Homework 3 Complex Manifolds Math 241, Spring 1996, Berkeley CA C. McMullen Due Tuesday, 5 March 96

1. Let X be a compact Riemann surface. An endomorphism of X is a nonconstant holomorphic map f : X → X; the set of all endomorphisms forms a semigroup End(X) under composition, whose invertible elements are the group Aut(X).

Let X = ℂ/(ℤ ⊕ τℤ). Then X is both a Riemann surface and a group under addition.

(a) Describe Aut(X) for τ = i, 2i and e^{2πi/3}.

(b) Let End_{0}(X) denote the endomorphisms of X that fix the identity element. Show every f ∈ End_{0}(X) is a group homomorphism, and End_{0}(X) is a commutative ring with respect to the operations (fg)(x) = f ∘ g(x), and (f + g)(x) = f(x) + g(x).

(c) Describe End_{0}(X) for τ = i, 2i and e^{2πi/3}.

(d) For what values of τ is End_{0}(X) bigger than ℤ?

2. Show End(X) = Aut(X) if the genus of X is 2 or more.

3. Let X be a connected compact complex manifold X that is also a Lie group, such that multiplication m : X × X → X is holomorphic. Show X is abelian. (Hints: show any holomorphic group homomorphism X → GL_{n}(ℂ) is trivial; then consider the adjoint action of X on its Lie algebra.)

4. Find nonconstant meromorphic functions f(z) and g(z) on ℂ such that f(z)^{3} + g(z)^{3} = 1. You may use the fact that for any a, b ∈ ℂ such that the discriminant of the polynomial 4x^{3} + ax + b is nonzero, there is a lattice Λ whose Weierstrass ℘-function satisfies

℘′(z)^{2} = 4℘(z)^{3} + a℘(z) + b.  (0.1)

5. (a) Define the push-forward f_{∗}(ω) of a meromorphic 1-form ω under a proper holomorphic map f : X → Y between Riemann surfaces, by analogy with pushforward of meromorphic functions. (It should have the property that f_{∗}f^{∗}ω = deg(f)ω.)

(b) Given ω(z) = ∑_{0}^{∞} a_{n}z^{n} dz on the disk, compute f_{∗}(ω) for f(z) = z^{d}.

(c) Let X be a compact Riemann surface. Show for any nonconstant meromorphic function f : X → ℂ̂, and any holomorphic 1-form ω on X, we have f_{∗}(ω) = 0.

6. Let ϕ_{U} : F(U) → G(U) be a morphism of sheaves of groups (defined for every open set U in a space X). Show K(U) = Ker ϕ_{U} is a sheaf, and R(U) = G(U)/ϕ_{U}(F(U)) is a presheaf. Given an example where R(U) is not a sheaf.

7. Give an example of a germ f ∈ O_{a}, a ∈ X such that π : (Y, b) → (X, a), g ∈ O(Y) gives the maximal analytic continuation of f, but π is not a covering map to its image.