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 "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 subsheaf of the meromorphic "p"-forms on "X", denoted
formula_1
The name comes from the fact that in complex analysis, formula_2; here formula_3 is a typical example of a 1-form on the complex numbers C with a logarithmic pole at the origin. Differential forms such as formula_3 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". By definition of formula_5 and the fact that the exterior derivative "d" satisfies "d"2 = 0, one has
formula_6
for every open subset "U" of "X". Thus the logarithmic differentials form a complex of sheaves formula_7, known as the logarithmic de Rham complex associated to the divisor "D". This is a subcomplex of the direct image formula_8, where formula_9 is the inclusion and formula_10 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" 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 formula_11 generated by the holomorphic differential forms formula_12 together with the 1-forms formula_13 for holomorphic functions formula_14 that are nonzero outside "D". Note that
formula_15
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 formula_16 such that "x" is the origin and "D" is defined by the equation formula_17 for some formula_18. On the open set "U", sections of formula_19 are given by
formula_20
This describes the holomorphic vector bundle formula_21 on formula_22. Then, for any formula_23, the vector bundle formula_24 is the "k"th exterior power,
formula_25
The logarithmic tangent bundle formula_26 means the dual vector bundle to formula_27. Explicitly, a section of formula_26 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,
formula_29
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
formula_30
being a quasi-isomorphism.
Logarithmic differentials in algebraic geometry.
In algebraic geometry, the vector bundle of logarithmic differential "p"-forms formula_5 on a smooth scheme "X" over a field, with respect to a divisor formula_32 with simple normal crossings, is defined as above: sections of formula_5 are (algebraic) differential forms ω on formula_34 such that both ω and "d"ω have a pole of order at most one along "D". Explicitly, for a closed point "x" that lies in formula_35 for formula_36 and not in formula_35 for formula_38, let formula_39 be regular functions on some open neighborhood "U" of "x" such that formula_35 is the closed subscheme defined by formula_41 inside "U" for formula_36, and "x" is the closed subscheme of "U" defined by formula_43. Then a basis of sections of formula_27 on "U" is given by:
formula_45
This describes the vector bundle formula_27 on "X", and then formula_5 is the "p"th exterior power of formula_27.
There is an exact sequence of coherent sheaves on "X":
formula_49
where formula_50 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":
 formula_51
where the sums run over all irreducible components of given dimension of intersections of the divisors "D""j". Here again, β is called the residue map.
Explicitly, on an open subset of formula_22 that only meets one component formula_35 of formula_54, with formula_35 locally defined by formula_56, the residue of a logarithmic formula_57-form along formula_35 is determined by: the residue of a regular "p"-form is zero, whereas
formula_59
for any regular formula_60-form formula_61. Some authors define the residue by saying that formula_62 has residue formula_63, which differs from the definition here by the sign formula_64.
Example of the residue.
Over the complex numbers, the residue of a differential form with log poles along a divisor formula_35 can be viewed as the result of integration over loops in formula_22 around formula_35. 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 formula_68, defined in affine coordinates formula_69 by the equation formula_70 where formula_71 and formula_72 is a complex number. Then "D" is a smooth hypersurface of degree 3 in formula_73 and, in particular, a divisor with simple normal crossings. There is a meromorphic 2-form on formula_73 given in affine coordinates by
formula_75
which has log poles along "D". Because the canonical bundle formula_76 is isomorphic to the line bundle formula_77, the divisor of poles of formula_78 must have degree 3. So the divisor of poles of formula_78 consists only of "D" (in particular, formula_78 does not have a pole along the line formula_81 at infinity). The residue of ω along "D" is given by the holomorphic 1-form
formula_82
It follows that formula_83 extends to a holomorphic one-form on the projective curve "D" in formula_73, an elliptic curve.
The residue map formula_85 considered here is part of a linear map formula_86, 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":
formula_87
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 formula_88 in C was called an "integral of the second kind" to mean that it could be written
formula_89
In modern terms, it follows that formula_90 is a 1-form on C with logarithmic poles on formula_88, since formula_88 is the zero set of the Weierstrass sigma function formula_93
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
formula_94
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
formula_95
degenerates at formula_96. So the cohomology of formula_34 with complex coefficients has a decreasing filtration, the Hodge filtration, whose associated graded vector spaces are the algebraically defined groups formula_98.
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 on the rational cohomology of formula_34. The resulting filtration on formula_100 can be constructed using the logarithmic de Rham complex. Namely, define an increasing filtration formula_101 by
formula_102
The resulting filtration on cohomology is the weight filtration:
formula_103
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
formula_104
and
formula_105
for all formula_106.
Notes.
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