UNIVERSIT ´ E NICE SOPHIA ANTIPOLIS Ann´ ee 2016/2017
Master 2 MPA Complex geometry
Exercise sheet 1
Exercise 1. Let U ⊂ C n be an open set, and let f : U → C a differentiable function. Denote by z 1 , . . . , z n the standard coordinates on C n , and set x j and y j for their real and imaginary parts. For a ∈ U we consider the R -linear map given by the differential
df a = ( ∂f
∂x 1
(a), ∂f
∂y 1
(a), . . . , ∂f
∂x n
(a), ∂f
∂y n
(a)) : C n = R 2n → C . Show that f is holomorphic in a if and only if df a is C -linear.
Exercise 2. Maximum principle.
a) Let U ⊂ C n 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| D 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 n . Show that X has dimension zero.
Exercise 3.
a) Let X ⊂ C 2 be the analytic set defined by z 2 1 − z 2 3 = 0. Show that X has dimension one.
b) Let X ⊂ C 3 be the analytic set defined by
z 1 z 3 = 0, z 2 z 3 = 0.
Determine the codimension of X in (0, 0, 0).
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 U V : F(U ) → F(V ),
for every inclusion V ⊂ U of open sets, which satisfy the following conditions:
1. r U U is the identity map F(U ) → F(U).
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2. If W ⊂ V ⊂ U are three open sets, then r U W = r V W ◦ r U V .
3. If U is an open set, s ∈ F(U ), and (V i ) i∈I is an open covering of U such that r U V
i(s) = 0 for every i ∈ I, then s = 0.
4. (Glueing property) If U is an open set, (V i ) i∈I is an open covering of U and s i ∈ F(V i ) satisfy
r V
i(V
i∩V
j) (s i ) = r V
j(V
i∩V
j) (s j )
for every i, j ∈ I, then there exists a unique s ∈ F(U) such that r U V
i(s) = s i . 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 U V : F(U ) → F(V ), f 7→ f | V ,
the restriction map. Show that F is a sheaf of abelian groups 1 . 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 U V 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 X ∗ (U) := {f : U → C ∗ | f is holomorphic}
and for an inclusion V ⊂ U we define r U V to be the restriction map. Show that O ∗ X is a sheaf of abelian groups (for the natural multiplicative structure).
Exercise 5. 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 1 × S 1 .
b) Let ϕ : C /Λ → C /Λ 0 be a biholomorphic map such that ϕ(0) = 0. Show that there exists a unique α ∈ C ∗ such that αΛ = Λ 0 and such that the diagram
C
π
z7→αz
˜ ϕ // C
π
0C /Λ ϕ // C /Λ 0
is commutative. For the proof you can admit 2 the following properties:
- Let ϕ : C /Λ → C /Λ 0 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 0 ◦ ϕ. ˜
- Aut( C ) = {z 7→ αz + β | α ∈ C ∗ , β ∈ C }.
c) Set Λ := Z ⊕ Z i and Λ 0 := Z ⊕ Z ζ where ζ is primitive third root of unity. Show that C /Λ and C /Λ 0 are not biholomorphic.
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The group structure on F(U) is the natural one given by addition of functions
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