Editing Logarithmic form (section)

==Logarithmic de Rham complex==
Let ''X'' be a complex manifold and ''D'' a reduced divisor on ''X''. By definition of <math>\Omega^p_X(\log D)</math> and the fact that the [[exterior derivative]] ''d'' satisfies ''d''<sup>2</sup> = 0, one has
:<math> d\Omega^p_X(\log D)(U)\subset \Omega^{p+1}_X(\log D)(U)</math>
for every open subset ''U'' of ''X''. Thus the logarithmic differentials form a [[chain complex|complex]] of sheaves <math>( \Omega^{\bullet}_X(\log D), d) </math>, known as the '''logarithmic de Rham complex''' associated to the divisor ''D''. This is a subcomplex of the [[direct image]] <math> j_*(\Omega^{\bullet}_{X-D}) </math>, where <math> j:X-D\rightarrow X </math> is the inclusion and <math> \Omega^{\bullet}_{X-D} </math> is the complex of sheaves of holomorphic forms on ''X''−''D''.

Of special interest is the case where ''D'' has [[normal crossings]]: that is, ''D'' is locally a sum of codimension-1 complex submanifolds that intersect transversely. In this case, the sheaf of logarithmic differential forms is the subalgebra of <math>j_*(\Omega^{\bullet}_{X-D})</math> generated by the holomorphic differential forms <math>\Omega^{\bullet}_X</math> together with the 1-forms <math>df/f</math> for holomorphic functions <math>f</math> that are nonzero outside ''D''.<ref>Deligne (1970), Definition II.3.1.</ref> Note that
:<math>\frac{d(fg)}{fg}=\frac{df}{f}+\frac{dg}{g}.</math>

Concretely, if ''D'' is a divisor with normal crossings on a complex manifold ''X'', then each point ''x'' has an open neighborhood ''U'' on which there are holomorphic coordinate functions <math>z_1,\ldots,z_n</math> such that ''x'' is the origin and ''D'' is defined by the equation <math> z_1\cdots z_k = 0 </math> for some <math>0\leq k\leq n</math>. On the open set ''U'', sections of <math> \Omega^1_X(\log D) </math> are given by<ref>Peters & Steenbrink (2008), section 4.1.</ref>
:<math>\Omega_X^1(\log D) = \mathcal{O}_{X}\frac{dz_1}{z_1}\oplus\cdots\oplus\mathcal{O}_{X}\frac{dz_k}{z_k} \oplus \mathcal{O}_{X}dz_{k+1} \oplus \cdots \oplus \mathcal{O}_{X}dz_n.</math>
This describes the holomorphic vector bundle <math>\Omega_X^1(\log D)</math> on <math>X</math>. Then, for any <math>k\geq 0</math>, the vector bundle <math>\Omega^k_X(\log D)</math> is the ''k''th [[exterior power]],
:<math> \Omega_X^k(\log D) = \bigwedge^k \Omega_X^1(\log D).</math>

The '''logarithmic tangent bundle''' <math>TX(-\log D)</math> means the dual vector bundle to <math>\Omega^1_X(\log D)</math>. Explicitly, a section of <math>TX(-\log D)</math> is a holomorphic [[vector field]] on ''X'' that is tangent to ''D'' at all smooth points of ''D''.<ref>Deligne (1970), section II.3.9.</ref>

===Logarithmic differentials and singular cohomology===
Let ''X'' be a complex manifold and ''D'' a divisor with normal crossings on ''X''. Deligne proved a holomorphic analog of de Rham's theorem in terms of logarithmic differentials. Namely,
:<math> H^k(X, \Omega^{\bullet}_X(\log D))\cong H^k(X-D,\mathbf{C}),</math>
where the left side denotes the cohomology of ''X'' with coefficients in a complex of sheaves, sometimes called [[hypercohomology]]. This follows from the natural inclusion of complexes of sheaves
:<math> \Omega^{\bullet}_X(\log D)\rightarrow j_*\Omega_{X-D}^{\bullet} </math>
being a [[quasi-isomorphism]].<ref>Deligne (1970), Proposition II.3.13.</ref>