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.

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About Charles Siegel

Charles Siegel is currently a postdoc at Kavli IPMU in Japan. He works on the geometry of the moduli space of curves.
This entry was posted in AG From the Beginning, Algebraic Geometry, Curves, Abelian Varieties, Complex Analysis. Bookmark the permalink.

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