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In contexts including [[complex manifold]]s and [[algebraic geometry]], a '''logarithmic''' [[differential form]] is a meromorphic differential form with [[pole (complex analysis)|poles]] of a certain kind.
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Let ''X'' be a complex manifold, and ''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 [[Sheaf (mathematics)|subsheaf]] of the meromorphic ''p''-forms on ''X'' with a pole along ''D'', denoted
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:<math>\Omega^p_X(\log D).</math>
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In the theory of [[Riemann surfaces]], one encounters logarithmic one-forms which have the local expression
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


:<math>\omega = \frac{df}{f} =\left(\frac{m}{z} + \frac{g'(z)}{g(z)}\right)dz</math>
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


for some [[meromorphic function]] (resp. [[rational function]]) <math> f(z) = z^mg(z) </math>, 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.
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


==Holomorphic log complex==
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].
By definition of <math>\Omega^p_X(\log D)</math> and the fact that exterior differentiation ''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>.  
This implies that there is a complex of sheaves <math>( \Omega^{\bullet}_X(\log D), d) </math>, known as the ''holomorphic log complex'' corresponding to the divisor ''D''. This is a subcomplex  of <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 simple [[normal crossings]]. Then if <math> \{D_{\nu}\} </math> are the smooth, irreducible components of ''D'', one has <math> D = \sum D_{\nu} </math> with the <math> D_{\nu} </math> meeting transversely. Locally ''D'' is the union of hyperplanes, with local defining equations of the form <math> z_1\cdots z_k = 0 </math> in some holomorphic coordinates. One can show that the stalk of <math> \Omega^1_X(\log D) </math> at ''p'' satisfies<ref name="foo">Chris A.M. Peters; Joseph H.M. Steenbrink (2007). Mixed Hodge Structures. Springer. ISBN 978-3-540-77015-6 {{Please check ISBN|reason=Check digit (6) does not correspond to calculated figure.}}</ref>
==Demos==
:<math>\Omega_X^1(\log D)_p = \mathcal{O}_{X,p}\frac{dz_1}{z_1}\oplus\cdots\oplus\mathcal{O}_{X,p}\frac{dz_k}{z_k} \oplus \mathcal{O}_{X,p}dz_{k+1} \oplus \cdots \oplus \mathcal{O}_{X,p}dz_n</math>
and that
:<math> \Omega_X^k(\log D)_p = \bigwedge^k_{j=1} \Omega_X^1(\log D)_p </math>.
Some authors, e.g.,<ref name = "foo2">Phillip A. Griffiths; Joseph Harris (1979). Principles of Algebraic Geometry. Wiley-Interscience. ISBN 0-471-05059-8.</ref> use the term ''log complex'' to refer to the holomorphic log complex corresponding to a divisor with normal crossings.


===Higher-dimensional example===
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:
Consider a once-punctured elliptic curve, given as the locus ''D'' of complex points (''x'',''y'') 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 '''C'''<sup>2</sup> and, in particular, a divisor with simple normal crossings. There is a meromorphic two-form on '''C'''<sup>2</sup>
:<math> \omega =\frac{dx\wedge dy}{g(x,y)} </math>
which has a simple pole along ''D''. The Poincaré residue <ref name = "foo2"/> of ω along ''D'' is given by the holomorphic one-form
:<math> \text{Res}_D(\omega) = \frac{dy}{\partial g/\partial x}|_D =-\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]], which is in some sense a generalization of the [[Residue Theorem]] for compact Riemann surfaces. This can be used to show, for example, that <math>dx/y|_D </math> extends to a holomorphic one-form on the [[Projective space#Projective space and affine space|projective closure]] of ''D'' in '''P'''<sup>2</sup>, 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 <math> j: X\hookrightarrow Y </math> 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
:<math> \Omega^{\bullet}_Y(\log D)\rightarrow j_*\Omega_{X}^{\bullet} </math>
turns out to be a quasi-isomorphism.  Thus
:<math> H^k(X;\mathbf{C}) = \mathbb{H}^k(Y, \Omega^{\bullet}_Y(\log D))</math>
where <math>\mathbb{H}^{\bullet}</math> denotes [[hypercohomology]] of a complex of abelian sheaves. There is<ref name="foo"/> a decreasing filtration <math>W_{\bullet} \Omega^p_Y(\log D) </math> given by
:<math>W_{m}\Omega^p_Y(\log D) =  \begin{cases}
0 & m < 0\\
\Omega^p_Y(\log D) & m\geq p \\
\Omega^{p-m}_Y\wedge \Omega^m_Y(\log D) & 0\leq m \leq p
\end{cases} </math>
which, along with the trivial  increasing filtration <math>F^{\bullet}\Omega^p_Y(\log D) </math> on logarithmic ''p''-forms, produces filtrations on cohomology
:<math> W_mH^k(X; \mathbf{C}) = \text{Im}(\mathbb{H}^k(Y, W_{m-k}\Omega^{\bullet}_Y(\log D))\rightarrow H^k(X; \mathbf{C})) </math>
:<math> F^pH^k(X; \mathbf{C}) = \text{Im}(\mathbb{H}^k(Y, F^p\Omega^{\bullet}_Y(\log D))\rightarrow H^k(X; \mathbf{C})) </math>.
One shows<ref name="foo"/> that <math> W_mH^k(X; \mathbf{C}) </math> can actually be defined over '''Q'''. Then the filtrations <math> W_{\bullet}, F^{\bullet}</math> on cohomology give rise to a mixed Hodge structure on <math> H^k(X; \mathbf{Z}) </math>.


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''<sup>1,0</sup> in ''H''<sup>1</sup>(''S''), when by the [[Dolbeault isomorphism]] it is interpreted as the [[sheaf cohomology]] group ''H''<sup>0</sup>(''S'',Ω); this is tautologous considering their definition. The ''H''<sup>1,0</sup> direct summand in ''H''<sup>1</sup>(''S''), as well as being interpreted as ''H''<sup>1</sup>(''S'',O) where O is the sheaf of [[holomorphic function]]s on ''S'', can be identified more concretely with a vector space of logarithmic differentials.
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==See also==
==Test pages ==
*[[Algebraic Geometry]]
*[[Adjunction formula]]
*[[Differential of the first kind]]
*[[Residue Theorem]]


==References==
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==Bug reporting==
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Latest revision as of 23:52, 15 September 2019

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