Algebraic geometry | Complex manifolds

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 h1,0. The differentials of the first kind, when integrated along paths, give rise to integrals that generalise the elliptic integrals to all curves over the complex numbers. They include for example the hyperelliptic 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 (as g = [(1+ deg Q)/2]). 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. (Wikipedia).

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Related pages

Albanese variety | Elliptic integral | Coherent sheaf | Differential form | Elliptic function | Algebraic variety | Irregularity of a surface | Complex manifold | Algebraic surface | Hodge theory | Point at infinity | Genus (mathematics) | Hyperelliptic curve | Composition series | Poincaré residue | Riemann surface | Mathematics | Jacobian variety | Kähler differential | Square-free polynomial | Generalized Jacobian | Linear combination | Logarithmic derivative | Affine space | Algebraic group | Algebraic curve | Complex number | Theta function | Abelian integral | Logarithmic form