a) Let X be a topological space, and F a sheaf of abelian groups on X . Let U = (U i ) i∈I be an open cover. In the lecture we defined the maps

UNIVERSIT ´ E NICE SOPHIA ANTIPOLIS Ann´ ee 2016/2017

Master 2 MPA Complex geometry

Exercise sheet 4

Exercise 1.

a) Let X be a topological space, and F a sheaf of abelian groups on X . Let U = (U i ) _i∈I be an open cover. In the lecture we defined the maps

δ ₁ = C ⁰ (U, F ) → C ¹ (U, F) δ ₂ = C ¹ (U, F) → C ² (U, F).

Show that δ 2 ◦ δ 1 = 0.

b) Let X be a complex manifold and Y ⊂ X a hypersurface. Show that there exists a holomor- phic line bundle L → X and a global section s : X → L such that Y = {x ∈ X | s(x) = 0}.

Exercise 2. For this exercise you can admit the following result ¹ : let g : C → C be a differen- tiable function. Then there exists a differentiable function f : C → C such that g = ^∂f _{∂ z ¯} . a) Show that H ^0,1 ( C ) = 0 and H ^1,1 ( C ) = 0.

b) Show that H ^1,1 ( P ¹ ) ' C . Hint: let U be the open cover by the standard open sets U ₀ and U 1 . Show that H ^1,1 ( P ¹ ) ' H ˇ ¹ (U, Ω _P

) and ˇ H ¹ (U, Ω _P

) is generated by ¹ _z dz.

Exercise 3. Show that ˇ H ¹ ( P ¹ , O _P

(−1)) = 0 in two different ways:

a) By using the Riemann-Roch theorem.

b) By a computation via ˇ Cech cohomology (use the open cover U by the standard open sets U ₀ and U 1 ).

c) Use a computation in ˇ Cech cohomology to show that ˇ H ¹ ( P ¹ , O _P

(−2)) 6= 0.

Exercise 4. Let X = C /Λ be a one-dimensional torus, so Ω X ' O X (as proven in the lecture) a) Show that H ^1,0 (X) ' C , so X is a compact curve of genus one.

b) Let D = x be an effective divisor, where x ∈ X is an arbitrary point, and denote by O X (D) the associated sheaf. Show that h ⁰ (X, O X (D)) = 1.

c) Deduce that h ¹ (X, O X (D)) = 0.

Exercise 5. (Picard group)

Let X be a complex manifold, and let π : L → X be a holomorphic line bundle, that is a holomorphic vector bundle of rank one. Let (U α ) _α∈A be an open covering of X such that L| U

is trivial. The transition functions of L are thus holomorphic maps g αβ : U α ∩ U β → C ^∗ .

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You can find the proof in Forster, Riemann surfaces, §13.

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a) Suppose that there exist holomorphic functions s α : U α → C ^∗ such that g _αβ = s _β

s α

on U α ∩ U β . Show that there exists a global section s : X → L such that s(x) 6= 0 for all x ∈ X.

Deduce that L is isomorphic to the trivial line bundle X .

b) Show that the isomorphism classes of holomorphic line bundles have a natural group structure given by the tensor product. We call this group the Picard group Pic(X).

c) Show that the Picard group is isomorphic to the ˇ Cech cohomology group ˇ H ¹ (X, O ^∗ _X ) ² . Hint: start by showing that a ˇ Cech cocycle defines transition functions of a line bundle.

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We compute the ˇ Cech cohomology for some sufficiently fine cover of X by polydiscs.

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