Logarithmic form

In algebraic geometry and the theory of complex manifolds, a logarithmic differential form is a differential form with poles of a certain kind. The concept was introduced by Pierre Deligne. In short, logarithmic differentials have the mildest possible singularities needed in order to give information about an open submanifold (the complement of the divisor of poles). (This idea is made precise by several versions of de Rham's theorem discussed below.)

Let X be a complex manifold, D ⊂ X a reduced divisor (a sum of distinct codimension-1 complex subspaces), and ω a holomorphic p-form on X−D. If both ω and dω have a pole of order at most 1 along D, then ω is said to have a logarithmic pole along D. ω is also known as a logarithmic p-form. The p-forms with log poles along D form a subsheaf of the meromorphic p-forms on X, denoted

${\displaystyle \Omega _{X}^{p}(\log D).}$

The name comes from the fact that in complex analysis, ${\displaystyle d(\log z)=dz/z}$; here ${\displaystyle dz/z}$ is a typical example of a 1-form on the complex numbers C with a logarithmic pole at the origin. Differential forms such as ${\displaystyle dz/z}$ make sense in a purely algebraic context, where there is no analog of the logarithm function.