Riemann Surfaces (Pt. II)
Point of Post: This is a continuation of this post.
Let be a lattice in (i.e. a discrete subgroup of of rank ), then we know that for -independent complex numbers and (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 ). Consider then the quotient group with the quotient topology inherited by the canonical projection –note that is connected, being the quotient of a connected space. We’d like to endow with a complex structure. To do this, we call , an open set in , small if for any .
We note now that if is small, then is bijective and thus is a homeomorphism . Now, if then we define . We claim that is an atlas for . 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 and in are such that . Now, we note that if then there exists a unique such that . Thus, we see is given by
Now, since is continuous, so is and by inspection we see that this is the mapping . Now, since is discrete this mapping must be locally constant. Thus, we see that for every point there exists a neighborhood for which is just for some . Thus, we see that is locally holomorphic, and thus is holomorphic.
We call the -torus. The terminology is appropriate, because topologically is just . Indeed, as topological groups we have that
Now, it is very tempting to try to conclude from this that and are conformally equivalent (the isomorphism in the category of complex manifolds) for any lattices and . 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 with . 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 and then if and only if and differ by an action (where we are letting act on the upper half-plane in the usual way). We shall prove this in a later post.
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 Forster, Otto. Lectures on Riemann Surfaces. New York: Springer-Verlag, 1981. Print.
 Conway, John B. Functions of One Complex Variable. New York: Springer-Verlag, 1978. Print.
 Gong, Sheng. Concise Complex Analysis. Singapore: World Scientific, 2001. Print
 Markley, Nelson Groh. Topological Groups: An Introduction. Hoboken, NJ: Wiley, 2010. Print.