Homework 5 Complex Manifolds Math 241, Spring 1996, Berkeley CA C. McMullen Due Tuesday, 9 April 96

1. Let X = ℂ/(ℤ⊕ℤτ), where τ = 2i, and let π : X → ℂ̂ be a degree two holomorphic branched cover, branched over {0, 1, ∞, λ}, where 1 < λ. Compute λ. (Hint: first find the answer for τ = i, then use the fact that there is a covering map from X to ℂ/(ℤ ⊕ ℤi).)

2. Given an explicit formula for the quadratic differential ϕ = Sπ^{−1} on X = ℂ̂ − {z : z^{4} = 1}, where S is the Schwarzian derivative and π : ℍ → X is the universal covering map.

3. Let f : X → Y be a continuous map between topological spaces, and let F and G be sheaves on X and Y.

(a) Let (f_{∗}F)(U) = F(f^{−1}(U)) Show f_{∗}F is a sheaf.

(b) Give a sensible definition of the sheaf f^{−1}(G), and show (f^{−1}G)_{x} ≅ G_{f(x)}. (Hint: pull back the espace étalé of G and consider its space of sections.)

(c) For F and G sheaves of abelian groups, show that there is a natural

bijection

Hom_{X}(f^{−1}G, F) = Hom_{Y} (G, f_{∗}F),

where Hom denotes the set of sheaf homomorphisms.

(Cf. Hartshorne, Algebraic Geometry, pp.65-68.)

4. Let X be a smooth manifold. Prove that the exact 1-forms on X form a closed subset among all smooth 1-forms in the C^{∞} topology.

5. Let ϕ be the distribution on ℝ defined by ∫ f(x)ϕ(x) dx = f′(0). Construct a sequence of explicit smooth functions ϕ_{n}(x) converging to ϕ in D(ℝ). (Recall this means ∫ f(x)ϕ_{n}(x) dx → f′(0) for all smooth, compactly supported f.)