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Wikipedia / 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

The name comes from the fact that in complex analysis is a typical example of a 1-form on the complex numbers C make sense in a purely algebraic context, where there is no analog of the logarithm function.

Logarithmic de Rham complex

Let X be a complex manifold and D a reduced divisor on X and the fact that the exterior derivative d satisfies d2 = 0, one has

for every open subset U of X. Thus the logarithmic differentials form a complex, known as the logarithmic de Rham complex associated to the divisor D. This is a subcomplex of the direct image is the complex of sheaves of holomorphic forms on X−D.

Of special interest is the case where D has normal crossings: that is, D that are nonzero outside D. 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 such that x is the origin and D. On the open set U are given by

This describes the holomorphic vector bundle 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.

Logarithmic differentials and singular cohomology

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. This follows from the natural inclusion of complexes of sheaves

being a quasi-isomorphism.

Logarithmic differentials in algebraic geometry

In algebraic geometry, the vector bundle of logarithmic differential p-forms on a smooth scheme X over a field, with respect to a divisor such that both ω and dω have a pole of order at most one along D. Explicitly, for a closed point x be regular functions on some open neighborhood U of x inside U, and x is the closed subscheme of U on U is given by:

on X is the p.

There is an exact sequence of coherent sheaves on X:

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. Here again, β is called the residue map.

is determined by: the residue of a regular p-form is zero, whereas

..

Example of the residue

can be viewed as the result of integration. In this context, the residue may be called the Poincaré residue.

For an explicit example, consider an elliptic curve D in the complex projective plane is a complex number. Then D is a smooth hypersurface given in affine coordinates by

which has log poles along D. Because the canonical bundle consists only of D at infinity). The residue of ω along D is given by the holomorphic 1-form

extends to a holomorphic one-form on the projective curve D, an elliptic curve.

considered here is part of a linear map, which may be called the "Gysin map". This is part of the Gysin sequence associated to any smooth divisor D in a complex manifold X:

Historical terminology

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). For example, the Weierstrass zeta function associated to a lattice in C was called an "integral of the second kind" to mean that it could be written

is a 1-form on C

Mixed Hodge theory for smooth varieties

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.

Moreover, when X is smooth and proper over C, the resulting spectral sequence

. with complex coefficients has a decreasing filtration, the Hodge filtration.

This is part of the mixed Hodge structure which Deligne defined on the cohomology of any complex algebraic variety. In particular, there is also a weight filtration by

The resulting filtration on cohomology is the weight filtration:

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. Then

and

.

See also

Notes

References