Logarithmic Form

Logarithmic Form

In contexts including complex manifolds and algebraic geometry, a logarithmic differential form is a meromorphic differential form with poles of a certain kind.

Let X be a complex manifold, and a divisor and a holomorphic p-form on . If and have a pole of order at most one along D, then is said to have a logarithmic pole along D. is also known as a logarithmic p-form. The logarithmic p-forms make up a subsheaf of the meromorphic p-forms on X with a pole along D, denoted .

In the theory of Riemann surfaces, one encounters logarithmic one-forms which have the local expression

for some meromorphic function (resp. rational function), where g is holomorphic and non-vanishing at 0, and m is the order of f at 0.. That is, for some open covering, there are local representations of this differential form as a logarithmic derivative (modified slightly with the exterior derivative d in place of the usual differential operator d/dz). Observe that has only simple poles with integer residues. On higher dimensional complex manifolds, the Poincaré residue is used to describe the distinctive behavior of logarithmic forms along poles.