its image in P2 is the zero locus of a homogeneous
polynomial; the degree d of this polynomial is by definition the degree of C. As a
divisor, C is linearly equivalent to dH (indeed, since Pic(P2 ) ' Z, any divisor D on P2
is linearly equivalent to mH for some m; if D is effective, m is the number of intersection
points between D and a generic hyperplane in P2 , and this is given by the degree of the
polynomial cutting D). 8
We want to show that for smooth plane curves the following relation between genus
and degree holds:
(8.6) g(C) = 21 (d − 1)(d − 2).
(For singular plane curves this formula must be modified.) We may prove this equa-
tion by using the adjunction formula: C is imbedded into P2 as a smooth analytic
hypersurface, so that
KC = ι∗ (KP2 + C),
where ι : C → P2 . Recalling that KP2 = −3H we then have KC = (d − 3)ι∗ H.
8We are actually using here a piece of intersection theory. The fact is that any k-dimensional
deg ι∗ H = d,
and
deg KC = d(d − 3) = 2g − 2
y 2 = x6 − 1 .
so that the number a does not depend on the representation of ϕ. We call it the residue
of ϕ at p, and denote it by Resp (ϕ).
P
Given a meromorphic 1-form ϕ its polar divisor is D = i pi , where the pi ’s are the
points where the local representatives of ϕ have poles of order 1.
P
Proposition 8.3. Let D = i pi be the polar divisor of a meromorphic 1-form ϕ.
P
Then i Respi (ϕ) = 0.
3. GENERAL RESULTS 115
3.5. The g = 0 case. We shall now show that all algebraic curves of genus zero
are isomorphic to the Riemann sphere P1 . Pick a point p ∈ C; the line bundle [p] is
trivial on C − {p}, and has a holomorphic section s0 which is nonzero on C − {p} and
has a simple zero at p (this means of course that (s0 ) = p). On the other hand, since by
Serre duality h1 (O) = h0 (K) = 0, by taking the cohomology exact sequence associated
with the sequence
0 → O → O(p) → kp → 0
we obtain the existence of a global section s of [p] which does not vanish at p. Of course
s vanishes at some other point s0 . Then the quotient f = s/s0 is a global meromorphic
function, with a simple pole at p and a zero at p0 .9 By considering ∞ as the value of f
at p, we may think of f as a holomorphic nonconstant map f : C → P1 ; this map takes
the value ∞ only once. Suppose that f takes the same value α at two distinct points
of C; then then function f − α has two zeroes and only one simple pole, which is not
possible. Thus f is injective. The following Lemma implies that f is surjective as well,
so that it is an isomorphism.
Lemma 8.4. Any holomorphic map between compact complex manifolds of the same
dimension whose Jacobian determinant is not everywhere zero is surjective.
9Otherwise one can directly identify the sections of L with meromorphic functions having (only) a
single pole at p, since such functions can be developed around p in the form
a
f (z) = + g(z) ,
z
where g is a holomorphic function. a ∈ C should be indentified with the projection of f onto kp . (Here
z is a local complex coordinate such that z(p) = 0.)
CHAPTER 9
Algebraic curves II
In this chapter we further study the geometry of algebraic curves. Topics covered
include the Jacobian variety of an algebraic curve, some theory of elliptic curves, and
the desingularization of nodal plane singular curves (this will involve the introduction
of the notion of blowup of a complex surface at a point).
A fundamental tool for the study of an algebraic curve C is its Jacobian variety
J(C), which we proceed now to define. Let V be an m-dimensional complex vector
space, and think of it as an abelian group. A lattice Λ in V is a subgroup of V of the
form
( 2m )
X
(9.1) Λ= ni vi , n i ∈ Z
i=1
where {vi }i=1,...,2m is a basis of V as a real vector space. The quotient space T = V /Λ
has a natural structure of complex manifold, and one of abelian group, and the two
structures are compatible, i.e. T is a compact abelian complex Lie group. We shall
call T a complex torus. Notice that by varying the lattice Λ one gets another complex
torus which may not be isomorphic to the previous one (the complex structure may be
different), even though the two tori are obviously diffeomorphic as real manifolds.
Example 9.1. If C is an algebraic curve of genus g, the group Pic0 (C), classifying
the line bundles on C with vanishing first Chern class, has a structure of complex torus
of dimension g, since it can be represented as H 1 (C, O)/H 1 (C, Z), and H 1 (C, Z) is a
lattice in H 1 (C, O). This is the Jacobian variety of C. In what follows we shall construct
this variety in a more explicit way.
the homology class of γ and the cohomology class of ω, and expresses the pairing < , >
between the Poincaré dual spaces H1 (C, C) = H1 (C, Z) ⊗Z C and H 1 (C, C).
Pick up a basis {[γ1 ], . . . , [γ2g ]} of the 2g-dimensional free Z-module H1 (C, Z), where
the γi ’s are smooth loops in C, and a basis {ω1 , . . . , ωg } of H 0 (C, K). We associate with
these data the g × 2g matrix Ω whose entries are the numbers
Z
Ωij = ωi .
γj
This is called the period matrix. Its columns Ωj are linearly independent over R: if for
all i = 1, . . . g
2g
X 2g
X Z
0= λj Ωij = λj ωi
j=1 j=1 γj
P2g R 1
then also j=1 λj γj ω̄i = 0. Since {ωi , ω̄i } is a basis for H (C, C), this implies
P2g
j=1 λj [γj ] = 0, that is, λj = 0. So the columns of the period matrix generate a
lattice Λ in Cg . The quotient complex torus J(C) = Cg /Λ is the Jacobian variety of C.
Define now the intersection matrix Q by letting Q−1 ij = [γj ] ∩ [γi ] (this is the Z-
valued “cap” or “intersection” product in homology). Notice that Q is antisymmetric.
Intrinsically, Q is an element in HomZ (H 1 (C, Z), H1 (C, Z)). Since the cup product in
cohomology is Poincaré dual to the cap product in homology, for any abelian differentials
ω, τ we have
[ω] ∪ [τ ] =< Q[ω], [τ ] > .
The relations (9.2), (9.3) can then be written in the form
Ω Q Ω̃ = 0, i Ω Q Ω† > 0
(here ˜ denotes transposition, and † hermitian conjugation). In this form they are called
Riemann bilinear relations.
A way to check that the construction of the Jacobi variety does not depend on the
choices we have made is to restate it invariantly. Integration over cycles defines a map
Z
0 ∗
i : H1 (C, Z) → H (C, K) , i([γ])(ω) = ω.
γ
This map is injective: if i([γ])(ω) = 0 for a given γ and all ω then γ is homologous to
the constant loop. Then we have the representation J(C) = H 0 (C, K)∗ /H1 (C, Z).
Exercise 9.2. By regarding J(C) as H 0 (C, K)∗ /H1 (C, Z), show that Serre and
Poincaré dualities establish an isomorphism J(C) ' Pic0 (C).
1. THE JACOBIAN VARIETY 119
1.1. The Abel map. After fixing a point p0 in C and a basis {ω1 , . . . , ωg } in
H 0 (C, K)
we define a map
(9.4) µ : C → J(C)
by letting Z p Z p
µ(p) = ω1 , . . . , ωg .
p0 p0
Actually the value of µ(p) in Cg
will depend on the choice of the path from p0 to p;
however, if δ1 and δ2 are two paths, the oriented sum δ1 − δ2 will define a cycle in
homology, the two values will differ by an element in the lattice, and µ(p) is a well-
defined point in J(C).
From (9.4) we may get a group homomorphism
by letting
X X X X
µ(D) = µ(pi ) − µ(qj ) if D= pi − qj .
i j i j
All of this depends on the choice of the base point p0 , note however that if deg D = 0
then the choice of p0 is immaterial.
Proposition 9.3. (Abel’s theorem) Two divisors D, D0 ∈ Div(C) are linearly equiv-
alent if and only if µ(D) = µ(D0 ).
Abel’s theorem may be stated in a fancier language as follows. Let Divd (C) be
the subset of Div(C) formed by the divisors of degree d, and let Picd (C) be the set of
line bundles of degree d.1 One has a surjective map ` : = Divd (C) → Picd (C) whose
kernel is isomorphic to H 0 (C, M∗ )/H 0 (C, O∗ ). Then µ filters through a morphism
ν : Picd (C) → J(C), and one has a commutative diagram
1Notice that Picd (C) ' Picd0 (C) as sets for all values of d and d0 .
120 9. ALGEBRAIC CURVES II
moreover, the morphism ν is injective (if ν(L) = 0, set L = `(D) (i.e. L = [D]); then
µ(L) = 0, that is, L is trivial).
We can actually say more about the morphism ν, namely, that it is a bijection. It
is enough to prove that ν is surjective for a fixed value of d (cf. previous footnote).
Let C d be the d-fold cartesian product of C with itself. The symmetric group Sd of
order d acts on C d ; we call the quotient Symd (C) = C d /Sd the d-fold symmetric product
of C. Symd (C) can be identified with the set of effective divisors of C of degree d. The
map µ defines a map µd : Symd (C) → J(C).
Any local coordinate z on C yields a local coordinate system {z 1 , . . . , z d } on C d ,
z i (p1 , . . . , pd ) = z(pi ),
and the elementary symmetric functions of the coordinates z i yield a local coordinate
system for Symd (C). Therefore the latter is a d-dimensional complex manifold. More-
over, the holomorphic map
is Sd -invariant, hence it descends to a map Symd (C) → J(C), which coincides with µd .
So the latter is holomorphic.
Exercise 9.5. Prove that Symd (P1 ) ' Pd . (Hint: write explicitly a morphism in
homogeneous coordinates.)
The surjectivity of ν follows from the following fact, usually called Jacobi inversion
theorem.
ωg (p1 ) . . . ωg (pg )
We may choose p1 so that ω1 (p1 ) 6= 0, and then subtracting a suitable multiple of ω1
from ω2 , . . . , ωg we may arrange that ω2 (p1 ) = · · · = ωg (p1 ) = 0. We next choose p2 so
that ω2 (p2 ) 6= 0, and arrange that ω3 (p2 ) = · · · = ωg (p3 ) = 0, and so on. In this way the
matrix (9.7) is upper triangular. With these choices of the abelian differentials ωi and of
1. THE JACOBIAN VARIETY 121
the points pi the Jacobian matrix {hji } is upper triangular as well, and since ωi (pi ) 6= 0,
its diagonal elements hii are nonzero at D, so that at the point D corresponding to our
choices the Jacobian determinant is nonzero. This means that the determinant is not
everywhere zero, and by Lemma 8.4 one concludes.
Proof. Let u ∈ J(C), and choose a divisor D ∈ µ−1 g (u). By Abel’s theorem the
−1
fibre µg (u) is formed by all effective divisors linearly equivalent to D, hence it is a
projective space. But since dim J(C) = dim Symd (C) the fibre of µg is generically
0-dimensional, so that generically it is a point.
Proof. Let D ∈ Divd (C) with d ≥ g. We may write D = D0 + D00 with deg D0 = g
and D00 ≥ 0. By mapping D0 to J(C) by Abel’s map and taking a counterimage in
Symg (C) we obtain an effective divisor E linearly equivalent to D0 . Then E + D00 is
effective and linearly equivalent to D.
Corollary 9.9. Every elliptic smooth algebraic curve (i.e. every smooth algebraic
curve of genus 1) is of the form C/Λ for some lattice Λ ⊂ C.
1.2. Jacobian varieties are algebraic. According to our previous discussion, any
1-dimensional complex torus is algebraic. This is no longer true for higher dimensional
tori. However, the Jacobian variety of an algebraic curve is always algebraic.
Let Λ be a lattice in Cn . Any point in the lattice singles out univoquely a cell in the
lattice, and two opposite sides of the cell determine after identification a closed smooth
loop in the quotient torus T = Cn /Λ. This provides an identification Λ ' H1 (T, Z).
Let now ξ be a skew-symmetric Z-bilinear form on H1 (T, Z). Since HomZ (Λ2 H1 (T,
Z), Z) ' H 2 (T, Z) canonically (check this isomorphism as an exercise), ξ may be re-
garded as a smooth complex-valued differential 2-form on T .
Proposition 9.11. The 2-form ξ which on the basis {ej } is represented by the
intersection matrix Q−1 is a positive (1,1) form.
Proof. If {ej , j = 1 . . . 2n} are the real basis vectors in Cn generating the lattice,
they can be regarded as basis in H1 (T, Z). They also generate 2n real vector fields on
T (after identifying Cn with its tangent space at 0 the ej yield tangent vectors to T at
the point corresponding to 0; by transporting them in all points of T by left transport
one gets 2n vector fields, which we still denote by ej ). Let {z 1 , . . . , z n } be the natural
local complex coordinates in T ; the period matrix may be described as
Z
Ωij = dz i .
ej
After writing ξ on the basis {dz i , dz̄ j } one can check that the stated properties of ξ are
equivalent to the Riemann bilinear relations.2
There exists on J(C) a (in principle smooth) line bundle L whose first Chern class
is the cohomology class of ξ. This line bundle has a connection whose curvature is
(cohomologous to) 2πi ξ; since this form is of type (1,1), L may be given a holomorphic
structure. With this structure, it is ample by Proposition 8.3.3 This defines a projective
imbedding of J(C), so that the latter is algebraic.
2. Elliptic curves
(9.8) y 2 = P (x),
2So we are not only proving that the Jacobian variety of an algebraic curve is algebraic, but, more
generally, that any complex torus satisfying the Riemann bilinear relations is algebraic.
3We are using the fact that if a smooth complex vector bundle E on a complex manifold X has a
connection whose curvature has no (0,2) part, then the complex structure of X can be “lifted” to E.
Cf. [19]. Otherwise, we may use the fact that the image of the map c1 in H 2 (J(C), Z) (the Néron-Severi
group of J(C), cf. subsection 6.5.1) may be represented as H 2 (J(C), Z) ∩ H 1,1 (J(C), Z), i.e., as the
group of integral 2-classes that are of Hodge type (1,1). The class of ξ is clearly of this type.
2. ELLIPTIC CURVES 123
so that, setting δ = α − β,
1
f˜2 + β f˜ − f 3 + δf = O( ) .
z
So the meromorphic function in the left-hand side is holomorphic away from p, and has
at p a simple pole. Such a function must be constant, otherwise it would provide an
isomorphism of C with the Riemann sphere.
Thus C may be described as a locus in P2 whose equation in affine coordinates is
(9.9) y 2 + βy = x3 − δx +
Exercise 9.2. Determine for what values of the parameter λ the curve (9.10) is
smooth.
(P 0 )2 − 4P 3 + 20 a = constant0
one usually writes g2 for 20 a and g3 for the constant in the right-hand side.
In terms of this representation we may introduce a map j : M1 → C, where M1 is
the set of isomorphism classes of smooth elliptic curves (the moduli space of genus one
4Even though the Weierstraß representation only provides the equation of the affine part of an
elliptic curve, the latter is nevertheless completely characterized. It is indeed true that any affine plane
curve can be uniquely extended to a compact curve by adding points at infinity, as one can check by
elementary considerations.
2. ELLIPTIC CURVES 125
5
curves)
1728 g23
j(C) = .
g23 − 27 g32
One shows that this map is bijective; in particular M1 gets a structure of complex
manifold. The number j(C) is called the j-invariant of the curve C. We may therefore
say that the moduli space M1 is isomorphic to C. 6
F = y 2 − 4x3 + g2 x + g3
(we use the same affine coordinates as in the previous representation). Since f˜ = df /ω
we have
dx
ω=
y
and the inverse of ψ is the Abel map,7
Z p
dx
ψ −1 (p) = mod Λ
p0 y
having chosen p0 at the point at infinity, p0 = ψ(0) = [0, 0, 1].
In terms of this construction we may give an elementary geometric visualization of
the group law in an elliptic curve. Let us choose p0 as the identity element in C. We
shall denote by p̄ the element p ∈ C regarded as a group element (so p̄0 = 0). By Abel’s
theorem, Proposition 9.3, we have that
(indeed one may think that p̄ = µ(p), and one has µ(p1 + p2 + p3 − 3 p0 ) = 0).
Let M (x, y) = mx + ny + q be the equation of the line in P2 through the points
p1 , p2 , and let p4 be the further intersection of this line with C ⊂ P2 . The function
M (z) = M (P(z), P 0 (z)) on C vanishes (of order one) only at the points p1 , p2 , p4 , and
has a pole at p0 . This pole must be of order three, so that the divisor of M (z) is
p1 + p2 + p4 − 3 p0 , i.e,̇ p1 + p2 + p4 − 3 p0 ∼ 0.
5The fancy coefficient 1728 comes from arithmetic geometry, where the theory is tailored to work
the upper half complex plane. This is not contradictory in that the quotient H/Sl(2, Z) is biholomorphic
to C! (Notice that on the contrary, H and C are not biholomorphic). Cf. [11].
7One should bear in mind that we have identified C with a quotient C/Λ.
126 9. ALGEBRAIC CURVES II
(where α = e2πi/3 ) and at the point at infinity p0 . The points p1 , p2 , p3 are collinear, so
that p̄1 + p̄2 + p̄3 = 0.
The two points q1 = (0, i), q2 = (0, −i) lie on C. The line through q1 , q2 intersects
C at the point at infinity, as one may check in homogeneous coordinates. So in this case
the elements q̄1 , q̄2 are one the inverse of the other, and q1 + q2 ∼ 2 p0 . More generally,
if q ∈ C is such that q̄ = −p̄, then p + q ∼ 2 p0 , and q is the further intersection of C
with the line going through p, p0 ; if p = (a, b), then q = (a, −b). So the branch points
pi are 2-torsion elements in the group, 2 p̄i = 0.
3. Nodal curves
In this section we show how (plane) curve singularities may be resolved by a proce-
dure called blowup.
3.1. Blowup. Blowing up a point in a variety8 means replacing the point with all
possible directions along which one can approach it while moving in the variety. We
shall at first consider the blowup of C2 at the origin; since this space is 2-dimensional,
the set of all possible directions is a copy of P1 . Let x, y be the standard coordinates in
C2 , and w0 , w1 homogeneous coordinates in P1 . The blowup of C2 at the origin is the
subvariety Γ of C2 × P1 defined by the equation
x w1 − y w0 = 0 .
8Our treatment of the blowup of an algebraic variety is basically taken from [1].
3. NODAL CURVES 127
(9.11) Γ
π / P1
σ
C2
which are holomorphic. If p ∈ C2 − {0} then σ −1 (p) is a point (which means that there
is a unique line through p and 0), so that
σ : Γ − σ −1 (0) → C2 − {0}
is a biholomorphism.9 On the contrary σ −1 (0) ' P1 is the set of lines through the origin
in C2 .
The fibre of π over a point (w0 , w1 ) ∈ P1 is the line x w1 − y w0 = 0, so that π makes
Γ into the total space of a line bundle over P1 . This bundle trivializes over the cover
{U0 , U1 }, and the transition function g : U0 ∩ U1 → C∗ is g(w0 , w1 ) = w0 /w1 , so that
the line bundle is actually the tautological bundle OP1 (−1).
This construction is local in nature and therefore can be applied to any complex
surface X (two-dimensional complex manifold) at any point p. Let U be a chart around
p, with complex coordinates (x, y). By repeating the same construction we get a complex
manifold U 0 with projections
π
U 0 −−−−→ P1
σy
U
and
σ : U 0 − σ −1 (p) → U − {p}
is a biholomorphism, so that one can replace U by U 0 inside X, and get a complex
manifold X 0 with a projection σ : X 0 → X which is a biholomorphism outside σ −1 (p).
The manifold X 0 is the blowup of X at p. The inverse image E = σ −1 (p) is a divisor
in X 0 , called the exceptional divisor, and is isomorphic to P1 . The construction of the
blowup Γ shows that X 0 is algebraic if X is.
morphism.
128 9. ALGEBRAIC CURVES II
Definition 9.2. The curve σ −1 (C) ⊂ Γ is the total transform of C. The curve
obtained by taking the topological closure of σ −1 (C) \ E in Γ is the strict transform of
C.
We want to check what points are added to σ −1 (C) \ E when taking the topological
closure. To this end we must understand what are the sequences in C2 which converge to
0 that are lifted by σ to convergent sequences. Let {pk = (xk , yk )}k∈N be a sequence of
points in C2 converging to 0; then σ −1 (xk , yk ) is the point (xk , yk , w0 , w1 ) with xk w1 −
yk w0 = 0. Assume that for k big enough one has w0 6= 0 (otherwise we would assume
w1 6= 0 and would make a similar argument). Then w1 /w0 = yk /xk , and {σ −1 (pk )}
converges if and only if {yk /xk } has a limit, say h; in that case {σ −1 (pk )} converges to
the point (0, 0, 1, h) of E. This means that the lines rk joining 0 to pk approach the
limit line r having equation y = hk. So a sequence {pk = (xk , yk )} convergent to 0 lifts
to a convergent sequence in Γ if and only if the lines rk admit a limit line r; in that
case, the lifted sequence converges to the point of E representing the line r.
The strict transform C 0 of C meets the exceptional divisor in as many points as
are the directions along which one can approach 0 on C, namely, as are the tangents
at C at 0. So, if C is smooth at 0, its strict transform meets E at one point. Every
intersection point must be counted with its multiplicity: if at the point 0 the curve C
has m coinciding tangents, then the strict transform meets the exceptional divisor at a
point of multiplicity m.
Definition 9.3. Let the (affine plane) curve C be given by the equation f (x, y) = 0.
We say that C has multiplicity m at 0 if the Taylor expansion of f at 0 starts at degree
m.
This means that the curve has m tangents at the point 0 (but some of them might
coincide). By choosing suitable coordinates one can apply this notion to any point of a
plane curve.
If the curve C has multiplicity m at 0 than it has m tangents at 0, and its strict
transform meets the exceptional divisor of Γ at m points (notice however that these
points are all distinct only if the m tangents are distincts).
Exercise 9.6. With reference to equation (9.10), determine for what values of λ
the curve has a nodal singularity.
3. NODAL CURVES 129
{(u, v, w0 , w1 ) ∈ C2 × P1 | u w0 = v w1 } .
Exercise 9.9. Repeat the previous calculations for the nodal curve xy = 0. In
particular show that the total transform is a reducible curve, consisting of the excep-
tional divisor and two more genus zero components, each of which meets the exceptional
divisor at a point.
Example 9.10. (The cusp) Let C be curve with equation y 2 = x3 . This curve
has multiplicity 2 at the origin where it has a double tangent.10 Proceeding as in the
previous example we get the equation v t3 = 1 for C 0 in Γ ∩ V0 , so that C 0 does not meet
E in this chart. In the other chart the equation of C 0 is t2 = u, so that C 0 meets E at
the point (0, 0, 0, 1); we have one intersection point because the two tangents to C at
the origin coincide.
The strict transform is an irreducible curve, and the total transform is a reducible
curve with two components meeting at a (double) point.
10Indeed this curve can be regarded as the limit for α → 0 of the family of nodal curves x3 + α2 x2 −
2
y = 0, which at the origin are tangent to the two lines y = ±α x.
130 9. ALGEBRAIC CURVES II
3.3. Normalization of a nodal plane curve. It is clear from the previous ex-
amples that the strict transform of a plane nodal curve C (i.e., a plane curve with only
nodal singularities) is again a nodal curve, with one less singular point. Therefore after
a finite number of blowups we obtain a smooth curve N , together with a birational
morphism π : N → C. N is called the normalization of C.
A genus formula. We give here, without proof, a formula which can be used to
compute the genus of the normalization N of a nodal curve C. Assume that N has t
irreducible components N1 , . . . , Nt , and that C has δ singular points. Then:
t
X
g(C) = g(Ni ) + 1 − t + δ.
1
For instance, by applying this formula to Example 9.8, we obtain that the normalization
is a projective line.
Bibliography
131
132 BIBLIOGRAPHY