Exercise 1. Let X be a real manifold given by some atlas (U i , ϕ i : U i → V i ⊂ R n ) i∈I . The atlas is oriented if the coordinates changes ϕ i ◦ ϕ −1 j have positive determinant, i.e.

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

Master 2 MPA Complex geometry

Exercise sheet 3

Exercise 1. Let X be a real manifold given by some atlas (U i , ϕ i : U i → V i ⊂ R ⁿ ) i∈I . The atlas is oriented if the coordinates changes ϕ i ◦ ϕ ⁻¹ _j have positive determinant, i.e.

det(Jac _ϕ

i

◦ϕ

(x)) > 0 ∀ x ∈ ϕ _j (U _i ∩ U _j ).

Let X be a complex curve given by some atlas (U i , ϕ i : U i → V i ⊂ C ) i∈I . Show that the corresponding real atlas is oriented. Generalise to complex manifolds X of arbitrary dimension.

Exercise 2. Let X be a complex curve.

a) Let ω ∈ H ⁰ (X, Ω _X ) be a holomorphic one-form. Show that ω is d-closed.

b) Let ω ∈ C ^∞ (X, Ω ^1,0 _X ) be a differential form of type (1, 0) that is d-closed. Show that ω is holomorphic.

Exercise 3. Let ω be the (1, 1)-form on P ¹ that is given locally in the standard open sets U 0

(resp. U 1 ) by ω = _(1+|z| ⁱ

)

dz ∧ d¯ z (resp. ω = _(1+|w| ⁱ

)

dw ∧ d w). ¯

a) Show that ω is well-defined, i.e. the local descriptions coincide on U 0 ∩U 1 (recall that w = _z ¹ ).

b) Show that ω is d-closed. Show that R

P

ω = 2π (use that idz ∧ d¯ z = 2dx ∧ dy).

c) Deduce that H ² ( P ¹ , C ) 6= 0 and H ^1,1 ( P ¹ ) 6= 0.

Exercise 4. Prove the generalised form of Cauchy’s formula: let U ⊂ C be an open set, and let f : U → C be a differentiable complex-valued function. Let D ⊂ U be the closure of a disc contained in U . Then for every w ∈ D, we have

f(w) = 1 2πi

Z

∂D

f (z) z − w dz −

Z

D

i 2π(z − w)

∂f

∂z dz ∧ dz.

Exercise 5. a) Let f : X → Y be a differentiable map between real manifolds. Show that the pull-back

f ^∗ : C ^∞ (Y, Ω ^k _{Y, C} ) → C ^∞ (X, Ω ^k _{X, C} )

induces a linear map between the de Rham cohomology groups f ^∗ : H ^k (Y, C ) → H ^k (X, C ).

b) Let now f : X → Y be a holomorphic map between complex manifolds. Show that the pull-back

f ^∗ : C ^∞ (Y, Ω ^k _{Y, C} ) → C ^∞ (X, Ω ^k _{X, C} )

1

maps forms of type (p, q) to forms of type (p, q). Hint: prove the claim for forms of type (1, 0) and (0, 1), then generalise.

Show that for all p, q ∈ N the pull-back induces a linear map between the Dolbeault cohomology groups f ^∗ : H ^p,q (Y ) → H ^p,q (X ).

Exercise 6. (Bonus exercise, probably no time for discussion in the exercise class.) We consider the differentiable 1-form on R ² \ 0 given by

α = 1

x ² + y ² (−ydx + xdy).

a) Show that α is d-closed.

b) Let i : S ¹ , → R ² \ 0 be the inclusion of the unit cercle. Show that Z

S

i ^∗ α = 2π.

Deduce that α and i ^∗ α are not d-exact.

c) Let ω be a differentiable 1-form on S ¹ . For every x ∈ S ¹ we define

f (x) :=

Z x

1

ω,

where we integrate over the segment of the circle from 1 to x in counterclockwise direction.

Show that lim _x→1 f (x) = 0 if R

S

ω = 0. Deduce that ω is d-exact if and only if R

S

ω = 0.

d) Show that H ¹ (S ¹ , R ) ' R .

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