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In contexts including [[complexalgebraic manifoldgeometry]]s and the theory of [[algebraiccomplex geometrymanifold]]s, a '''logarithmic''' [[differential form]] is a meromorphic differential form with [[pole (complex analysis)|poles]] of a certain kind. The concept was introduced by [[Pierre Deligne|Deligne]].<ref>Deligne (1970), Pierresection II.3.</ref> In ''Equationsshort, différentielleslogarithmic àdifferentials pointshave singuliersthe réguliers''.mildest Lecturepossible Notessingularities needed in Mathematics.order 163to give information about an open [[submanifold]] (the complement of the divisor of poles). Berlin-Heidelberg-New(This York:idea Springer-Verlagis made precise by several versions of [[de Rham's theorem]] discussed below.</ref>)

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

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 1 along ''D'', then ω is said to have a logarithmic pole along ''D''. ω is also known as a logarithmic ''p''-form. The ''p''-forms with log poles along ''D'' form a [[Sheaf (mathematics)|subsheaf]] of the meromorphic ''p''-forms on ''X'', denoted

:<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.

In the theory of [[Riemann surfaces]], one encounters logarithmic one-forms which have the local expression

==Logarithmic de Rham complex==

:<math>\omega = \frac{df}{f} =\left(\frac{m}{z} + \frac{g'(z)}{g(z)}\right)dz</math>

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''² = 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

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.

:<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>

==Holomorphic log complex==

:<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>

By definition of <math>\Omega^p_X(\log D)</math> and the fact that exterior differentiation ''d'' satisfies ''d''² = 0, one has

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> d\Omega^p_X(\log D)(U)\subset \Omega^{p+1}_X(\log D)(U) </math>.

:<math> \Omega_X^k(\log D) = \bigwedge^k \Omega_X^1(\log D).</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''.

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>

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 978-3-540-77017-6</ref>

:<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.

===Logarithmic differentials and singular cohomology===

===Higher-dimensional example===

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,

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'''² and, in particular, a divisor with simple normal crossings. There is a meromorphic two-form on '''C'''²

:<math> \omegaH^k(X, =\fracOmega^{dx\wedge dybullet}{g_X(x,y\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

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> \textOmega^{Res\bullet}_D_X(\omegalog D) = \frac{dy}{\partialrightarrow g/j_*\partial x}|_D =-\fracOmega_{dx}{\partial g/\partial y}|_D = X-\frac{1D}^{2}\frac{dx}{ybullet}|_D </math>.

being a [[quasi-isomorphism]].<ref>Deligne (1970), Proposition II.3.13.</ref>

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'''², a smooth elliptic curve.

==Logarithmic differentials in algebraic geometry==

=== Hodge theory ===

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:

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>{du_1 \Omega^{\bulletover u_1}_Y(, \logdots, {du_k D)\rightarrowover u_k}, j_*\Omega_, du_{Xk+1}^{, \bullet}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>.

turns out to be a quasi-isomorphism. Thus

:<math> H^k(X;\mathbf{C}) = \mathbb{H}^k(Y, \Omega^{\bullet}_Y(\log D))</math>

There is an [[exact sequence]] of [[coherent sheaves]] on ''X'':

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}0 \to \Omega^p_Y1_X \to \Omega^1_X(\log D) =\overset{\beta}\to \oplus_j ({i_j})_*\beginmathcal{casesO}_{D_j} \to 0,</math>

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>

where the sums run over all irreducible components of given dimension of intersections of the divisors ''D''_''j''. 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^p_Y{p-m}_X\cdot \Omega^m_X(\log D) & 0\leq m \geqleq p \\

\Omega^{p-m}_Y\wedge \Omega^m_Yp_X(\log D) & 0\leq m \leqgeq p.

\end{cases} </math>

The resulting filtration on cohomology is the weight filtration:<ref>Peters & Steenbrink (2008), Theorem 4.2.</ref>

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;-D, \mathbf{C}) = \text{Im}(\mathbb{H}^k(YX, W_{m-k}\Omega^{\bullet}_Y_X(\log D))\rightarrow H^k(X; -D,\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>.

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

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''^1,0 in ''H''¹(''S''), when by the [[Dolbeault isomorphism]] it is interpreted as the [[sheaf cohomology]] group ''H''⁰(''S'',Ω); this is tautologous considering their definition. The ''H''^1,0 direct summand in ''H''¹(''S''), as well as being interpreted as ''H''¹(''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.

:<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>

==See also==

*[[Algebraic Geometry]]

*[[Adjunction formula]]

*[[Borel–Moore homology]]

*[[Differential of the first kind]]

*[[ResidueLog Theoremstructure]]

*[[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 }}

{{Reflist}}

* {{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]]