Homework 7 Complex Manifolds Math 241, Spring 1996, Berkeley CA C. McMullen Due Thursday, 2 May 96

1. Let X be a compact complex manifold, and let E → X be a holomorphic line bundle over X. Suppose there is a holomorphic vector bundle F → X such that E ⊕ F is isomorphic to X × ℂ^{n}, the trivial bundle of dimension n = 1 + dim(F).

Show that either E is trivial, or E has no holomorphic sections other than the zero section.

2. Let X = ℙ^{1} × ℙ^{1} be a product of two projective lines, and let π_{1} : X → ℙ^{1} and π_{2} : X → ℙ^{1} be the two projection maps. Let O(d, e) be the line bundle π_{1}^{∗}O(d) ⊗ π_{2}^{∗}O(e), the product of the pullback of two standard line bundles on ℙ^{1}.

Describe the space of sections H^{0}(X, O(d, e)), and in particular compute its dimension.

3. Let X be a compact Riemann surface X of genus g ≥ 1. Show the map f : X → Pic(X) sending P to the line bundle L_{P} is injective.