Exercises
Exercise 1. Let f : C → C be a holomorphic function. Assume that there exists
a constant M, such that Re(f (z)) ≤ M, for all z ∈ C. Show that f is constant.
Exercise 2.
Prove Liouville’s theorem: a bounded holomorphic function defined
on the complex plane, f : C → C, is constant.
Exercise 3.
Prove the Fundamental Theorem of Algebra: any polynomial with
complex coefficients has at least one root in C.
Exercise 4.
Let f : D → C be a holomorphic function defined on an open set
of the complex plane. Write f = u + iv (u and v are the real and imaginary parts
of f , respectively). Using the Cauchy-Riemann equations show that u and v are
harmonic functions.
Exercise 5. Prove by direct computation that if X is a simply connected surface
then H1 (X, Z) is trivial.
Exercise 6. Show that any connected, locally path connected topological space is
path connected.
Exercise 7. Let p : X → Y a covering. Show that X is a manifold if and only if Y
is a manifold.
Exercise 8. Let G be a group and H = [G, G] its commutator subgroup; that is,
H is generated by all elements of the form ghg −1h−1 , for g and h in G. Prove that
H is a normal subgroup and G/H is abelian.
Exercise 9. Show that the punctured plane C∗ is homeomorphic to a cylinder.
Exercise 10. Prove the following version of Van Kampen Theorem:
let M be a topological space, U and V open subsets of M satisfying M = U ∪ V
and such that the intersection U ∩ V is path connected and non-empty. If U and V
are simply connected then M is simply connected.
b is simply connected.
Use this result to show that C
Exercise 11. Consider the following subspace of R2 : X = A ∪ B ∪ C, where:
A = {(x, y); x ≥ 0, y = 1}, B = {(x, y); x ≥ 0, y = −1};
C = {(x, y); x < 0, y = 0}.
Define a topology on X by putting the subspace topology on A\{(0, 1)}, B\{(0, 1)}
and C, and for the points (0, 1) (and (0, −1)) use the basis of neighbourhoods given
by Nǫ = {(x, 1); 0 ≤ x < ǫ} ∪ {(x, 1); −ǫ ≤ x < 0} (similarly for (0, −1)). Show
that X is locally homeomorphic to R but it is not a manifold.
Exercise 12. Let p : X → Y be a covering mapping, where X and Y are manifolds, and let p∗ : π1 (X, x0 ) → π1 (Y, p(x0 )) be the induced mapping between the
fundamental groups. Show that p∗ is injective.
Exercise 13. Find an example of a covering map p : X → Y , and a continuous
function f : X → X such that f ◦ p = f but f is not a homeomorphism (this shows
that the condition of homeomorphism cannot be removed from the definition on
1.1.20.
Exercise 14. Show that the result of exercise 7 is true if you substitute manifold
for Riemann surface.
Exercise 15. Can you find a Riemann surface structure on C which is not compatible with the usual structure, induced by (C, Id)?
Hint: C is homeomorphic to D.
b given by
Exercise 16. Prove that the stereographic projection P : S 2 → C,
P (x1 , x2 , x3 ) =


 x1 +ix2 , if x3 6= 1,
1−x3

∞, if x3 = 1,
b
is a homeomorphism between the sphere S 2 of R3 and the Riemann sphere C.
Exercise 17. The projective line (or projective space of dimension 1) P1 is defined as
follows: consider the equivalence relation ∼ in M = C2 \{(0, 0)} given by (z1 , z2 ) ∼
(w1 , w2 ) if there exists a non-zero complex number λ, such that z1 = λw1 and
b is homeomorphic to P1 .
z2 = λw2 . Set P1 = M/ ∼. Prove that C
Hint: denote by [z : w] the points P1 (notation as in 2.7.1), consider the map from
b given by [z : w] 7→ z/w if w 6= 0 and [z : 0] 7→ ∞.
P1 to C
Exercise 18.
Let G be a group of Möbius transformations and D a connected
b Assume that the following conditions are satisfied:
open subset of C.
1. g(D) = D, for all g ∈ G.
2. For every point z in D, there exists a neighbourhood U of z, such that, if
g ∈ G satisfies g 6= ID, then g(U) ∩ U = ∅.
Denote by D/G the quotient space of D by the action of G; that is, z1 and z2 are
equivalent if there exists g ∈ G, such that g(z2 ) = z1 . Show that D/G carries a
unique Riemann surface structure such that the quotient mapping π : D → D/G is
holomorphic.
Exercise 19. Show that if f : X → C is holomorphic and X is a compact Riemann
surface then f is constant.
Exercise 20.
Show that the set of poles of a holomorphic function defined on a
Riemann surface is discrete.
Exercise 21. A mapping f : X → Y between topological spaces is called proper
if f −1 (K) is compact in X, for any K compact subset of Y . Extend the definition
of degree and proposition 1.3.11 to the case of proper mappings (not necessarily
between compact surfaces).
Exercise 22. Prove proposition 1.3.14
Exercise 23.
Let f : X → Y be a holomorphic mapping between compact
Riemann surfaces of genera g and g ′ respectively. Show that if g = g ′ then f is a
covering map. Show that if g ′ = 1 then g = 1.
Exercise 24. Prove that the residue of a polynomial p(z) = a0 + a1 z + · · · an z n at
b is equal to a1 .
the point ∞, when one considers p as a meromorphic mapping on C,
Exercise 25.
b → C
b on the Riemann
Prove that a meromorphic function f : C
sphere is rational, i.e. the quotient of two polynomials.
Exercise 26. Let R(z) =
p(z)
q(z)
be a rational function on the Riemann sphere, where
p and q are polynomials without common factors. Show that the point ∞ is a:
• pole if deg (p) > deg (q);
• zero if deg (p) < deg (q);
• regular point if deg (p) > deg (q).
b be a non-constant meromorphic function on a
Exercise 27.
Let f : X → C
Riemann surface. For a point p of X, let n denote the ramification index of f at
that point. Define
z(q) =


(f (q) − f (p))1/n , if f (p) 6= ∞,

(f (q))−1/n , if f (p) = ∞,
for q is a neighbourhood of p. Show that z is a local coordinate on X.
Exercise 28. Let X be a compact surface, p1 , . . . , pn distinct points of X. Prove
that any non-constant holomorphic function f : X ′ = X\{p1, . . . , pn } → C comes
arbitrarily close to any point z ∈ C. More precisely, show that for any z ∈ C, and
any ǫ > 0, there exists a point p ∈ X ′ such that |f (p) − z| < ǫ.
Exercise 29. Let τ and µ be two complex numbers with positive imaginary parts.
Let Tτ and Tµ denote the corresponding tori (see 1.3.6). Show that if λ is a complex
number satisfying λGτ ⊂ Gµ , then the function f : C → C, given by f (z) = λz,
induces a holomorphic mapping from F : Tτ → Tµ . Prove that F is biholomorphic
if and only if λ Gτ = Gµ .
Exercise 30.
Let f : X → Y be a non-constant holomorphic mapping between
Riemann surfaces. Show that the mapping f ∗ : O(Y ) → O(X), defined by f ∗ (g) =
g ◦ f , is a ring homomorphism.
Exercise 31. Let Tτ = C/Gτ be as in 1.3.6. Show that Tτ is a connected Hausdorff
space. Moreover, show that Tτ is a topological manifold of dimension 2.
Exercise 32. Show that the operators ∂ and ∂¯ satisfy ∂ ◦ ∂ = 0 and ∂¯ ◦ ∂¯ = 0, by
direct computation.
Exercise 33.
Let α be a smooth (complex valued) 2-form on a surface X with
compact support. Let U = {Ui }ni=1 and V = {Vj }m
j=1 be two finite open coverings
of the support of α by coordinate patches. Let {fn }ni=1 and {gj }m
j=1 be partitions of
unity subordinated to U and V respectively. Show that
n Z
m Z
X
X
fi α =
gj α.
i=1
Exercise 34.
X
j=1
X
Let X = X̃/G be a Riemann surface, where X̃ is the universal
covering space and G the group of deck transformations. Let p : X̃ → X be the
covering map. Assume ω is a closed 1-form on X, and F a primitive of p∗ (ω), that
is, F : X̃ → C satisfies dF = p∗ (ω). Show that for every g in G there exists a
complex number ag , such that F − F ◦ g −1 = ag .
Use the Residues theorem to show that there does not exist a
Exercise 35.
function on C/Gτ with a just pole of order 1.
Exercise 36. Let G be a group with at least two elements, and X a topological space
with at least two non-empty disjoint open sets. Define a presheaf F by F (U) = G
if U ⊂ X is a non-empty open set, and F (∅) the trivial group. The restriction
homomorphisms ρ : F (U) → F (V ) are given by the identity homomorphism if
V 6= ∅ and the trivial homomorphism if V = ∅. Show that F is a presheaf but not
a sheaf. Compute the associated sheaf.
Hint: consider two disjoint open sets U1 and U2 , and let f ∈ F (U1 ∪ U2 ) be given
by g1 in U1 and g2 in U2 .
Exercise 37. Show that if X is a simply connected surface, then H 1 (X, C) and
H 1 (X, Z) are trivial.
Exercise 38.
Let C∗ = C\{0}. Consider the covering given by the open sets
U1 = C∗ \R− and U2 = C∗ \R+ . Here R− and R+ are the negative and positive real
axis, respectively.
1. Prove that U = {U1 , U2 } is a Leray covering for the sheaf of locally constant
functions with integer values Z.
2. Show that Z 1 (U, Z) ∼
= Z(U1 ∩ U2 ) × Z(U1 ∩ U2 ).
3. Show that Z(Ui ) ∼
= Z, for i = 1, 2.
4. Prove that the boundary operator δ : C 0 (U, Z) → C 1 (U, Z) is given by
(a1 , a2 ) 7→ (a2 − a1 , a2 − a1 ).
5. Conclude that H 1 (C∗ , Z) ∼
= Z.
Exercise 39.
Show that if X is a compact surface, the mapping H 1 (X, Z) →
H 1 (X, C) induced by the inclusion Z ֒→ C is injective. Here Z and C denote the
sheaves of locally constant functions with values in the integer and complex numbers
respectively.
Exercise 40. Show that the sheaf sequence:
β
0 → C∗ → O∗ → Ω → 0,
where β(f ) = df /f , is exact.
Exercise 41. Show that the mapping δ ∗ of 1.5.15 is well defined.
Exercise 42. Show that two divisors on the Riemann sphere are linearly equivalent
if and only if they have the same degree.
Exercise 43. Let g : C → C be a smooth function with compact support. Show
that g is uniformly continuous. Use this to prove that if ξ ∈ C, then the integral
Z 2π
g(ξ + reiθ ) dθ
0
converges to 2πg(ξ) when r → 0 (see 2.2.1).
Exercise 44. Use the inequality |(hn+1 − hn − pn )(z)| < 2−n to prove that the
function in 2.2.2 (pg. 64) is well defined.
b a non-constant
Exercise 45. Let X be a compact Riemann surface and f : X → C
meromorphic function of degree d. Show that the sheaf of meromorphic functions
on X is a finite algebraic extension field of C(f ) (the field of rational functions on
f ) as follows:
a) Let C be the set of critical points of f ; that is, C consists of the points p ∈ X
such that dp (f ) = 0. Set B = f (C) (critical values of f ) and A = f −1 (C). Show
b
that f : X\A → C\B
is a covering of degree d.
b be a non-constant meromorphic function. Let S be the set
b) Let g : X → C
b
of poles of g, S = g −1 (∞). If z ∈ C\B\S
and f −1 (z) = {p1 , . . . , pd } (these points
are distinct), define aj as the j-th elementary function on g(p1 ), . . . , g(pd ). These
means that the aj , for j = 1, . . . , d satisfy the following equation:
(g(p))d + a1 (f (p)) (g(p))d−1 + · · · + ad (f (p)) ≡ 0,
for all p in X\A\f −1(f (S)). Show that aj are holomorphic functions.
c) We want to show that aj extend to meromorphic functions on X. That will
prove that the equation displayed above holds for all points of X and will finish the
exercise. To show that, let a be a point in B ∪ f (S), choose a neighbourhood U of
a such that the poles of g on U lie in f −1 (a). Let w be a holomorphic function on
U, not identically 0 but satisfying w(a) = 0. Show that (w ◦ f )n g is holomorphic
for some non-negative integer n (on f −1 (U)). Let bj be the elementary symmetric
functions of (w ◦ f )n g on f −1 (z) = {p1 , . . . , pd }, for z ∈ W \{a}, where W is an open
subset of U. Show that functions bj extend to holomorphic functions at a. Use the
relation aj = bj /w jn to show that a − j is meromorphic on X.
Exercise 46. Prove the Riemann-Hurwitz formula using the Riemann-Roch theorem.
Exercise 47. Prove the assertions of 2.4.7
Let L be a line bundle on a Riemann surface X and L−1 the dual
Exercise 48.
bundle. Prove that for any point p ∈ X there is an isomorphism between (Lp )∗ , the
dual space of Lp , and (L−1 )p .
Exercise 49. Show that the mapping between the Picard group X and H 1 (X, O∗ )
of 2.4.14 is a group isomorphism.
Exercise 50. Let K be a canonical divisor on a compact surface X of genus g.
Show that the dimension of H 0 (X, O(m =, K)) is given as in the table 1 at the end
of the exercises.
Exercise 51.
Let K be a canonical divisor on a compact surface X of genus g.
Show that
dim H 0 (X, O(m K − p)) = dim H 0 (X, O(m K)) − 1,
for any point p of X.
Exercise 52. Compute the dimension of the spaces of higher order differentials.
Exercise 53.
Prove that projective embeddings im (see 2.7.4 and 2.7.8) are well
defined.
Exercise 54. Recall the Weierstrass ℘ function for the torus Tτ (that is, Tτ = C/Gτ
where ℑ(τ ) > 0) is given by the expression (2.6.6):
1
1
1 X
.
−
℘(z) = +
z λ6=0 (z − λ)2 λ2
Here λ is of the form λ = n + mτ with n and m integers.
a) Prove that ℘ is an even function (℘(z) = ℘(−z)).
b) Show that the derivative of ℘ is given by
(℘′ )(z) =
X
1
,
(z − λ)3
where the sum is taken over all points of the form λ = n + mτ , including λ = 0.
Use this expression to show that ℘′ is an odd function (℘′ (−z) = −℘′ (z)).
b be the mapping (of degree 2) induced by ℘ on the torus Tτ .
c) Let f℘ : T τ → C
Show that the ramification points of f℘ are given by the points corresponding to 0,
1/2, τ /2 and (1 + τ )/2.
d) Show that ℘′ vanishes (or it has a singularity if you do not take local coordinates) precisely at the four points given above (and the corresponding points on C
congruent via the group Gτ ).
e) Show that the mapping i : Tτ → P2 given by p 7→ [1 : ℘(p) : ℘′ (p)] defines an
embedding of Tτ into the 2-dimensional projective space.
Exercise 55. Complete the proof of theorem 2.7.11 as follows:
a) To define the map φL : X → Pn in a neighbourhood of a point p we have used
a section σ that does not vanish at p. Show that if σ1 is another section defined in
a neighbourhood of p, say U ′ , with σ1 (p) 6= 0 then for every point q ∈ U ∩ U ′ we
have that
sj (q)
σ(q)
s (q)
= h(q) σj1 (q) , where h is a non-zero holomorphic section. This shows
that φL does not depend on the choice of σ as long as φL (q) is in Pn (see below).
b) Use 2.6.5 to prove that the sections sj do not have common zeroes and therefore φL maps X into Pn .
c) In the text we have shown that if p and q are two distinct points of X then
there exists a section of L which vanishes at p but it does not vanish at q. Use this
fact to prove that φL is injective.
d) We have also shown that if p ∈ X then there exists a section of L with a
simple zero at p. Using this fact show that the differential of φL does not vanish.
This completes the proof of the fact that φL : X → Pn is an embedding.
Exercise 56.
Let z1 , . . . , z2g+2 be distinct points on the Riemann sphere. Prove
that the Riemann surface of the polynomial
w 2 = (z − z1 ), . . . , (z − z2g+g )
is a compact surface X of genus g. Let z denote the function on X, and pj the
preimage of zj (j = 1, . . . , 2g + g). Without loss of generality we can assume that
the points pj are not equal to ∞ or 0. Assume the divisor of z is given by div(z) =
q3 + q4 − q1 − q2 ; prove that the divisor of w is given by div(w) = (p1 + · · · + p2g+2 ) −
(g + 1)q1 − (g + 1)q2 . Use this to prove that the 1-forms
z j dz
,
w
for j = 0, . . . , g − 1
form a basis of the space of 1-forms on X. Basis for holomorphic differentials on a
hyperelliptic surface.
Exercise 57. Show that if f : X → C is a meromorphic function of degree d on a
hyperelliptic surface of genus g, and d ≤ g, then d is even.
Exercise 58. Prove that if U is an open, connected subset of C, and A is a discrete
subset of U, then U\A is connected.
Exercise 59. Let 1 < α1 < · · · < αg 2g be the complementary set of the numbers
n1 , . . . , ng of Weierstrass’ theorem (2.8.1) in {1, . . . , 2g}. Call these numbers the
non-gaps at the point p (the nj ’s are called the gaps). Prove the following statement
regarding the non-gaps:
1. If αj and αk are two “non-gaps” with αj + αk ≤ 2g, then αj + αk is also a
“non-gap”.
2. For each integer j with 0 < j < g one has αj + αg−j ≥ 2g.
3. If α1 = 2 then αj = 2j and αj + αg−1 = 2g, for 0 < j < g.
4. If α1 > 2, then for some j with 0 < j < 2g one has αj + αg−j > 2g.
P
5. g−1
j=1 αj ≥ g(g − 1), and the equality occurs if and only if α1 = 2.
Exercise 60. Prove lemma 2.8.4 for the case of m ≥ 3.
b
Exercise 61. Let X be a hyperelliptic compact surface of genus g and f : X → C
a function of degree 2. Let j : X → X denote the hyperelliptic involution. This
exercise shows an explicit basis of H 0(X, Ω). Assume that f is not branched over
∞; that is that the fixed points of j are not poles of f .
a) Choose a point p in X not fixed by j; show that there exists a meromorphic
b such that u(q) 6= u(j(q)) for q in a neighbourhood of p, and
function u : X → C
that we can choose u to be holomorphic at p and j(p).
b) It follows from exercise 45 that there is an equation of the following form:
u2 (p) + 2a1 (f (p)) u(p) + a2 (f (p)) = 0.
Write (u+a1 )2 = p/q, where p and q are polynomials on z. Then show that pq = p1 q12 ,
where p1 does not have multiple roots; moreover, p1 (z) = c(z − z1 ) · · · (z − z2g+2 ),
where c 6= is a complex number and zj = f (pj ), for pj , j = 1, . . . , 2g + 2 the
Weierstrass points of X. Show that zj 6= zk if j 6= k.
c) Define w = (u + a1 )q/q1 . Show that w(p1) 6= w(p2 ) for f −1 (z) = {p1 , p2 } in
an open set of X. Show that w = p21 .
d) Show that w(p) =6= w(j(p)) for p in an open set of X but w(p)2 = w(j(p))2
in that open set, and therefore on X by the identity principle. This implies that
w(p) = −w(j(p)) on X. Show that w is a function of degree g + 1 on X.
e) Define forms for j = 0, . . . , g − 1 by
ωj (p) =
f j (p) f ′(p) dz
,
w(f (p))
where z is a local coordinate on X. Use the identity w = p21 to show that ωj =
2f j dw/p′1 on f −1 (C) and thus ωj is holomorphic at those points.
f) To prove that ωj is holomorphic show that near a pole of f we have w =
c1 f g+1 O(1 + z1 ) and therefore ωj = c1 f j−g−1(1 + O( z1 )) is also holomorphic.
b be a function of degree 2 on a hyperelliptic Riemann
Exercise 62. Let f : X → C
surface X of genus g. Denote by p1 , . . . , p2g+2 the Weierstrass points of X. Show
that the images of these points under f are distinct; that is, f (pj ) 6= f (pk ), for
1 ≤ j < k ≤ 2g + 2.
Exercise 63. Prove Noether Gap Theorem (2.8.2).
Exercise 64.
If D is a divisor on a surface X we say that O(D) is globally
generated if there exists a function f ∈ H 0 (X, O(D)) such that ordp (f ) = −D(p)
for every p in X. Equivalently, every function g defined on a neighbourhood of p with
div(g) ≥ −D can be written as g = h f , where h is holomorphic on a neighbourhood
of p. Show that if D has degree greater than 2g − 1 and X is compact of genus g
then O(D) is globally generated.
Exercise 65. Let X = C/Gτ , p0 a point of X, Dn the divisor np0 . Show that



0, n < 0,



dim H 0(X, O(Dn )) = 1, n = 0,




n, n > 0.
Exercise 66. Consider the homeomorphism w : C → D given by
w(z) =
z
.
1 + |z|
This mapping induces a Riemann surface structure on C, given by a single chart
(C, w). Denote by C1 the complex plane with this structure. Let u : C → R be
defined by u(z) =
x
,
1+|z|
where z = x + iy. Show that u is harmonic on C1 , but it is
not when we consider the standard structure of the complex plane.
Exercise 67.
Let A be an open subset of the complex plane, p a point in the
boundary of A. Suppose that there exists a disc D centred at a point q ∈ A such
that D ∩ A {p} and D ⊂ A. Show that p is a regular point of A.
Hint: let c be the middle point of the segment joining p and q. Show that the
function β(z) = log(r/2) − log(|z − c|) is a barrier at p.
e its universal covering space (idenExercise 68. Let X be a Riemann surface, X
b C or H) and G the group of deck transformations (identified with a
tified with C,
group of Möbius transformations). Show that X is biholomorphic to the quotient
e
X/G.
Exercise 69. Let A(z) =
az+b
cz+d
be a Möbius transformation with ad − bc = 1.
1. Show that A has order 2 (that is, A ◦ A = Id) if and only if a + d = 0.
b if and only if |a + d| = 2.
2. Prove that A has only one fixed point in C
Exercise 70.
of the form z 7→
b is given by the transformations
The Picard subgroup G of Aut(C)
az+b
,
cz+d
satisfying:
1. ad − bc = 1.
2. a, b, c, d are in Z[i]; that is, their real and imaginary parts are integers.
Show that G is discrete but it does not act properly discontinuously at any point of
the Riemann sphere.
Exercise 71. Prove the second lemma in 3.3.5
Exercise 72. Show that |z − w|2 − |1 − w̄z|2 | = |z|2 + |w|2 − |z|2 |w|2 − 1. Use this
z−w
| < 1. Prove proposition 3.3.8
to show that if |z| = 1 and |w| < 1 then | 1−
w̄z
Exercise 73.
Let G be a group of Möbius transformations of the form Tλ : z 7→
z + λ, where λ is a complex number. Let r = inf{|λ}; Tλ ∈ G}. Show that if there
does not exist Tµ in G with r = |µ| then G cannot be discrete.
Exercise 74. Compute the area of a hyperbolic pentagon with angles given as in
picture 13. More generally, compute the area of a hyperbolic pentagon (or convex
polyhedron).
Exercise 75. Show that if X = H/G is a compact surface of genus 2 then N(G)
strictly contains G.
Exercise 76.
Let zi , i = 1, . . . , 4 be four distinct point in the Riemann sphere.
Compute the cross ratios (zτ (1) , zτ (2) , : zτ (3) , zτ (4) ) as τ varies over all 24 permutations
of four symbols. Show that there are only 6 distinct values among all these cross
ratios.
Exercise 77. Let C be a line or circle orthogonal to S 1 and A an automorphism
of the unit disc. Show that A(C) is a line or circle orthogonal to S 1 .
Exercise 78. Show that the mapping z 7→ −z̄ is an isometry in the hyperbolic
metric of H.
Exercise 79. Prove lemma 3.4.4 by direct computation.
Exercise 80. Prove lemma 3.4.17.
Exercise 81. Prove that the exponential sequence
exp
0 → Z → C → C∗ → 0
is exact.
Exercise 82. Show that the field of meromorphic functions on a compact surface
b be a non-constant
is an algebraic field of one variable. More precisely, let f : X → C
meromorphic function on a compact surface X; let d denote the degree of f . Let g be
another meromorphic function on X. Prove that there exist meromorphic functions
b → C,
b j = 1, . . . , d, such that
aj : C
(g(p))d + a1 (f (p)) (g(p))d−1 + · · · + ad (f (p)) = 0,
for all p in X.
Exercise 83. Show that a discrete subset of the complex plane is either finite or
infinite countable.
Hint: cover C by a countable increasing sequence of compact sets.
g
m
dimension
0
m≤0
1 − 2m
m>0
0
1
0
g≥1 m<0
0
m=0
1
m=1
g
m > 1 (2g − 1) (g − 1)
Table 1.
φ
Figure 15. Hyperbolic pentagon.