HW205 - Differential Geometry 2 Homework 05 1. Let Λ ⊂...

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Unformatted text preview: Differential Geometry 2 Homework 05 1. Let Λ ⊂ V and Λ ⊂ V be lattices in finite-dimensional complex vector spaces. Show that if F ∈ O(V /Λ, V /Λ ) is a holomorphic map between the corresponding tori then there exist a linear map A ∈ L(V, V ) such that A(Λ) ⊂ Λ and a vector b ∈ V such that z ∈ V ⇒ F [z ] = [Az + b]. 2. Let 0 < λ < 1 and let Z act on C − {0} by k ∈ Z, z ∈ C − {0} ⇒ k · z = λk z. Explicitly find a lattice Λ ⊂ C such that the Hopf manifold (C − {0})/Z and the torus C/Λ are biholomorphic. 1 ...
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