Logarithmic form: Difference between revisions

Revision as of 16:32, 17 August 2011

Revision as of 18:26, 17 August 2011

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which has a simple pole along ''D''. The Poincaré residue 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]]. This can be used to show, for example, that <math>\frac{dx}{/y}|_D </math> extends to a holomorphic one-form on the projective closure of ''D'' in <math>\mathbb{P}^2 </math>, a smooth elliptic curve.

=== Hodge Theory ===