Exercise 1. Let X be a complex manifold.

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

Master 2 MPA Complex geometry

Exercise sheet 2

Exercise 1. Let X be a complex manifold.

a) Let E → X be a holomorphic vector bundle of rank r. Let (U

)

be a trivialising cover for the vector bundle E, and let

g

: U

∩ U

→ GL( C , r)

be the transition functions of E. Show that the transition functions satisfy the cocycle relations g

(z) = id ∀ z ∈ U

,

g

(z) ◦ g

(z) = g

(z) ∀ z ∈ U

∩ U

∩ U

. b) Let (U

)

be an open cover of X and for every i, j ∈ I let

g

: U

∩ U

→ GL( C , r)

be a holomorphic maps satisfying the cocycle relations. Show that there exists a holomorphic vector bundle E → X of rank r having g

as transition functions.

Exercise 2. Let X be a complex manifold, and let E → X be a holomorphic vector bundle of rank r. For every open set U ⊂ X we denote by H

(U, E) the sections of E|

→ U .

a) Show that H

(U, E) is a C -vector space (in particular it has an abelian group structure).

b) If V ⊂ U is an inclusion of open sets, we define

r

: H

(U, E) → H

(V, E), s 7→ s|

the restriction map. Show that the sections of E → X form a sheaf of abelian groups. We will denote this sheaf by O

(E) (In the case where E = X × C is the trivial bundle, we obtain the sheaf of holomorphic functions O

.).

Exercise 3. We consider the projective line P

and the rank one vector bundle O

(k) → P

. We denote by H

( P

, O

(k)) the space of global sections. Show that

H

( P

, O

(k)) '







0 if k < 0,

C if k = 0,

homogeneous polynomials of degree k in two variables if k > 0.

Exercise 4. Let X be a complex manifold, and let L → X be a holomorphic line bundle (i.e. a holomorphic vector bundle of rank one). Denote by H

(X, L) the vector space of global sections of L, and suppose that this space has finite dimension. Suppose also that for every x ∈ X there

1

exists a global section s ∈ H

(X, L) such that s(x) 6= 0. Let s

, . . . s

be a basis of H

(X, L).

We set

X

:= {x ∈ X | s

(x) 6= 0}.

a) Show that for every j ∈ {1, . . . , n}, we have a well-defined holomorphic function s

s

: X

→ C , x 7→ s

(x) s

(x) . b) Show that the holomorphic map

ϕ

: X

→ P

, x 7→ (1 : s

(x)

s

(x) : . . . : s

(x) s

(x) ) extends to a holomorphic map ϕ : X → P

.

Exercise 5. Let X be a real manifold of dimension n, and let

d : C

(X, Ω

) → C

(X, Ω

) be the exterior derivative.

a) By definition we have d(d(f )) = 0 for every differentiable function f : X → R . Deduce that we have

d(d(ω)) = 0 ∀ ω ∈ C

(X, Ω

).

b) Let Y be a real manifold of dimension m, and let ϕ : X → Y be a morphism which in some local charts U ⊂ R

and V ⊂ R

is given by a differentiable map

(ϕ

, . . . , ϕ

) : U → V.

We define a pull-back map

ϕ

: C

(Y, Ω

) → C

(X, Ω

), ω 7→ ϕ

ω as follows

: suppose that in the local chart V the 1-form ω is defined by

m

X

j=1

α

dy

.

Then ϕ

ω is defined on U by

m

X

j=1

α

◦ ϕ d(ϕ

).

Show that pull-back commutes with the exterior differential, i.e. for a differentiable function f : Y → R we have

d(ϕ

(f )) = ϕ

(d(f )),

where ϕ

f is simply the composition f ◦ ϕ. Generalise the construction for arbitrary k-forms.

1It would be more precise to writedk :C^∞(X,Ω^k_X)→C^∞(X,Ω^k+1_X ), but we forget the indexkto have a simpler notation.

2If you want you can check thatϕ^∗is well-defined, i.e.ϕ^∗ωdoes not depend on the choice of local coordinates.

2