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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,
:<math>\Omega^p_X(\log D).</math>
The name comes from the fact that in [[complex analysis]], <math>d(\log z)=dz/z</math>; here <math>dz/z</math> is a typical example of a 1-form on the [[complex number]]s '''C''' with a logarithmic pole at the origin. Differential forms such as <math>dz/z</math> make sense in a purely algebraic context, where there is no analog of the [[logarithm]] function.
==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>
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> \
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'':
:<math>0 \to \Omega^
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
where the sums run over all irreducible components of given dimension of intersections of the divisors ''D''<sub>''j''</sub>. Here again, β is called the residue map.
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|_{D_j}</math>
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 [[integral|integration]] over loops in <math>X</math> around <math>D_j</math>. In this context, the residue may be called the [[Poincaré residue]].
For an explicit example,<ref>Griffiths & Harris (1994), section 2.1.</ref> consider an elliptic curve ''D'' in the complex [[projective plane]] <math>\mathbf{P}^2=\{ [x,y,z]\}</math>, defined in affine coordinates <math>z=1</math> by the equation <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 [[hypersurface]] of degree 3 in <math>\mathbf{P}^2</math> and, in particular, a divisor with simple normal crossings. There is a meromorphic 2-form on <math>\mathbf{P}^2</math> given in affine coordinates by
:<math>\omega =\frac{dx\wedge dy}{g(x,y)},</math>
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>
==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=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 [[coherent sheaf cohomology]] with [[singular cohomology]]. Namely, for any smooth scheme ''X'' over '''C''' with a divisor with simple normal crossings ''D'', there is a natural isomorphism
:<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>
Moreover, when ''X'' is smooth and [[proper morphism|proper]] over '''C''', the resulting [[spectral sequence]]
:<math>E_1^{pq} = H^q(X,\Omega^p_X(\log D)) \Rightarrow H^{p+q}(X-D,\mathbf{C})</math>
degenerates at <math>E_1</math>.<ref>Deligne (1971), Corollaire 3.2.13.</ref> So the cohomology of <math>X-D</math> with complex coefficients has a decreasing filtration, the '''Hodge filtration''', whose associated graded vector spaces are the algebraically defined groups <math>H^q(X,\Omega^p_X(\log D))</math>.
This is part of the [[mixed Hodge structure]] which Deligne defined on the cohomology of any [[complex algebraic variety]]. In particular, there is also a '''weight filtration''' on the rational cohomology of <math>X-D</math>. The resulting filtration on <math>H^*(X-D,\mathbf{C})</math> can be constructed using the logarithmic de Rham complex. Namely, define an increasing filtration <math>W_{\bullet} \Omega^p_X(\log D) </math> by
:<math>W_{m}\Omega^p_X(\log D) = \begin{cases}
0 & m < 0\\
\Omega^
\Omega^
\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
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
:<math>H^q(X,\Omega^p_X(\log D)\otimes O_X(-D)\otimes L)=0</math>
for all <math>p+q>n</math>.<ref>Esnault & Viehweg (1992), Corollary 6.4.</ref>
==
*[[Adjunction formula]]
*[[Borel–Moore homology]]
*[[Differential of the first kind]]
*[[Log structure]]
*[[Mixed Hodge structure]]
*[[Residue theorem]]
*[[Poincaré residue]]
==Notes==
{{reflist|30em}}
==References==
* {{Citation | last1=Deligne | first1=Pierre | title=Equations Différentielles à Points Singuliers Réguliers | author1-link=Pierre Deligne | 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| url-access=subscription }}
* {{citation|author1-last=Deligne|author1-first=Pierre|author1-link=Pierre Deligne|title=Théorie de Hodge II|journal=Publ. Math. IHÉS |volume=40|pages=5–57|year=1971|doi=10.1007/BF02684692 |mr=0498551|s2cid=118967613 |url=http://www.numdam.org/item/PMIHES_1971__40__5_0/|url-access=subscription}}
*{{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|last1=Griffiths |first1=Phillip | author-link1=Phillip Griffiths |last2=Harris |first2=Joseph |author-link2=Joe Harris (mathematician) | title=Principles of algebraic geometry | series=Wiley Classics Library | publisher=Wiley Interscience | year=1994 | orig-year=1978 | isbn=0-471-05059-8 | mr=0507725|doi=10.1002/9781118032527}}
* {{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|series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics |publisher=Springer|year=2008|volume=52 |isbn=978-3-540-77017-6|mr=2393625|doi=10.1007/978-3-540-77017-6}}
==External links==
*[[Aise Johan de Jong]], [http://math.columbia.edu/~dejong/seminar/note_on_algebraic_de_Rham_cohomology.pdf Algebraic de Rham cohomology].
[[Category:Complex analysis]]
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