Last time, we discussed integration theory of functions along paths on Riemann surfaces, and then we decided that we wanted to talk about compact Riemann surfaces.  Unfortunately, there aren’t any holomorphic functions on them, and meromorphic functions are the wrong choice about what to integrate along curves.  Today, we’ll talk about the correct things to integrate, and some of their properties.

The issue of what the proper thing to integrate really boils down to understanding a bit of notation that most people ignore or at least don’t think much about in calculus: dz

On a first pass, dz is just something you write so that the profession doesn’t mark down your score.  On a second, it’s an infinitesimal that you can manipulate if you’re careful.  But really, it’s something else entirely.  One thing that can be agreed on is that if f(z) is a differentiable (holomorphic) function, then d(f(z))=f'(z)dz.  This is the familiar chain rule, if we say we can divide, we get that \frac{df}{dz}=f'(z).  But it turns out, unsurprisingly in many ways, that the chain rule is the key to everything.

Let X be any Riemann surface, it doesn’t have to be compact.  It can be covered by charts U_\alpha each of which is biholomorphic to an open subset of the complex plane, and with transition maps.  We define a holomorphic 1 form to be on each chart an object f(z)dz, where f(z) is holomorphic, such that if z=T(w) is a transition function, then we have that g(w)=f(T(w))T'(w).  This means that T transforms between g(w)dw and f(z)dz, so our actual object is coordinate independent, because it is well-defined on the overlaps.  THIS is what you want to integrate around curves.

And now, we integrate in exactly the usual way.  We pick a path to integrate along, parametrize, and then substitute into the 1-form and do the integral.  Of course, the story is hardly simple. First off, there’s the question of existence of holomorphic differentials! If we are really worried, we can define meromorphic 1-forms in the same way, but assuming that the function f(z) or g(w) is meromorphic rather than actually being holomorphic, but there turn out to be plenty of holomorphic 1-forms to use.  In fact, it’s a consequence of the Riemann-Roch theorem (and many other methods of proof) that the dimension of the space of holomorphic 1-forms on a compact Riemann surface, denoted by \Omega^1_X, is equal to the genus, and we can just take this to be the definition of the genus, if we so choose.

Now, of course, there’s a problem.  As discussed before, we get the same integral if we change to any homologous loop.  So integrals are path dependent.  If we integrate along a curve from P to Q, we get a different answer than if we first go in a loop, then integrate along the path from P to Q.  So integrals from one point to another are only well-defined up to the loops, and the key to understanding integration on a compact Riemann surface is to understand the integrals along loops.

So, we pick a basis for H_1(X,\mathbb{Z}).  And instead of picking an arbitrary basis, we pick a symplectic one.  We can always find a basis that makes the intersection form \left[\begin{array}{cc}0&-I\\I&0\end{array}\right], and we label the loops \alpha_i and \beta_j so that \alpha_i\cdot \alpha_j=0, \beta_i\cdot \beta_j=0 and \alpha_i\cdot \beta_j=\delta_{ij}.  Then we note that the pairing \Omega^1_X\times H_1(X,\mathbb{Z}) is non-degenerate, so we can take a basis for the 1-forms dual to the \alpha_i‘s, \omega_i.  This leaves only the values on the \beta_j‘s unknown, and we arrange them as \Omega_{ij}=\int_{\beta_j}\omega_i and call the matrix \Omega the period matrix of the Riemann surface X with respect to the basis \alpha_1,\ldots,\alpha_g,\beta_1,\ldots,\beta_g.

The Period Matrix will take a starring role in the next few posts, as we develop a few of its properties and use it to understand curves themselves a bit better directly, as well as in more complex constructions.

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