Logarithmic form
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.^{[1]}
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) , where g is holomorphic and non-vanishing at 0, and m is the order of f at 0. That is, for some open covering, there are local representations of this differential form as a logarithmic derivative (modified slightly with the exterior derivative d in place of the usual differential operator d/dz). Observe that ω has only simple poles with integer residues. On higher-dimensional complex manifolds, the Poincaré residue is used to describe the distinctive behavior of logarithmic forms along poles.
Contents
Holomorphic log complex
By definition of and the fact that exterior differentiation d satisfies d^{2} = 0, one has
- .
This implies that there is a complex of sheaves , known as the holomorphic log complex corresponding to the divisor D. This is a subcomplex of , where is the inclusion and is the complex of sheaves of holomorphic forms on X−D.
Of special interest is the case where D has simple normal crossings. Then if are the smooth, irreducible components of D, one has with the meeting transversely. Locally D is the union of hyperplanes, with local defining equations of the form in some holomorphic coordinates. One can show that the stalk of at p satisfies^{[2]}
and that
- .
Some authors, e.g.,^{[3]} 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 where and is a complex number. Then D is a smooth irreducible hypersurface in C^{2} and, in particular, a divisor with simple normal crossings. There is a meromorphic two-form on C^{2}
which has a simple pole along D. The Poincaré residue ^{[3]} 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 extends to a holomorphic one-form on the projective closure of D in P^{2}, a smooth elliptic curve.
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 a good compactification. This means that Y is a compact algebraic manifold and D = Y−X is a divisor on Y with simple normal crossings. The natural inclusion of complexes of sheaves
turns out to be a quasi-isomorphism. Thus
where denotes hypercohomology of a complex of abelian sheaves. There is^{[2]} a decreasing filtration given by
which, along with the trivial increasing filtration on logarithmic p-forms, produces filtrations on cohomology
- .
One shows^{[2]} that can actually be defined over Q. Then the filtrations on cohomology give rise to a mixed Hodge structure on .
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 H^{1,0} in H^{1}(S), when by the Dolbeault isomorphism it is interpreted as the sheaf cohomology group H^{0}(S,Ω); this is tautologous considering their definition. The H^{1,0} direct summand in H^{1}(S), as well as being interpreted as H^{1}(S,O) where O is the sheaf of holomorphic functions on S, can be identified more concretely with a vector space of logarithmic differentials.
Algebraic Geometry
In algebraic geometry, the sheaf of logarithmic differential p-forms on a smooth projective variety X along a smooth divisor is defined and fits into the exact sequence of locally free sheaves:
where are the inclusions of irreducible divisors (and the pushforwards along them are extension by zero), and β is called the residue map when p is 1.
For example,^{[4]} if x is a closed point on and not on , then
form a basis of at x, where are local coordinates around x such that are local parameters for .
See also
- Borel–Moore homology
- Algebraic Geometry
- Adjunction formula
- Differential of the first kind
- Residue Theorem
- Poincaré residue
References
- ^ Deligne, Pierre. Equations différentielles à points singuliers réguliers. Lecture Notes in Mathematics. 163. Berlin-Heidelberg-New York: Springer-Verlag.
- ^ ^{a} ^{b} ^{c} Chris A.M. Peters; Joseph H.M. Steenbrink (2007). Mixed Hodge Structures. Springer. ISBN 978-3-540-77017-6
- ^ ^{a} ^{b} Phillip A. Griffiths; Joseph Harris (1979). Principles of Algebraic Geometry. Wiley-Interscience. ISBN 0-471-05059-8.
- ^ Deligne, Part II, Lemma 3.2.1.
- Aise Johan de Jong, Algebraic de Rham cohomology.
- Pierre Deligne, Equations Différentielles à Points Singuliers Réguliers. Lecture Notes in Math. 163.