Abstract Nonsense

Riemann Surfaces (Pt. II)

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

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$\Lambda$-tori

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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$z\overset{\varphi^{-1}}{\mapsto}z+\Lambda\overset{\psi}{\mapsto}\omega_z+z$

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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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References;

[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.