Editing Logarithmic form (section)

In [[algebraic geometry]] and the theory of [[complex manifold]]s, a '''logarithmic''' [[differential form]] is a differential form with [[pole (complex analysis)|poles]] of a certain kind. The concept was introduced by [[Pierre Deligne]].<ref>Deligne (1970), section II.3.</ref> 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 (algebraic geometry)|divisor]] (a sum of distinct codimension-1 complex subspaces), and ω a holomorphic ''p''-form on ''X''−''D''. If both ω and ''d''ω 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 [[Sheaf (mathematics)|subsheaf]] of the meromorphic ''p''-forms on ''X'', denoted
:<math>\Omega^p_X(\log D).</math>

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