Homework 2 Complex Manifolds Math 241, Spring 1996, Berkeley CA C. McMullen Due Tuesday, 20 February 96

1. For a smooth function f(z) on ℂ with z = x + iy, let

Δf = \frac{∂^{2}f}{∂x^{2}} + \frac{∂^{2}f}{∂y^{2}} and

∂\overline{∂}f(z) = \frac{∂^{2}f(z)}{∂z∂\overline{z}} dz ∧ d\overline{z}.

(a) Show ∂\overline{∂}f is proportional to Δfdx ∧ dy, and find the constant of proportionality.

(b) Using the idea ∂\overline{∂} log |z| is a δ-mass, give an integral formula to solve the equation ∂\overline{∂}f = g, where g(z) dz ∧ d\overline{z} is a smooth, compactly supported (1, 1)-form on ℂ.

(c) Using Stokes’ theorem, prove your solution is correct.

2. What is the Galois group of ℂ(z) over ℂ?

3. Show every proper field extension K of ℂ contains a subfield isomorphic to ℂ(z). In particular dim_{ℂ} K = ∞.

4. (a) Show the maps f : ℂ̂ → ℂ̂ given by f(z) = z^{4} and f(z) = z^{2} + 1/z^{2} determine Galois field extensions K/k, where K and k are isomorphic to ℂ(z).

(b) Determine their Galois groups.

(c) Show any degree 4 Galois extension K/k with K, k ≅ ℂ(z) is isomorphic to one of these examples.

5. For U an open set in a Riemann surface X, let F(U) be the vector space of bounded analytic functions, with the obvious restriction maps when V ⊂ U. Show F is a presheaf but not a sheaf.

6. Similarly, let F(U) = O^{∗}(U)/ exp O(U), the quotient of the group of nowhere vanishing holomorphic functions on U by those of the form e^{f}. Show F is a presheaf but not a sheaf.

7. (a) Show that on ℂ̂ one can find a meromorphic function with prescribed Laurent tails at a finite set of points z_{1}, …, z_{n}; that is, such that f has poles only at the points z_{i} and for a prespecified b_{ij}, i = 1, …, n, j = 1, …, m we have

f(z) = ∑_{j=1}^{m} \frac{b_{ij}}{(z − z_{i})^{j}} + something holomorphic

near each z_{i}.

(b) Show that a system of Laurent tails is the same as a global section of the quotient sheaf M/O, where M and O are the sheaves of meromorphic and holomorphic functions.

(c) Show that on a complex torus X = ℂ/Λ the Laurent tails of a meromorphic function cannot be prescribed arbitrarily.

(d) Show that the Laurent tails of a meromorphic 1-form ω(z)dz on ℂ̂ cannot be prescribed arbitrarily, and characterize those tails that can be realized.