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}

so

\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).

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