Homework 4 Complex Manifolds Math 241, Spring 1996, Berkeley CA C. McMullen Due Tuesday, 19 March 96

1. Let m_{0} be the ideal of germs of smooth functions vanishing at the origin in ℝ^{2}. Let m_{0}^{2} be the ideal generated by the products {fg : f, g ∈ m_{0}}. Prove m_{0}/m_{0}^{2} is 2-dimensional as a vector space over ℂ. (Hint: if f(x, y) vanishes to order 2, show f(x, 0) and f(x, y) − f(x, 0) both belong to m_{0}^{2}.)

2. Let π : X → ℂ̂ be a compact Riemann surface presented as a degree two covering of the sphere branched over a_{1}, …, a_{2n} ∈ ℂ, n > 0.

(a) Find the zeros and poles of π^{∗}(dz).

(b) Show for any i, the algebraic function

\frac{z^{i}}{\sqrt{(z − a_{1}) ⋯ (z − a_{2n})}}

determines a single-valued meromorphic function f_{i}(w) on X, z = π(w), and find its zeros and poles on X (with multiplicities).

(c) Find the greatest integer g such that the forms ω_{i}(w) = f_{i}(w)π^{∗}(dz), i = 0, 1, …, g − 1, are globally holomorphic on X.

(d) Where does ω_{i} vanish on X?

(e) Show these ω_{i} give a basis for Ω(X).

(f) Show H^{1}(X) = Ω(X) ⊕ \overline{Ω}(X).

(g) Show π_{∗}(ω) = 0 for any holomorphic 1-form ω on X.

(h) Suppose ω is a nonzero meromorphic 1-form on X. How many more zeros than poles does it have (with multiplicity)?

3. (a) Find two linearly independent solutions g_{1}(z), g_{2}(z) to the differential equation

g′′(z) + \frac{1}{4z^{2}} g(z) = 0

on ℂ^{∗}. (Hint: given g_{1}, you can find g_{2} by writting g_{2}(z) = g_{1}(z)f(z) and examining the equation satisfied by f.)

(b) Compute the monodromy matrix M = [m_{ij}] for your solutions. This is the matrix defined by h_{i} = ∑ m_{ij}g_{j}, where g_{i}, i = 1, 2 are the germs of solutions at z = 1 you just found, and h_{i} is the result of analytically continuing g_{i} once around S^{1}.

(c) More generally, find a basis of solutions and the monodromy matrix for the equation

g′′(z) + \frac{α}{z^{2}} g(z) = 0,

α an arbitrary complex number.

(d) Show that solutions to the nonlinear equation

η′(z) − \frac{1}{2}η(z)^{2} = ϕ(z)

correspond to solutions of the linear equation

g′′(z) + \frac{1}{2}ϕ(z)g(z) = 0

via η(z) = −2g′(z)/g(z).

(e) The Schwarzian derivative of an analytic function f(z) is defined by

Sf(z) = (\frac{f′′(z)}{f′(z)})′ − \frac{1}{2} (\frac{f′′(z)}{f′(z)})^{2} .

Describe how to find a solution to the equation

Sf(z) = ϕ(z)

for an arbitrary holomorphic function ϕ(z).

(f) Find a solution to the equation

Sf(z) = \frac{α}{z^{2}} .