In algebraic geometry and the theory of complex manifolds, a logarithmic differential form is a differential form with poles of a certain kind.
[1] 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.)
is a typical example of a 1-form on the complex numbers C with a logarithmic pole at the origin.
make sense in a purely algebraic context, where there is no analog of the logarithm function.
and the fact that the exterior derivative d satisfies d2 = 0, one has for every open subset U of X.
, known as the logarithmic de Rham complex associated to the divisor D. This is a subcomplex of the direct image
Of special interest is the case where D has normal crossings: that is, D is locally a sum of codimension-1 complex submanifolds that intersect transversely.
In this case, the sheaf of logarithmic differential forms is the subalgebra of
that are nonzero outside D.[2] Note that Concretely, if D is a divisor with normal crossings on a complex manifold X, then each point x has an open neighborhood U on which there are holomorphic coordinate functions
is the kth exterior power, The logarithmic tangent bundle
is a holomorphic vector field on X that is tangent to D at all smooth points of D.[4] Let X be a complex manifold and D a divisor with normal crossings on X. Deligne proved a holomorphic analog of de Rham's theorem in terms of logarithmic differentials.
Namely, where the left side denotes the cohomology of X with coefficients in a complex of sheaves, sometimes called hypercohomology.
[5] In algebraic geometry, the vector bundle of logarithmic differential p-forms
with simple normal crossings, is defined as above: sections of
such that both ω and dω have a pole of order at most one along D.[6] Explicitly, for a closed point x that lies in
is the inclusion of an irreducible component of D. Here β is called the residue map; so this sequence says that a 1-form with log poles along D is regular (that is, has no poles) if and only if its residues are zero.
More generally, for any p ≥ 0, there is an exact sequence of coherent sheaves on X: where the sums run over all irreducible components of given dimension of intersections of the divisors Dj.
Over the complex numbers, the residue of a differential form with log poles along a divisor
For an explicit example,[8] consider an elliptic curve D in the complex projective plane
given in affine coordinates by which has log poles along D. Because the canonical bundle
extends to a holomorphic one-form on the projective curve D in
This is part of the Gysin sequence associated to any smooth divisor D in a complex manifold X: In the 19th-century theory of elliptic functions, 1-forms with logarithmic poles were sometimes called integrals of the second kind (and, with an unfortunate inconsistency, sometimes differentials of the third kind).
in C was called an "integral of the second kind" to mean that it could be written In modern terms, it follows that
Over the complex numbers, Deligne proved a strengthening of Alexander Grothendieck's algebraic de Rham theorem, relating coherent sheaf cohomology with singular cohomology.
Namely, for any smooth scheme X over C with a divisor with simple normal crossings D, there is a natural isomorphism for each integer k, where the groups on the left are defined using the Zariski topology and the groups on the right use the classical (Euclidean) topology.
[9] Moreover, when X is smooth and proper over C, the resulting spectral sequence degenerates at
This is part of the mixed Hodge structure which Deligne defined on the cohomology of any complex algebraic variety.
by The resulting filtration on cohomology is the weight filtration:[11] Building on these results, Hélène Esnault and Eckart Viehweg generalized the Kodaira–Akizuki–Nakano vanishing theorem in terms of logarithmic differentials.
Namely, let X be a smooth complex projective variety of dimension n, D a divisor with simple normal crossings on X, and L an ample line bundle on X.