HW205sol - 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- { } by k ∈ Z , z ∈ C- { } ⇒ k · z = λ k z. Explicitly find a lattice Λ ⊂ C such that the Hopf manifold ( C-{ } ) / Z and the torus C / Λ are biholomorphic. Solutions (1) Let π and π be the quotient maps V → V/ Λ and V → V / Λ respectively. As a first step, note that there exists a continuous map e F from V to V such that π ◦ e F = F ◦ π because V is simply-connected and π a covering map. Note that e F is holo- morphic, as the tori carry charts that locally invert their covering maps....
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This note was uploaded on 07/08/2011 for the course MTG 6256 taught by Professor Robinson during the Spring '09 term at University of Florida.

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HW205sol - Differential Geometry 2 Homework 05 1. Let Λ...

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