HW205sol - Dierential Geometry 2 Homework 05 1 Let V and V...

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Differential Geometry 2 Homework 05 1. Let Λ V and Λ 0 V 0 be lattices in finite-dimensional complex vector spaces. Show that if F ∈ O ( V/ Λ , V 0 / Λ 0 ) is a holomorphic map between the corresponding tori then there exist a linear map A L ( V, V 0 ) such that A (Λ) Λ 0 and a vector b V 0 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 = λ k z. Explicitly find a lattice Λ C such that the Hopf manifold ( C - { 0 } ) / Z and the torus C / Λ are biholomorphic. Solutions (1) Let π and π 0 be the quotient maps V V/ Λ and V 0 V 0 / Λ 0 respectively. As a first step, note that there exists a continuous map e F from V to V 0 such that π 0 e F = F π because V is simply-connected and π 0 a covering map. Note that e F is holo- morphic, as the tori carry charts that locally invert their covering maps. Now, if z V and λ Λ then π 0 e F ( z + λ ) = F ( π ( z + λ )) = F ( π ( z )) = π 0 e F ( z ) so that e F ( z + λ ) = e F ( z ) + μ ( λ, z ) where μ ( λ, z ) Λ ostensibly depends on λ and z . In fact, the dependence on
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