that has residue , which differs from the definition here by the sign . Example of the residue Over the complex numbers, the residue of a differential form with log poles along a divisor can be viewed as the result of integration over loops in around . In this context, the residue may be called the Poincaré residue. For an explicit example, consider an elliptic curve D in the complex projective plane , defined in affine coordinates by the equation where and is a complex number. Then D is a smooth hypersurface of degree 3 in and, in particular, a divisor with simple normal crossings. There is a meromorphic 2-form on given in affine coordinates by which has log poles along D. Because the canonical bundle is isomorphic to the line bundle , the divisor of poles of must have degree 3. So the divisor of poles of consists only of D (in particular, does not have a pole along the line at infinity). The residue of ω along D is given by the holomorphic 1-form It follows that extends to a holomorphic one-form on the projective curve D in , an elliptic curve. The residue map considered here is part of a linear map , which may be called the "Gysin map". This is part of the Gysin sequence associated to any smooth divisor D in a complex manifold X: Historical terminology In the 19th-century theory of elliptic functions, 1-forms with logarithmic poles were sometimes called integrals of the second kind (and, with an unfortunate inconsistency, sometimes differentials of the third kind). For example, the Weierstrass zeta function associated to a lattice in C was called an "integral of the second kind" to mean that it could be written In modern terms, it follows that is a 1-form on C with logarithmic poles on , since is the zero set of the Weierstrass sigma function Mixed Hodge theory for smooth varieties Over the complex numbers, Deligne proved a strengthening of Alexander Grothendieck's algebraic de Rham theorem, relating coherent sheaf cohomology with singular cohomology. Namely, for any smooth scheme X over C with a divisor with simple normal crossings D, there is a natural isomorphism for each integer k, where the groups on the left are defined using the Zariski topology and the groups on the right use the classical (Euclidean) topology. Moreover, when X is smooth and proper over