Before we go on to discuss what makes a complex torus algebraic, let us perhaps return to the case of elliptic curves. The claim is that all elliptic curves are algebraic; other than the one case explicitly provided in the first post, this has by no means been shown, so let us dwell on this a little further.
There are a few ways that we can see this fact. We will go over the most obvious one first.
Fix an element in the upper half-plane (and so in effect, fix a lattice and hence an elliptic curve). Consider the complex function
where by convention the primed summation means that we sum over all non-zero elements of the lattice. This function is called the Weirstrass -function. Note that this function satisfies
for all and . While it is not holomorphic (it has a pole of order 2 at every lattice point), it is meromorphic, and it is translation invariant for every element in the lattice. It thus yields a well-defined meromorphic function on the elliptic curve .
We claim the following relationship holds:
where are given by the expressions
It is interesting to remark that these expressions are in fact the Eisenstein series of weights 4 and 6, respectively. This can be checked by verifying how they transform as functions of under the two transformations
and noting that they are thus modular forms of weights 4 and 6, respectively; if you happen to know that these spaces are one-dimensional, then you are done. If not, then the following exercise is somewhat instructive.
Exercise Show that the function (where we now mention its explicit -dependence) can also be written as
It is helpful to start with the identity
which can be checked by looking at the zeros and poles of these two functions.
Now, the shown identity can be verified by comparing the poles on the left- and right-hand sides of the expression; as they match up, the two expressions must be equal (why?)
The point of all this, of course, is that the expression above provides us an explicit description of our elliptic curve as the affine plane curve
and in particular, we see that every elliptic curve can be written as the curve cut out by a cubic polynomial.
Now, while this is all true, it is not particularly enlightening. If, after reading this, someone were to ask you “Why are elliptic curves projective?”, all you could answer would be “Because the Wierstrass -function exists and satisfies a certain differential equation”. It would not answer, in any way, the question as to why complex tori are not necessarily projective. So if our goal was to answer that question, then this approach, while interesting, fails.
For us to move on, we need to perhaps consider more what exactly it means for a variety to be projective.