## Understanding Integration III: Jacobians

Now, we’re going to talk a bit about the geometry of the periods, which were completely analytic in nature.  As we mentioned, for a compact Riemann surface $X$, we have a period matrix $\Omega$ that encodes the complex integration theory on the surface.

We can use $\Omega$ to construct a lattice.  Inside of $\mathbb{C}^g$, there’s naturally a lattice $\mathbb{Z}^g$, but it’s not of full rank, the quotient isn’t compact.  However, if we add in another rank $g$ lattice that is independent from the natural one, that will be full rank.  So now, we take the lattice $\mathbb{Z}^g\oplus\Omega\mathbb{Z}^g$.  This is then a lattice in $\mathbb{C}^g$ of full rank that varies holomorphically with the pair of Riemann surface and symplectic basis of $H_1(X,\mathbb{Z})$.

This means that the quotient $\mathbb{C}^g/\mathbb{Z}^g\oplus\Omega\mathbb{Z}^g$ varies holomorphically with that data.  In fact, because a change of basis is linear, the transformation extends to all of $\mathbb{C}^g$, and so the quotient doesn’t depend on which symplectic basis we chose! Thus, this torus, which we will denote by $J(X)$ only depends on the Riemann surface, not on the symplectic basis.

We can also define a function on $\mathbb{C}^g$ out of $\Omega$, and it will behave decently with respect to the lattice.  Define $\theta(\Omega,z)$, the Riemann theta function, to be the multivariate Fourier series $\sum_{n\in\mathbb{Z}^g} \exp[\pi i(n^t\Omega n+2n^tz)]$.  It’s not hard to see from this definition that if we translate $z$ by an element of $\mathbb{Z}^g$, the function is invariant.  And it’s holomorphic (convergence is guaranteed because $\Omega$ is symmetric and has positive definite imaginary part) everywhere.  So it can’t be periodic in the other directions.  In those, you pick up an exponential factor.

Because of this, $\theta(\Omega,z)=0$ is periodic, and so we get a divisor on $J(X)$ called the Theta divisor, $\Theta$.  The geometry of this divisor is a rich and detailed subject of study, and we’ll talk about it a bit in later posts.  For now, the main point is that for any Riemann surface $X$, this divisor defines what is called a principal polarization.  One way to see that is because the bilinear form on the lattice is unimodular.  Another way is by directly computing that $\mathscr{O}(\Theta)$ has a unique (up to scaling) global section.  The third, and in many ways most geometric, way is by looking at the map $J(X)\to \hat{J(X)}$ given by $a\mapsto \Theta_a-\Theta$ where $\Theta_a$ is the translate of $\Theta$ by $a$.  This gives an isomorphism between $J(X)$ and the dual torus of degree zero divisors on $J(X)$.

Next time, we’ll have an interlude with an application of the geometry of the theta divisor, and then we’ll get back to constructing things with 1-forms on curves.