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. 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 dω 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) f ( z ) = z m g ( z ) {displaystyle f(z)=z^{m}g(z)} , 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. By definition of Ω X p ( log ⁡ D ) {displaystyle Omega _{X}^{p}(log D)} and the fact that exterior differentiation d satisfies d2 = 0, one has This implies that there is a complex of sheaves ( Ω X ∙ ( log ⁡ D ) , d ) {displaystyle (Omega _{X}^{ullet }(log D),d)} , known as the holomorphic log complex corresponding to the divisor D. This is a subcomplex of j ∗ Ω X − D ∙ {displaystyle j_{*}Omega _{X-D}^{ullet }} , where j : X − D → X {displaystyle j:X-D ightarrow X} is the inclusion and Ω X − D ∙ {displaystyle Omega _{X-D}^{ullet }} 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 { D ν } {displaystyle {D_{ u }}} are the smooth, irreducible components of D, one has D = ∑ D ν {displaystyle D=sum D_{ u }} with the D ν {displaystyle D_{ u }} meeting transversely. Locally D is the union of hyperplanes, with local defining equations of the form z 1 ⋯ z k = 0 {displaystyle z_{1}cdots z_{k}=0} in some holomorphic coordinates. One can show that the stalk of Ω X 1 ( log ⁡ D ) {displaystyle Omega _{X}^{1}(log D)} at p satisfies