More about 1-forms on Riemann surfaces

It’s now time to do some more manipulations with differential forms on a Riemann surface. This will establish several notions we will need in the future.

The Hodge star

Given the 1-form ${\omega}$ in local coordinates as ${u dz + v d\bar{z}}$ , define

$\displaystyle ^*{\omega} := -iu dz + iv d\bar{z} .$

In other words, given the decomposition ${T^*(X) = T^{*(1,0)}(X) \oplus T^{*(0,1)}(X)}$ , we act by ${-i}$ on the first sumamand and by ${i}$ on the second. This shows that the operation is well-defined. Note that ${^*{}}$ is conjugate-linear and ${^*{}^2 = -1}$ . Also, we see that ${^*{} dx = dy, ^*{dy} = -dx}$ if ${z = x + iy}$ . This operation is called the Hodge star.

From the latter description of the Hodge star we see that for any smooth ${f}$ ,

$\displaystyle d ^*{} df = d( -if_z dz + if_{\bar{z}} d\bar{z}) = 2i f_{z \bar{z}} dz \wedge d\bar{z}.$

From the definitions of ${f_{z}, f_{\bar{z}}}$ , this can be written as ${-2i \Delta f dz \wedge d\bar{z}}$ if ${\Delta}$ is the usual Laplacian with respect to the local coordinates ${x,y}$ .

The Hodge star allows us to define co things. A form ${\omega}$ is co-closed if ${d ^*{} \omega = 0}$ ; it is co-exact if ${\omega = ^* df}$ for ${f}$ smooth.

Conjugation

Given ${\omega = u dz + v d \bar{z}}$ , define

$\displaystyle \bar{\omega} := \bar{v} dz + \bar{u} d \bar{z} .$

It can be checked that, since the transition functions are holomorphic, the definition transforms appropriately. Moreover, writing ${\omega = f dx + gdy}$ , we have ${\bar{\omega} = \bar{f} dx + \bar{g} dy}$ ; this is another way of seeing that conjugation is well-defined (i.e., it comes from the map ${\mathbb{C} \otimes T^*_{\mathbb{R}}(X) \rightarrow \mathbb{C} \otimes T^*_{\mathbb{R}}(X)}$ via complex conjugation on the first factor, where the ${\mathbb{R}}$ subscript refers to the real tangent space).

Moreover, the operators ${^*}$ and ${\bar{}}$ anticommute.

The conjugation will be necessary when we place a Hermitian inner product on the space of complex-valued differentials.

Harmonic and holomorphic differentials

The reason we need these is that globally defined harmonic or holomorphic functions are constant on a compact surface; this is basically the maximum principle. So a differential ${\omega}$ is harmonic if locally ${\omega = df}$ for ${f}$ harmonic (i.e., ${\Delta f = 0}$ in local coordinates—note that the choice of local coordinates does not matter by the discussion above on the Hodge star!).

There is a basic and standard criterion for harmonicity:

Proposition 1 ${\omega}$ is harmonic iff it is closed and co-closed.

If ${\omega = df}$ with ${f}$ harmonic locally, then ${\omega}$ is evidently closed, and in local coordinates ${z=x+iy}$ , ${d ^*{} \omega = d ^*{} df = \Delta f = 0}$ . Conversely, a closed form is locally exact (Poincare lemma), so locally ${\omega = df}$ for some smooth ${f}$ . If ${\omega}$ is co-closed then reversing the reasoning shows ${f}$ is harmonic.

Next, we do the same for holomorphic differentials. ${\omega}$ is holomorphic if locally ${\omega = df}$ for ${f}$ holomorphic. Note in particular that in local coordinates ${\omega}$ is holomorphic iff it is of the form

$\displaystyle \omega = u(z) dz$

for ${u}$ holomorphic. In particular, ${^*{} \omega = -i \omega}$ . Also, ${\omega}$ is harmonic.

This is actually a sufficient condition as well for holomorphicity.

Proposition 2 ${\omega}$ is holomorphic iff ${\omega}$ is closed and ${^*{} \omega = -i \omega}$ .

We have proved the “only if” statement. Assume ${\omega}$ satisfies the latter two conditions; then, by the second, locally

$\displaystyle \omega = u(z) dz$

and by the first, ${u_{\bar{z}} = 0}$ , which is the Cauchy-Riemann equation for ${u}$ . In particular, ${\partial f}$ is holomorphic if ${f}$ is harmonic.

Integration

A Riemann surface is an oriented two-manifold: the holomorphic changes of coordinates have Jacobians of the form

$\displaystyle \begin{bmatrix} A & B \\ -B & A \end{bmatrix}$

and consequently have positive determinants. Thus we can talk about integration.

If ${D}$ is a compact submanifold with boundary ${\partial D}$ and ${\omega}$ a 1-form on ${D}$ , then

$\displaystyle \int_{\partial D} \omega = \int_{D} d \omega$

by Stokes’ theorem.

It is useful to collect some corollaries of this formula.

First, replacing ${\omega}$ by ${f \omega}$ :

$\displaystyle \int_{\partial D} f \omega = \int_{D} f d \omega + \int_D df \wedge \omega.$

If ${f}$ or ${\omega}$ has compact support, then we can replace ${D}$ by ${X}$ —so the formula becomes

$\displaystyle \int_X \omega \wedge df = \int_{X} f d \omega.$

The inner product

Define a measurable 1-form to be a measurable section ${\omega}$ of the complexified tangent bundle, where measurable means that in local coordinates ${\omega = u dz + v d\bar{z}}$ for ${u,v}$ measurable. Define the inner product of 1-forms ${\omega, \Omega}$ by

$\displaystyle (\omega_1, \omega_2) = \int_X \omega_1 \wedge ^*{}( \bar{\omega_2}) ,$

whenever the integral is defined. Say ${\omega_1, \omega_2}$ are supported in a coordinate neighborhood ${U}$ with coordinate ${z}$ . Then if ${\omega_1 = u_1 dz + v_1 d\bar{z}, \omega_2 = u_2 dz + v_2 d \bar{z}}$ , then

$\displaystyle ^*{} \bar{\omega_2} = -i \bar{v_2} dz + i \bar{u_2} d \bar{z}$

$\displaystyle (\omega_1, \omega_2) = \int_U i( u_1 \bar{u_2 } + v_1 \bar{v_2} dz \wedge d \bar{z} = \frac{1}{2} \int_U ( u_1 \bar{u_2 } + v_1 \bar{v_2} dx \wedge dy .$

In particular, by using a partition of unity, we see that the inner product ${(.,.)}$ is Hermitian and positive-definite. Define ${L^2(X)}$ to be the set of 1-forms ${\omega}$ with ${||\omega|| := (\omega, \omega)^{1/2}< \infty}$ , modulo those which coincide almost everywhere. This is actually a Hilbert space; the proof is the same as it is for the normal ${L^p}$ spaces. Namely, given differentials ${\omega_i}$ with ${||\omega_i - \omega_{i+1}||< 2^{-i}}$ (we can always extract from a Cauchy sequence such a subsequence), then the sum

$\displaystyle \omega_1 + \sum ( \omega_i - \omega_{i+1})$

converges almost everywhere; this is seen by writing in local coordinates and using Fatou’s lemma. Similarly as in the usual case, the aforementioned sum is the limit of the sequence ${\{\omega_i\}}$ .

Next, I will explain the decomposition of $L^2(X)$ .

Climbing Mount Bourbaki