Exercise 1. Let U ⊂ C

UNIVERSIT ´ E NICE SOPHIA ANTIPOLIS Ann´ ee 2017/2018

Master 2 MPA Complex geometry

Exercise sheet 1

Exercise 1. Let U ⊂ C

be an open set, and let f : U → C a differentiable function. Denote by z

, . . . , z

the standard coordinates on C

, and set x

and y

for their real and imaginary parts. For a ∈ U we consider the R -linear map given by the differential

df

= ( ∂f

∂x

(a), ∂f

∂y

(a), . . . , ∂f

∂x

(a), ∂f

∂y

(a)) : C

= R

→ C . Show that f is holomorphic in a if and only if df

is C -linear.

Exercise 2. Maximum principle.

a) Let U ⊂ C

be an open set, and let f : U → C be a holomorphic function such that |f | has a local maximum in some point w ∈ U . Show that there exists a polydisc D ⊂ U around w such that f|

is constant.

b) Let X be a compact complex manifold. Show that a holomorphic function f : X → C is constant.

c) Let X be compact complex manifold that is a submanifold of C

. Show that X has dimension zero.

Exercise 3. 1-dimensional tori.

Let Λ ⊂ C be a lattice of rank 2, and let X := C /Λ be the associated torus.

a) Show that X is diffeomorphic to S

× S

.

b) Let ϕ : C /Λ → C /Λ

be a biholomorphic map such that ϕ(0) = 0. Show that there exists a unique α ∈ C

such that αΛ = Λ

and such that the diagram

C

π

z7→αz

˜ ϕ

// C

π⁰

C /Λ

// C /Λ

is commutative. For the proof you can admit

the following properties:

- Let ϕ : C /Λ → C /Λ

be a biholomorphic map such that ϕ(0) = 0. Then there exists a holomorphic map ˜ ϕ : C → C such that ˜ ϕ(0) = 0 and ϕ ◦ π = π

◦ ϕ. ˜

- Aut( C ) = {z 7→ αz + β | α ∈ C

, β ∈ C }.

c) Set Λ := Z ⊕ Z i and Λ

:= Z ⊕ Z ζ where ζ is primitive third root of unity. Show that C /Λ and C /Λ

are not biholomorphic.

1You will see the proofs in the course by Cazanave and Waschkies.

1

Exercise 4. Sheaves of abelian groups. Let X be a topological space. A sheaf of abelian groups on X is the following data:

a) an abelian group F (U ) for every open set U ⊂ X and b) a morphism of abelian groups

r

: F(U ) → F(V ),

for every inclusion V ⊂ U of open sets, which satisfy the following conditions:

1. r

is the identity map F(U ) → F(U).

2. If W ⊂ V ⊂ U are three open sets, then r

= r

◦ r

.

3. If U is an open set, s ∈ F(U ), and (V

)

is an open covering of U such that r

(s) = 0 for every i ∈ I, then s = 0.

4. (Glueing property) If U is an open set, (V

)

is an open covering of U and s

∈ F(V

) satisfy

r

(s

) = r

(s

)

for every i, j ∈ I, then there exists a unique s ∈ F(U) such that r

(s) = s

. a) For every open set U ⊂ X we denote by

F(U ) := {f : U → C }

the set of all (complex-valued) functions on U . For an inclusion V ⊂ U of open sets, we define r

: F(U ) → F(V ), f 7→ f |

,

the restriction map. Show that F is a sheaf of abelian groups

. b) For every open set U ⊂ X we denote by

C (U ) := {f : U → C | f is locally constant}

and for an inclusion V ⊂ U we define r

to be the restriction map. Show that C is a sheaf of abelian groups. If we replace ‘locally constant’ by ‘constant’, do we still obtain a sheaf ? c) Let now X be a complex manifold. For every open set U ⊂ X we denote by

O

(U) := {f : U → C

| f is holomorphic}

and for an inclusion V ⊂ U we define r

to be the restriction map. Show that O

is a sheaf of abelian groups (for the natural multiplicative structure).

d) Let A ⊂ C

be an analytic subset. For every open set U ⊂ C

we denote by I (U ) := {f : U → C | f is holomorphic and f (z) = 0 ∀ z ∈ U ∩ A}

and for an inclusion V ⊂ U we define r

to be the restriction map. Show that I is a sheaf of abelian groups

.

2The group structure onF(U) is the natural one given by addition of functions

3For the proof you can admit thatO_Cnis a sheaf.

2