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. The concept was introduced by Deligne.
Let X be a complex manifold, D ⊂ X a divisor, and ω a holomorphic p-form on X−D. If ω and dω have a pole of order at most one along D, then ω is said to have a logarithmic pole along D. ω 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 ω 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.
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