Homework 6 Complex Manifolds Math 241, Spring 1996, Berkeley CA C. McMullen Due Tuesday, 23 April 96

1. Let X be a smooth manifold, and let E^{∗} be the multipicative sheaf of nowhere vanishing smooth functions. Relate H^{p}(X, E^{∗}) to cohomology with coefficients in a constant sheaf.

2. Compute dim H^{0}(ℂ̂, O_{D}) and dim H^{1}(ℂ̂, O_{D}) for an arbitrary divisor D on ℂ̂, and verify the Riemann-Roch theorem.

3. Let X be a complex torus. Find two divisors D and E on X of the same degree, such that dim H^{0}(X, O_{D}) ≠ dim H^{0}(X, O_{E}).

4. Given a short exact sequence of sheaves 0 → A → B → C → 0, describe the connecting homomorphism

δ^{∗} : H^{1}(X, C) → H^{2}(X, A)

on the level of representative cocycles, and show it is well-defined on the level of cohomology.

5. (a) Prove that for the exponential sequence 0 → ℤ → O → O^{∗} → 0 on a compact Riemann surface X, the map

δ^{∗} : H^{1}(X, O^{∗}) → H^{2}(X, ℤ) ≅ ℤ

is surjective.

(b) Give an explicit cocycle in H^{1}(X, O^{∗}) mapping to a generator of H^{2}(X, ℤ). (Hint: consider the line bundle associated to a divisor).

6. Let D be a divisor on a compact Riemann surface X. The sheaf O_{D} is generated by global sections if for every x ∈ X, there is an f ∈ O_{D}(X) such that the stalk (O_{D})_{x} is generated by f_{x} as an O_{x}-module. Show O_{D} is generated by global sections ⟺ for all x ∈ X there is a meromorphic function f with (f) + D ≥ 0 and ord(f, x) = −D(x) ⟺ for all P ∈ X, 1 + dim O_{D−P} (X) = dim O_{D}(X) ⟺ for all P, dim Ω_{−D}(X) = dim Ω_{P −D}(X).

7. Let K be a canonical divisor and let P ∈ X. Prove that O_{K} is generated by global sections but O_{K+P} is not.