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===Higher Dimensional Example===
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 of <math>\omega </math> along ''D'' is given by the holomorphic one-form
:<math> \text{Res}_D(\omega) = \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'', a smooth elliptic curve.
=== Hodge Theory ===
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