Logarithmic form: Difference between revisions

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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 163.to Berlin-Heidelberg-Newgive York:information Springer-Verlagabout an open submanifold (the complement of the divisor of poles).</ref>

Let ''X'' be a complex manifold, ''D'' ⊂ ''X'' a [[Divisor (algebraic geometry)|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 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'', 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 numbers '''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 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>

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.

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 ''X''. 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.

===Higher-dimensional example===

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> \omega =\frac{dx\wedge dy}{g(x,y)} </math>

which has logarithmic poles along ''D''. The [[Poincaré residue]]<ref>Griffiths & Harris (1994), section 3.5.</ref> of ω along ''D'' is given by the holomorphic one-form

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) = \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>

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.

===Historical Hodge theory 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

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}_Yzeta(\log Dz)=\rightarrow j_*\Omega_frac{X\sigma'(z)}^{\bulletsigma(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>.

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>.

===Logarithmic differentials and singular cohomology===

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.

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-D,\mathbf{C}) \cong H^k(X, \Omega^{\bullet}_X(\log D)),</math>

where the right 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>

==Logarithmic differentials in algebraic geometry==

== Sheaf of logarithmic forms ==

In [[algebraic geometry]], the [[sheafvector (mathematics)|sheaf]]bundle of '''logarithmic differential ''p''-forms''' <math>\Omega^p_X(\log D)</math> on a [[Smoothsmooth scheme|smooth]] [[projective variety]] ''X'' alongover a smoothfield, 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 ω and fits''d''ω intohave 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 [[exactclosed sequence]]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 locally''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 freegiven sheavesby:

:<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^p_X \to \Omega^p_X(\log D) \overset{\beta}\to \oplus_j {i_j}_*\Omega^{p-1}_{D_j} \to 0, \, p \ge 1</math>

: <math>0 \to \Omega^1_X \to \Omega^1_X(\log D) \overset{\beta}\to \oplus_j ({i_j})_*\mathcal{O}_{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

where <math>i_j: D_j \to X</math> are the inclusions of irreducible divisors (and the pushforwards along them are extension by zero), and β is called the [[residue map]] when ''p'' is 1.

:<math>\text{Res}_{D_j}\bigg(\frac{df}{f}\wedge \alpha\bigg)=\alpha|_D</math>

for any regular <math>(p-1)</math>-form <math>\alpha</math>.<ref>Deligne (1970), sections II.3.5 to II.3.7.</ref>

==Mixed Hodge theory for smooth varieties==

For example,<ref>{{harvnb|Deligne|loc=Part II, Lemma 3.2.1.}}</ref> if ''x'' is a closed point on <math>D_j, 1 \le j \le k</math> and not on <math>D_j, j > k</math>, then

Over the complex numbers, Deligne proved a strengthening of [[Alexander Grothendieck]]'s algebraic de Rham theorem, relating [[singular cohomology]] with [[coherent sheaf cohomology]]. Namely, for any smooth scheme ''X'' over '''C''' with a divisor with simple normal crossings ''D'', there is a natural isomorphism

:<math>{du_1 \over u_1}, \dots, {du_k \over u_k}, \, du_{k+1}, \dots, du_n</math>

:<math> H^k(X, \Omega^{\bullet}_X(\log D)) \cong H^k(X-D,\mathbf{C})</math>

form a basis of <math>\Omega^1_X(\log D)</math> at ''x'', where <math>u_j</math> are local coordinates around ''x'' such that <math>u_j, 1 \le j \le k</math> are local parameters for <math>D_j, 1 \le j \le k</math>.<!--(Roughly, if <math>f = g \prod_1^k u^{a_j}</math> is a section around ''x'', then <math>d(\log(f)) = {dg \over g} + \sum_1^k a_i {du_j \over u_j}</math> and use this to get the claim.)-->

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-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>

[[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>

==See also==

Line 66 ⟶ 83:

*[[Borel–Moore homology]]

*[[Differential of the first kind]]

*[[ResidueMixed TheoremHodge structure]]

*[[Residue theorem]]

*[[Poincaré residue]]

Line 72 ⟶ 90:

{{Reflist}}

* {{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}}

*Aise Johan de Jong, [http://math.columbia.edu/~dejong/seminar/note_on_algebraic_de_Rham_cohomology.pdf Algebraic de Rham cohomology].

* {{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/}}

*[[Pierre Deligne]], Equations Différentielles à Points Singuliers Réguliers. Lecture Notes in Math. 163.

*{{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}}

*{{cite book |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|publisher=Springer|year=2008|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]]