Abstract Nonsense

Crushing one theorem at a time

Riemann Surfaces (Pt. II)

Point of Post: This is a continuation of this post.

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Let \Lambda be a lattice in \mathbb{C} (i.e. a discrete subgroup of \mathbb{R}^2 of rank 2), then we know that \Lambda=\mathbb{Z}\omega_1+\mathbb{Z}\omega_2 for \mathbb{R}-independent complex numbers \omega_1 and \omega_2 (if this isn’t known to you, it’s just a fact, you can see a proof of this, as well as a lot of other cool math, in [6]). Consider then the quotient group X_\Lambda:=\mathbb{C}/\Lambda with the quotient topology inherited by the canonical projection \pi:\mathbb{C}\to X_\Lambda–note that X_\Lambda is connected, being the quotient of a connected space. We’d like to endow X_\Lambda with a complex structure. To do this, we call U\subseteq\mathbb{C} , an open set in \mathbb{C}small if x\ne y\text{ mod }\Lambda for any x,y\in U.

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We note now that if U is small, then \pi\mid_U is bijective and thus is a homeomorphism U\to\pi(U)\subseteq X_\Lambda. Now, if A=\{(U,\pi\mid_U):U\text{ is small and open}\} then we define \mathfrak{A}=\left\{(\pi(U),\varphi^{-1}):(U,\varphi)\in A\right\}. We claim that \mathfrak{A} is an atlas for X_\Lambda. Indeed, it’s clear that it’s a topological atlas, so it suffices to prove that the overlap maps are biholomorphic. To see this we suppose that (U,\varphi) and (V,\psi) in \mathfrak{A} are such that U\cap V\ne\varnothing. Now, we note that if z+\Lambda\in V then there exists a unique \omega_z\in\Lambda such that \psi^{-1}(z+\Lambda)=z+\omega_z. Thus, we see \psi\circ\varphi^{-1}:\varphi(U\cap V)\to\psi(U\cap V) is given by

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Now, since \psi\circ\varphi^{-1} is continuous, so is \psi\circ\varphi^{-1}-\text{id}_{\varphi(U\cap V)} and by inspection we see that this is the mapping \varphi(U\cap V)\to\Lambda. Now, since \Lambda is discrete this mapping must be locally constant. Thus, we see that for every point p\in\varphi(U\cap V) there exists a neighborhood for which \psi\circ\varphi^{-1} is just z\mapsto z+\omega for some \omega\in\Lambda. Thus, we see that \psi\circ\varphi^{-1} is locally holomorphic, and thus is holomorphic.

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We call X_\Lambda the \Lambda-torus. The terminology is appropriate, because topologically X_\Lambda is just \mathbb{T}^2. Indeed, as topological groups we have that

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X_\Lambda\approx \mathbb{R}^2/\mathbb{Z}^2\approx(\mathbb{R}/\mathbb{Z})^2\approx\mathbb{S}^1\times\mathbb{S}^1\approx\mathbb{T}^2

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Now, it is very tempting to try to conclude from this that X_\Lambda and X_{\Lambda'} are conformally equivalent (the isomorphism in the category of complex manifolds) for any lattices \Lambda and \Lambda'. I mean, we know that they are homeomorphic, and since we are in low-dimension this implies that they are diffeomorphic (since in dimension at most three, two smooth manifolds are homeomorphic if and only if they are diffeomorphic). But, we shall see that any torus coming from a lattice is biholomorphic to a torus coming from a lattice of the form \mathbb{Z}+\mathbb{Z}\tau with \text{Im}(\tau)>0. Thus, it suffices to decide when two tori of that form are different in our definition of different. A fascinating theorem, and one which leads to some seriously interesting math, is that if \Lambda=\mathbb{Z}+\mathbb{Z}\tau and \Lambda'=\mathbb{Z}+\mathbb{Z}\tau' then X_\Lambda\cong X_{\Lambda'} if and only if \tau and \tau' differ by an \text{SL}_2(\mathbb{Z}) action (where we are letting \text{SL}_2(\mathbb{Z}) act on the upper half-plane \mathbb{H} in the usual way). We shall prove this in a later post.

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[1] Varolin, Dror. Riemann Surfaces by Way of Complex Analytic Geometry. Providence, RI: American Mathematical Society, 2011. Print.

[2] Miranda, Rick. Algebraic Curves and Riemann Surfaces. Providence, RI: American Mathematical Society, 1995. Print.

[3] Forster, Otto. Lectures on Riemann Surfaces. New York: Springer-Verlag, 1981. Print.

[4] Conway, John B. Functions of One Complex Variable. New York: Springer-Verlag, 1978. Print.

[5] Gong, Sheng. Concise Complex Analysis. Singapore: World Scientific, 2001. Print

[6] Markley, Nelson Groh. Topological Groups: An Introduction. Hoboken, NJ: Wiley, 2010. Print.


October 2, 2012 - Posted by | Complex Analysis, Riemann Surfaces, Uncategorized | , , , , , ,

1 Comment »

  1. […] Riemann Surfaces (Pt. III) Point of Post: This is a continuation of this post. […]

    Pingback by Riemann Surfaces (Pt. III) « Abstract Nonsense | October 2, 2012 | Reply

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