Logarithmic form

Summary

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.[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.)

Let X be a complex manifold, DX a reduced divisor (a sum of distinct codimension-1 complex subspaces), and ω a holomorphic p-form on XD. 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, ; here is a typical example of a 1-form on the complex numbers C with a logarithmic pole at the origin. Differential forms such as make sense in a purely algebraic context, where there is no analog of the logarithm function.

Logarithmic de Rham complex

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Let X be a complex manifold and D a reduced divisor on X. By definition of   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 of sheaves  , known as the logarithmic de Rham complex associated to the divisor D. This is a subcomplex of the direct image  , where   is the inclusion and   is the complex of sheaves of holomorphic forms on XD.

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   generated by the holomorphic differential forms   together with the 1-forms   for holomorphic functions   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   such that x is the origin and D is defined by the equation   for some  . On the open set U, sections of   are given by[3]

 

This describes the holomorphic vector bundle   on  . Then, for any  , the vector bundle   is the kth exterior power,

 

The logarithmic tangent bundle   means the dual vector bundle to  . Explicitly, a section of   is a holomorphic vector field on X that is tangent to D at all smooth points of D.[4]

Logarithmic differentials and singular cohomology

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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.[5]

Logarithmic differentials in algebraic geometry

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In algebraic geometry, the vector bundle of logarithmic differential p-forms   on a smooth scheme X over a field, with respect to a divisor   with simple normal crossings, is defined as above: sections of   are (algebraic) differential forms ω on   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   for   and not in   for  , let   be regular functions on some open neighborhood U of x such that   is the closed subscheme defined by   inside U for  , and x is the closed subscheme of U defined by  . Then a basis of sections of   on U is given by:

 

This describes the vector bundle   on X, and then   is the pth exterior power of  .

There is an exact sequence of coherent sheaves on X:

 

where   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.

Explicitly, on an open subset of   that only meets one component   of  , with   locally defined by  , the residue of a logarithmic  -form along   is determined by: the residue of a regular p-form is zero, whereas

 

for any regular  -form  .[7] Some authors define the residue by saying that   has residue  , which differs from the definition here by the sign  .

Example of the residue

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Over the complex numbers, the residue of a differential form with log poles along a divisor   can be viewed as the result of integration over loops in   around  . In this context, the residue may be called the Poincaré residue.

For an explicit example,[8] consider an elliptic curve D in the complex projective plane  , defined in affine coordinates   by the equation   where   and   is a complex number. Then D is a smooth hypersurface of degree 3 in   and, in particular, a divisor with simple normal crossings. There is a meromorphic 2-form on   given in affine coordinates by

 

which has log poles along D. Because the canonical bundle   is isomorphic to the line bundle  , the divisor of poles of   must have degree 3. So the divisor of poles of   consists only of D (in particular,   does not have a pole along the line   at infinity). The residue of ω along D is given by the holomorphic 1-form

 

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

The residue map   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

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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

 

In modern terms, it follows that   is a 1-form on C with logarithmic poles on  , since   is the zero set of the Weierstrass sigma function  

Mixed Hodge theory for smooth varieties

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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  .[10] So the cohomology of   with complex coefficients has a decreasing filtration, the Hodge filtration, whose associated graded vector spaces are the algebraically defined groups  .

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  . The resulting filtration on   can be constructed using the logarithmic de Rham complex. Namely, define an increasing filtration   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. Then

 

and

 

for all  .[12]

See also

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Notes

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  1. ^ Deligne (1970), section II.3.
  2. ^ Deligne (1970), Definition II.3.1.
  3. ^ Peters & Steenbrink (2008), section 4.1.
  4. ^ Deligne (1970), section II.3.9.
  5. ^ Deligne (1970), Proposition II.3.13.
  6. ^ Deligne (1970), Lemma II.3.2.1.
  7. ^ Deligne (1970), sections II.3.5 to II.3.7; Griffiths & Harris (1994), section 1.1.
  8. ^ Griffiths & Harris (1994), section 2.1.
  9. ^ Deligne (1970), Corollaire II.6.10.
  10. ^ Deligne (1971), Corollaire 3.2.13.
  11. ^ Peters & Steenbrink (2008), Theorem 4.2.
  12. ^ Esnault & Viehweg (1992), Corollary 6.4.

References

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  • Deligne, Pierre (1970), Equations Différentielles à Points Singuliers Réguliers, Lecture Notes in Mathematics, vol. 163, Springer-Verlag, doi:10.1007/BFb0061194, ISBN 3540051902, MR 0417174, OCLC 169357
  • Deligne, Pierre (1971), "Théorie de Hodge II", Publ. Math. IHÉS, 40: 5–57, doi:10.1007/BF02684692, MR 0498551, S2CID 118967613
  • Esnault, Hélène; Viehweg, Eckart (1992), Lectures on vanishing theorems, Birkhäuser, doi:10.1007/978-3-0348-8600-0, ISBN 978-3-7643-2822-1, MR 1193913
  • Griffiths, Phillip; Harris, Joseph (1994) [1978], Principles of algebraic geometry, Wiley Classics Library, Wiley Interscience, doi:10.1002/9781118032527, ISBN 0-471-05059-8, MR 0507725
  • Peters, Chris A.M.; Steenbrink, Joseph H. M. (2008), Mixed Hodge structures, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol. 52, Springer, doi:10.1007/978-3-540-77017-6, ISBN 978-3-540-77017-6, MR 2393625
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