Intersection Number - Definition For Riemann Surfaces

Definition For Riemann Surfaces

Let X be a Riemann surface. Then the intersection number of two closed curves on X has a simple definition in terms of an integral. For every closed curve c on X (i.e., smooth function ), we can associate a differential form with the pleasant property that integrals along c can be calculated by integrals over X:

, for every closed (1-)differential on X,

where is the wedge product of differentials, and is the hodge star. Then the intersection number of two closed curves, a and b, on X is defined as

.

The have an intuitive definition as follows. They are a sort of dirac delta along the curve c, accomplished by taking the differential of a unit step function that drops from 1 to 0 across c. More formally, we begin by defining for a simple closed curve c on X, a function f_c by letting be a small strip around c in the shape of an annulus. Name the left and right parts of as and . Then take a smaller sub-strip around c, with left and right parts and . Then define f_c by

.

The definition is then expanded to arbitrary closed curves. Every closed curve c on X is homologous to for some simple closed curves c_i, that is,

, for every differential .

Define the by

.