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In [[algebraic geometry]] and the theory of [[complex manifold]]s, a '''logarithmic''' [[differential form]] is a differential form with [[pole (complex analysis)|poles]] of a certain kind. The concept was introduced by [[Pierre Deligne]].<ref>Deligne (1970), section II.3.</ref> In short, logarithmic differentials have the mildest possible singularities needed in order to give information about an open submanifold (the complement of the divisor of poles). (This idea is made precise by several versions of [[de Rham's theorem]] discussed below.)
Let ''X'' be a complex manifold, ''D'' ⊂ ''X'' a reduced [[Divisor (algebraic geometry)|divisor]] (a sum of distinct codimension-1 complex subspaces), and ω a holomorphic ''p''-form on ''X''−''D''. If both ω and ''d''ω have a pole of order at most
:<math>\Omega^p_X(\log D).</math>
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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
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> \Omega_X^k(\log D) = \bigwedge^k \Omega_X^1(\log D).
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>
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>▼
:<math> \text{Res}_D(\omega) = \left. \frac{dy}{\partial g/\partial x} \right |_D =\left. -\frac{dx}{\partial g/\partial y} \right |_D = \left. -\frac{1}{2}\frac{dx}{y} \right |_D. </math>▼
===Historical terminology===▼
In the 19th-century theory of [[elliptic function]]s, 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_(group)|lattice]] <math>\Lambda</math> in '''C''' was called an "integral of the second kind" to mean that it could be written▼
▲:<math>\zeta(z)=\frac{\sigma'(z)}{\sigma(z)}.</math>
In modern terms, it follows that <math>\zeta(z)dz</math> is a 1-form on '''C''' with logarithmic poles on <math>\Lambda</math>, since <math>\Lambda</math> is the zero set of <math>\sigma(z)</math>.▼
===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
:<math> H^k(X
where the
:<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>
==Logarithmic differentials in algebraic geometry==
In algebraic geometry, the vector bundle of '''logarithmic differential ''p''-forms''' <math>\Omega^p_X(\log D)</math> on a [[smooth scheme]] ''X'' over a field, with respect to a [[divisor (algebraic geometry)|divisor]] <math>D = \sum D_j</math> with simple normal crossings, is defined as above: sections of <math>\Omega^p_X(\log D)</math> are (algebraic) differential forms ω on <math>X-D</math> such that both ω and ''d''ω have a pole of order at most one along ''D''.<ref>Deligne (1970), Lemma II.3.2.1.</ref> Explicitly, for a closed point ''x'' that lies in <math>D_j</math> for <math>1 \le j \le k</math> and not in <math>D_j</math> for <math>j > k</math>, let <math>u_j</math> be regular functions on some open neighborhood ''U'' of ''x'' such that <math>D_j</math> is the closed subscheme defined by <math>u_j=0</math> inside ''U'' for <math>1 \le j \le k</math>, and ''x'' is the closed subscheme of ''U'' defined by <math>u_1=\cdots=u_n=0</math>. Then a basis of sections of <math>\Omega^1_X(\log D)</math> on ''U'' is given by:
:<math>{du_1 \over u_1}, \dots, {du_k \over u_k}, \, du_{k+1}, \dots, du_n.</math>
This describes the vector bundle <math>\Omega^1_X(\log D)</math> on ''X'', and then <math>\Omega^p_X(\log D)</math> is the ''p''th exterior power of <math>\Omega^1_X(\log D)</math>.
There is an [[exact sequence]] of [[coherent sheaves]] on ''X'':
:
where <math>i_j: D_j \to X</math> is the inclusion of an irreducible component of ''D''. Here β is called the '''residue''' map; so this sequence says that a 1-form with log poles along ''D'' is regular (that is, has no poles) if and only if its residues are zero. More generally, for any ''p'' ≥ 0, there is an exact sequence of coherent sheaves on ''X'':
: <math>0 \to \Omega^p_X \to \Omega^p_X(\log D) \overset{\beta}\to \oplus_j ({i_j})_*\Omega^{p-1}_{D_j}(\log (D-D_j)) \to \cdots \to 0,</math>
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Explicitly, on an open subset of <math>X</math> that only meets one component <math>D_j</math> of <math>D</math>, with <math>D_j</math> locally defined by <math>f=0</math>, the residue of a logarithmic <math>p</math>-form along <math>D_j</math> is determined by: the residue of a regular ''p''-form is zero, whereas
:<math>\text{Res}_{D_j}\bigg(\frac{df}{f}\wedge \alpha\bigg)=\alpha|
for any regular <math>(p-1)</math>-form <math>\alpha</math>.<ref>Deligne (1970), sections II.3.5 to II.3.7; Griffiths & Harris (1994), section 1.1.</ref> Some authors define the residue by saying that <math>\alpha\wedge(df/f)</math> has residue <math>\alpha|_{D_j}</math>, which differs from the definition here by the sign <math>(-1)^{p-1}</math>.
===Example of the residue===
Over the complex numbers, the residue of a differential form with log poles along a divisor <math>D_j</math> can be viewed as the result of [[integration]] over loops in <math>X</math> around <math>D_j</math>. In this context, the residue may be called the [[Poincaré residue]].
▲
which has log poles along ''D''. Because the [[canonical bundle]] <math>K_{\mathbf{P}^2}=\Omega^2_{\mathbf{P}^2}</math> is isomorphic to the line bundle <math>\mathcal{O}(-3)</math>, the divisor of poles of <math>\omega</math> must have degree 3. So the divisor of poles of <math>\omega</math> consists only of ''D'' (in particular, <math>\omega</math> does not have a pole along the line <math>z=0</math> at infinity). The residue of ω along ''D'' is given by the holomorphic 1-form
▲:<math> \text{Res}_D(\omega) = \left. \frac{dy}{\partial g/\partial x} \right |_D =\left. -\frac{dx}{\partial g/\partial y} \right |_D = \left. -\frac{1}{2}\frac{dx}{y} \right |_D. </math>
It follows that <math>dx/y|_D </math> extends to a holomorphic one-form on the projective curve ''D'' in <math>\mathbf{P}^2</math>, an elliptic curve.
The residue map <math>H^0(\mathbf{P}^2,\Omega^2_{\mathbf{P}^2}(\log D))\to H^0(D,\Omega^1_D)</math> considered here is part of a linear map <math>H^2(\mathbf{P}^2-D,\mathbf{C})\to H^1(D,\mathbf{C})</math>, 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'':
:<math>\cdots \to H^{j-2}(D)\to H^j(X)\to H^j(X-D)\to H^{j-1}(D)\to\cdots.</math>
▲In the 19th-century theory of [[elliptic function]]s, 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_(group)|lattice]] <math>\Lambda</math> in '''C''' was called an "integral of the second kind" to mean that it could be written
:<math>\zeta(z)=\frac{\sigma'(z)}{\sigma(z)}.</math>
▲In modern terms, it follows that <math>\zeta(z)dz=d\sigma/\sigma</math> is a 1-form on '''C''' with logarithmic poles on <math>\Lambda</math>, since <math>\Lambda</math> is the zero set of the Weierstrass sigma function <math>\sigma(z).</math>
==Mixed Hodge theory for smooth varieties==
Over the complex numbers, Deligne proved a strengthening of [[Alexander Grothendieck]]'s algebraic de Rham theorem, relating [[
:<math> H^k(X, \Omega^{\bullet}_X(\log D)) \cong H^k(X-D,\mathbf{C})</math>
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.<ref>Deligne (1970), Corollaire II.6.10.</ref>
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:<math>W_{m}\Omega^p_X(\log D) = \begin{cases}
0 & m < 0\\
\Omega^{p-m}_X\cdot \Omega^m_X(\log D) & 0\leq m \leq p
\Omega^p_X(\log D) & m\geq p.
\end{cases} </math>
The resulting filtration on cohomology is the weight filtration:<ref>Peters & Steenbrink (2008), Theorem 4.2.</ref>
:<math> W_mH^k(X-D, \mathbf{C}) = \text{Im}(H^k(X, W_{m-k}\Omega^{\bullet}_X(\log D))\rightarrow H^k(X-D,\mathbf{C})).</math>
Building on these results, [[Hélène Esnault]] and [[Eckart Viehweg]] generalized the [[Nakano vanishing theorem|Kodaira–Akizuki–Nakano vanishing theorem]] in terms of logarithmic differentials. Namely, let ''X'' be a smooth complex projective variety of dimension ''n'', ''D'' a divisor with simple normal crossings on ''X'', and ''L'' an ample line bundle on ''X''. Then
:<math>H^q(X,\Omega^p_X(\log D)\otimes L)=0</math>
and
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*[[Borel–Moore homology]]
*[[Differential of the first kind]]
*[[Log structure]]
*[[Mixed Hodge structure]]
*[[Residue theorem]]
*[[Poincaré residue]]
==
{{
==References==
* {{Citation | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | title=Équations différentielles à points singuliers réguliers | series=Lecture Notes in Mathematics | publisher=[[Springer-Verlag]] | oclc=169357 | year=1970 | volume=163 | isbn=3540051902 | mr=0417174|doi=10.1007/BFb0061194|url=https://publications.ias.edu/node/355}}
* {{citation|author1-last=Deligne|author1-first=Pierre|author1-link=Pierre Deligne|title=Théorie de Hodge II|journal=Publ. Math. IHES |volume=40|pages=5–57|year=1971|mr=0498551|url=http://www.numdam.org/item/PMIHES_1971__40__5_0/}}
*{{Citation|author1-last=Esnault|author1-first=Hélène | author1-link=Hélène Esnault | author2-last=Viehweg | author2-first=Eckart | author2-link=Eckart Viehweg | title=Lectures on vanishing theorems | publisher=Birkhäuser| isbn=978-3-7643-2822-1 |mr=1193913 | year=1992|doi=10.1007/978-3-0348-8600-0}}
*{{
* {{citation|author1-last=Peters|author1-first=Chris A.M.|author2-last=Steenbrink|author2-first=Joseph H. M.|author2-link=Joseph H. M. Steenbrink|title=Mixed Hodge structures|publisher=Springer|year=2008|isbn=978-3-540-77017-6|mr=2393625|doi=10.1007/978-3-540-77017-6}}
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