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.
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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)
Holomorphic log complex
By definition of
This implies that there is a complex of sheaves
Of special interest is the case where D has simple normal crossings. Then if
and that
Some authors, e.g., use the term log complex to refer to the holomorphic log complex corresponding to a divisor with normal crossings.
Higher-dimensional example
Consider a once-punctured elliptic curve, given as the locus D of complex points (x,y) satisfying
which has a simple pole along D. The Poincaré residue of ω along D is given by the holomorphic one-form
Vital to the residue theory of logarithmic forms is the Gysin sequence, which is in some sense a generalization of the Residue Theorem for compact Riemann surfaces. This can be used to show, for example, that
Hodge theory
The holomorphic log complex can be brought to bear on the Hodge theory of complex algebraic varieties. Let X be a complex algebraic manifold and
turns out to be a quasi-isomorphism. Thus
where
which, along with the trivial increasing filtration
One shows that
Classically, for example in elliptic function theory, the logarithmic differential forms were recognised as complementary to the differentials of the first kind. They were sometimes called differentials of the second kind (and, with an unfortunate inconsistency, also sometimes of the third kind). The classical theory has now been subsumed as an aspect of Hodge theory. For a Riemann surface S, for example, the differentials of the first kind account for the term H1,0 in H1(S), when by the Dolbeault isomorphism it is interpreted as the sheaf cohomology group H0(S,Ω); this is tautologous considering their definition. The H1,0 direct summand in H1(S), as well as being interpreted as H1(S,O) where O is the sheaf of holomorphic functions on S, can be identified more concretely with a vector space of logarithmic differentials.