Logarithmic form

In <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a> and the theory of <a href="/facts/Complex_manifold/DlvPnpzm">complex manifolds</a>, a logarithmic <a href="/facts/Differential_form/T9gyHtMv">differential form</a> is a differential form with <a href="/facts/Pole_(complex_analysis)/gnA3U8VP">poles</a> of a certain kind. The concept was introduced by <a href="/facts/Pierre_Deligne/wYC8fhfq">Pierre Deligne</a>. 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 <a href="/facts/De_Rham%2527s_theorem/4qGKeJq0">de Rham's theorem</a> discussed below.)
Let X be a complex manifold, D ⊂ X a reduced <a href="/facts/Divisor_(algebraic_geometry)/V0QDypeo">divisor</a> (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 <a href="/facts/Sheaf_(mathematics)/MM3hcRw4">subsheaf</a> of the meromorphic p-forms on X, denoted

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

The name comes from the fact that in <a href="/facts/Complex_analysis/hgCoQlH0">complex analysis</a>, 
 
 
 
 d
 (
 log
 ⁡
 z
 )
 =
 d
 z
 
 /
 
 z
 
 
 {\displaystyle d(\log z)=dz/z}
 
; here 
 
 
 
 d
 z
 
 /
 
 z
 
 
 {\displaystyle dz/z}
 
 is a typical example of a 1-form on the <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a> C with a logarithmic pole at the origin. Differential forms such as 
 
 
 
 d
 z
 
 /
 
 z
 
 
 {\displaystyle dz/z}
 
 make sense in a purely algebraic context, where there is no analog of the <a href="/facts/Logarithm/QeXaV97q">logarithm</a> function.

Logarithmic form open-in-new

Logarithmic form