The following is bugging the heck out of me.
Let be a complete algebraic curve of genus defined over the complex numbers. Then is the abelian group gotten by taking formal sums of points on and modding out by the relation that, for any nonzero meromorphic function , the sum of the zeroes of equals the sum of the poles. Such a formal (finite) sum of points of is called a “divisor”. If is a divisor then the degree of is . Since any meromorphic function has equally many zeroes and poles, we can define the degree of a point in ; let be the component of coming from divisors of degree . Then is an abelian variety of dimension and is a principal homogeneous space for . Obviously, for any nonnegative integer , there is a map called “take the sum”. When , the image of this map is called the Theta divisor and it is a complex subvariety of codimension one in .
That’s one description of the Theta divisor. Here is another. Let be the vector space of global homolomorphic 1-forms on and let be the dual vector space. There is an injection of into . Namely, if is a 1-cycle in and is a global holomorphic 1-form, set . Then there is an isomorphism defined as follows: for any degree zero divisor , choose a 1-chain with and map to the functional .
One then writes down a holomorphic function on , whose definition I have not quite fully internalized. is not quite -periodic, but it has the property that, for any , the ratio for some functions and on . In particular, the zero locus of is -periodic. The Theta divisor is the image of this zero locus in .
So, in the first perspective I gave, the Theta divisor is a hypersurface in , whereas, in the second, it is a hypersurface in . There is no canonical isomorphism between these two, so what gives?
Now that I have written all this out, I see sort of what the answer must be. To define , one does something like choose a basis of , and this choice must somehow give me an isomorphism between and . But I’d still appreciate hearing from anyone who has thought through the details and has an elegant way to present them.