Logarithmic form: Difference between revisions

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Revision as of 18:09, 29 June 2021

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In contexts including [[complex manifold]]s and [[algebraic geometry]], a '''logarithmic''' [[differential form]] is a meromorphic differential form with [[pole (complex analysis)|poles]] of a certain kind. The concept was introduced by [[Pierre Deligne|Deligne]].<ref>Deligne, Pierre. ''Equations différentielles à points singuliers réguliers''. Lecture Notes in Mathematics. 163. Berlin-Heidelberg-New York: Springer-Verlag.</ref>

Let ''X'' be a complex manifold, ''D'' ⊂ ''X'' a [[Divisor_Divisor (algebraic_geometryalgebraic geometry)|divisor]], and ω a holomorphic ''p''-form on ''X''−''D''. If ω and ''d''ω 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 [[Sheaf (mathematics)|subsheaf]] of the meromorphic ''p''-forms on ''X'' with a pole along ''D'', denoted

:<math>\Omega^p_X(\log D).</math>

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===Higher-dimensional example===

Consider a once-punctured elliptic curve, given as the locus ''D'' of complex points (''x'',''y'') 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 '''C'''² and, in particular, a divisor with simple normal crossings. There is a meromorphic two-form on '''C'''²

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

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

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