Logarithmic form: Difference between revisions

Revision as of 06:47, 8 November 2012

Revision as of 01:27, 11 December 2012

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In contexts including [[complex manifold]]s and [[algebraic geometry]], a '''logarithmic''' [[differential form]] is a meromorphic differential form with [[pole]]s{{dndisambiguation needed|date=November 2012}} of a certain kind.

Let ''X'' be a complex manifold, and <math> D\subset X </math> a [[divisor]] and <math>\omega </math> a holomorphic ''p''-form on <math>X-D </math>. If <math>\omega</math> and <math>d\omega</math> have a pole of order at most one along ''D'', then <math>\omega</math> is said to have a logarithmic pole along ''D''. <math>\omega</math> is also known as a logarithmic ''p''-form. The logarithmic ''p''-forms make up a [[Sheaf_Sheaf (mathematics)|subsheaf]] of the meromorphic ''p''-forms on ''X'' with a pole along ''D'', denoted <math>\Omega^p_X(\log D)</math>.

In the theory of [[Riemann surfaces]], one encounters logarithmic one-forms which have the local expression

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:<math>\omega = \frac{df}{f} =\left(\frac{m}{z} + \frac{g'(z)}{g(z)}\right)dz</math>

for some [[meromorphic function]] (resp. [[rational function]]) <math> f(z) = z^mg(z) </math>, 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 <math> \omega </math> 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.

==Holomorphic Log Complex==

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and that

:<math> \Omega_X^k(\log D)_p = \bigwedge^k_{j=1} \Omega_X^1(\log D)_p </math>.

Some authors, e.g. ,<ref name = "foo2">Phillip A. Griffiths; Joseph Harris (1979). Principles of Algebraic Geometry. Wiley-Interscience. ISBN 0-471-05059-8.</ref>, use the term ''log complex'' to refer to the holomorphic log complex corresponding to a divisor with normal crossings.

===Higher Dimensional Example===

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which has a simple pole along ''D''. The Poincaré residue <ref name = "foo2"/> of <math>\omega </math> along ''D'' is given by the holomorphic one-form

:<math> \text{Res}_D(\omega) = \frac{dy}{\partial g/\partial x}|_D =-\frac{dx}{\partial g/\partial y}|_D = -\frac{1}{2}\frac{dx}{y}|_D </math>.

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 <math>dx/y|_D </math> extends to a holomorphic one-form on the [[Projective_space#Projective_space_and_affine_space|projective closure]] of ''D'' in <math>\mathbb{P}^2 </math>, a smooth elliptic curve.

=== Hodge Theory ===

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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'',&Omega;); 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 function]]s on ''S'', can be identified more concretely with a vector space of logarithmic differentials.

==See Alsoalso==

*[[Algebraic Geometry]]

*[[Adjunction formula]]