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Consider a complex lattice $\Lambda \subset \mathbb C$ generated by some real basis $\{w_1,w_2 \}$. I want to know why the quotient group $\mathbb C/ \Lambda$ has a unique structure as Riemann surface.

SamC
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  • Modding by the lattice identifies the left and right edge of the fundamental parallelogram making a tube and also identifies the top and bottom edges making a torus. – John McGee Sep 26 '15 at 12:13
  • @JohnMcGee I can picture how $\mathbb C / \Lambda \cong T $ a torus. But if I want to show something is a Riemann surface, I need to show that is a complex one manifold right? So I want to find charts and holomorphic transition maps. This is the part I can't understand, how to write down those maps. – SamC Sep 26 '15 at 12:23
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    I'm not sure what you mean by a 'unique' Riemann surface structure. When you present it as $\mathbb{C}/\Lambda$, it certainly has an obvious one, but as a manifold, it has a one-dimensional (over $\mathbb{C}) family of such structures. – Rhys Sep 26 '15 at 12:33
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    @Rhys Will that help if I add "such that the canonical projection $\mathbb C \to \mathbb C / \Lambda$ is holomorphic" at the end. – SamC Sep 26 '15 at 12:39
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    Yes that will help. And for the charts, just restrict the canonical projection to small open sets - small means they don't contain any pair of points whose difference lies in $\Lambda$ - and take their inverses as charts. The transition maps between different charts are translations on every connected component of the intersection of the coordinate neighbourhoods, hence that gives you a complex structure. – Daniel Fischer Sep 26 '15 at 12:54

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