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. (more…)