Differential of The First Kind

In mathematics, differential of the first kind is a traditional term used in the theories of Riemann surfaces (more generally, complex manifolds) and algebraic curves (more generally, algebraic varieties), for everywhere-regular differential 1-forms. Given a complex manifold M, a differential of the first kind ω is therefore the same thing as a 1-form that is everywhere holomorphic; on an algebraic variety V that is non-singular it would be a global section of the coherent sheaf Ω1 of Kähler differentials. In either case the definition has its origins in the theory of abelian integrals.

The dimension of the space of differentials of the first kind, by means of this identification, is the Hodge number

h0,1.

The differentials of the first kind, when integrated along paths, give rise to integrals that generalise to all curves over the complex numbers the elliptic integrals. They include for example the hyper-elliptic integrals of type

where Q is a square-free polynomial of any given degree > 4. The allowable power k has to be determined by analysis of the possible pole at the point at infinity on the corresponding hyperelliptic curve. When this is done, one finds that the condition is

k ≤ g − 1,

or in other words, k at most 1 for degree of Q 5 or 6, at most 2 for degree 7 or 8, and so on.

Quite generally, as this example illustrates, for a compact Riemann surface or algebraic curve, the Hodge number is the genus g. For the case of algebraic surfaces, this is the quantity known classically as the irregularity q. It is also, in general, the dimension of the Albanese variety, which takes the place of the Jacobian variety.