Logarithmic form
In contexts including complex manifolds and algebraic geometry, a logarithmic differential form is a meromorphic differential form with poles of a certain kind. The concept was introduced by Deligne. Let "X" be a complex manifold, "D" ⊂ "X" a 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 subsheaf of the meromorphic "p"-forms on "X" with a pole along "D , denoted In the theory of Riemann surfaces, one encounters logarithmic one-forms which have the local expression for some meromorphic function (resp. rational function) formula_3, where g" is holomorphic and non-vanishing at 0, and "m" is the order of "f" at "0". That is, for some open covering, there are local representations of this differential form as a logarithmic derivative (modified slightly with the exterior derivative "d" in place of the usual differential operator "d/dz"). Observe that ω has only simple poles with integer residues. On higher-dimensional complex manifolds, the Poincaré residue is used to describe the  distinctive behavior of logarithmic forms along poles. Holomorphic log complex. By definition of formula_4 and the fact that exterior differentiation "d" satisfies "d" = 0, one has This implies that there is a complex of sheaves formula_6, known as the "holomorphic log complex" corresponding to the divisor "D". This is a subcomplex of formula_7, where formula_8 is the inclusion and formula_9 is the complex of sheaves of holomorphic forms on "X"−"D". Of special interest is the case where "D" has simple normal crossings. Then if formula_10 are the smooth, irreducible components of "D", one has formula_11 with the formula_12 meeting  transversely. Locally "D" is the union of hyperplanes, with local defining equations of the form formula_13 in some holomorphic coordinates. One can show that the stalk of formula_14 at "p" satisfies and that Some authors, e.g., use the term "log complex" to refer to the holomorphic log complex corresponding to a divisor with normal crossings. Higher-dimensional example. Consider a once-punctured elliptic curve, given as the locus "D" of complex points ("x","y") satisfying formula_17 where formula_18 and formula_19 is a complex number. Then "D" is a smooth irreducible hypersurface in C and, in particular, a divisor with simple normal  crossings. There is a meromorphic two-form on C which has a simple pole along "D". The Poincaré residue of ω along "D" is given by the holomorphic one-form 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 formula_22 extends to a holomorphic one-form on the projective closure of "D" in P, a smooth elliptic curve. Hodge theory. The holomorphic log complex can be brought to bear on the Hodge theory  of complex algebraic varieties. Let "X" be a complex algebraic manifold and formula_23 a good compactification. This means that "Y" is a compact algebraic manifold and "D" = "Y"−"X" is a divisor on "Y" with simple normal crossings. The natural inclusion of complexes of sheaves turns out to be a quasi-isomorphism. Thus where formula_26 denotes hypercohomology of a complex of abelian sheaves. There is a decreasing filtration formula_27 given by which, along with the trivial increasing filtration formula_29 on logarithmic "p"-forms, produces filtrations on cohomology One shows that formula_32 can actually be defined over Q. Then the filtrations formula_33  on cohomology give rise to a mixed Hodge structure on formula_34. 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" in "H"("S"), when by the Dolbeault isomorphism it is interpreted as the sheaf cohomology group "H"("S",Ω ); this is tautologous considering their definition. The "H" direct summand in "H"("S"), as well as being interpreted as "H"("S",O) where O is the sheaf of holomorphic functions on "S", can be identified more concretely with a vector space of logarithmic differentials. Sheaf of logarithmic forms. In algebraic geometry, the sheaf of logarithmic differential "p"-forms formula_4 on a smooth projective variety "X" along a smooth divisor formula_36 is defined and fits into the exact sequence of locally free sheaves: where formula_38 are the inclusions of irreducible divisors (and the pushforwards along them are extension by zero), and β is called the  residue map when "p" is 1. For example, if "x" is a closed point on formula_39 and not on formula_40, then form a basis of formula_42 at "x", where formula_43 are local coordinates around "x" such that formula_44 are local parameters for formula_39.