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 in local coordinates as , define

In other words, given the decomposition , we act by on the first sumamand and by on the second. This shows that the operation is well-defined. Note that is conjugate-linear and . Also, we see that if . This operation is called the **Hodge star**.

From the latter description of the Hodge star we see that for any smooth ,

From the definitions of , this can be written as if is the usual Laplacian with respect to the local coordinates .

The Hodge star allows us to define co things. A form is **co-closed** if ; it is **co-exact** if for smooth.

**Conjugation **

Given , define

It can be checked that, since the transition functions are holomorphic, the definition transforms appropriately. Moreover, writing , we have ; this is another way of seeing that conjugation is well-defined (i.e., it comes from the map via complex conjugation on the first factor, where the subscript refers to the real tangent space).

Moreover, the operators and **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 is **harmonic** if locally for harmonic (i.e., 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 1is harmonic iff it is closed and co-closed.

If with harmonic locally, then is evidently closed, and in local coordinates , . Conversely, a closed form is locally exact (Poincare lemma), so locally for some smooth . If is co-closed then reversing the reasoning shows is harmonic.

Next, we do the same for holomorphic differentials. is **holomorphic** if locally for holomorphic. Note in particular that in local coordinates is holomorphic iff it is of the form

for holomorphic. In particular, . Also, is harmonic.

This is actually a sufficient condition as well for holomorphicity.

Proposition 2is holomorphic iff is closed and .

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

and by the first, , which is the Cauchy-Riemann equation for . In particular, is holomorphic if is harmonic.

**Integration **

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

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

If is a compact submanifold with boundary and a 1-form on , then

by Stokes’ theorem.

It is useful to collect some corollaries of this formula.

First, replacing by :

If or has compact support, then we can replace by —so the formula becomes

**The inner product **

Define a **measurable 1-form** to be a measurable section of the complexified tangent bundle, where measurable means that in local coordinates for measurable. Define the **inner product** of 1-forms by

whenever the integral is defined. Say are supported in a coordinate neighborhood with coordinate . Then if , then

so

In particular, by using a partition of unity, we see that the inner product is Hermitian and positive-definite. Define to be the set of 1-forms with , modulo those which coincide almost everywhere. This is actually a Hilbert space; the proof is the same as it is for the normal spaces. Namely, given differentials with (we can always extract from a Cauchy sequence such a subsequence), then the sum

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 .

Next, I will explain the decomposition of .

December 6, 2009 at 7:48 pm

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