Logarithmic form: Difference between revisions

Revision as of 18:26, 17 August 2011

Revision as of 19:06, 17 August 2011

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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 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 sub[[sheafSheaf_(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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Consider a once-punctured elliptic curve, given as the locus ''D'' of complex points <math> (x,y) </math> satisfying <math> g(x,y) = y^2 - f(x) = 0 </math>, where <math>f(x) = x(x-1)(x-\lambda) </math> and <math> \lambda\neq 0,1 </math> is a complex number. Then ''D'' is a smooth irreducible [[hypersurface]] in <math>\mathbb{C}^2</math> and, in particular, a divisor with simple normal crossings. There is a meromorphic two-form on <math>\mathbb{C}^2 </math>

:<math> \omega =\frac{dx\wedge dy}{g(x,y)} </math>

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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One shows<ref name="foo"/> that <math> W_mH^k(X; \mathbb{C}) </math> can actually be defined over <math>\mathbb{Q} </math>. Then the filtrations <math> W_{\bullet}, F^{\bullet} </math> on cohomology give rise to a mixed Hodge structure on <math> H^k(X; \mathbb{Z}) </math>.

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.