Logarithmic form

In contexts including complex manifolds and algebraic geometry, a logarithmic differential form is a meromorphic differential form with poles of a certain kind. The concept was introduced by Deligne.^[1]

Let X be a complex manifold, and D ⊂ X a divisor and ω a holomorphic p-form on X−D. If ω and dω 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 subsheaf of the meromorphic p-forms on X with a pole along D, denoted

$\Omega^p_X(\log D).$

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

$\omega = \frac{df}{f} =\left(\frac{m}{z} + \frac{g'(z)}{g(z)}\right)dz$

for some meromorphic function (resp. rational function) $f(z) = z^mg(z)$, 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.

Holomorphic log complex

By definition of $\Omega^p_X(\log D)$ and the fact that exterior differentiation d satisfies d² = 0, one has

$d\Omega^p_X(\log D)(U)\subset \Omega^{p+1}_X(\log D)(U)$.

This implies that there is a complex of sheaves $( \Omega^{\bullet}_X(\log D), d)$, known as the holomorphic log complex corresponding to the divisor D. This is a subcomplex of $j_*\Omega^{\bullet}_{X-D}$, where $j:X-D\rightarrow X$ is the inclusion and $\Omega^{\bullet}_{X-D}$ 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 $\{D_{\nu}\}$ are the smooth, irreducible components of D, one has $D = \sum D_{\nu}$ with the $D_{\nu}$ meeting transversely. Locally D is the union of hyperplanes, with local defining equations of the form $z_1\cdots z_k = 0$ in some holomorphic coordinates. One can show that the stalk of $\Omega^1_X(\log D)$ at p satisfies^[2]

$\Omega_X^1(\log D)_p = \mathcal{O}_{X,p}\frac{dz_1}{z_1}\oplus\cdots\oplus\mathcal{O}_{X,p}\frac{dz_k}{z_k} \oplus \mathcal{O}_{X,p}dz_{k+1} \oplus \cdots \oplus \mathcal{O}_{X,p}dz_n$

and that

$\Omega_X^k(\log D)_p = \bigwedge^k_{j=1} \Omega_X^1(\log D)_p$.

Some authors, e.g.,^[3] 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 $g(x,y) = y^2 - f(x) = 0$, where $f(x) = x(x-1)(x-\lambda)$ and $\lambda\neq 0,1$ is a complex number. Then D is a smooth irreducible hypersurface in C² and, in particular, a divisor with simple normal crossings. There is a meromorphic two-form on C²

$\omega =\frac{dx\wedge dy}{g(x,y)}$

which has a simple pole along D. The Poincaré residue ^[3] of ω along D is given by the holomorphic one-form

$\text{Res}_D(\omega) = \frac{dy}{\partial g/\partial x}|_D =-\frac{dx}{\partial g/\partial y}|_D = -\frac{1}{2}\frac{dx}{y}|_D$.

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 $dx/y|_D$ extends to a holomorphic one-form on the projective closure of D in P², 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 $j: X\hookrightarrow Y$ 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

$\Omega^{\bullet}_Y(\log D)\rightarrow j_*\Omega_{X}^{\bullet}$

turns out to be a quasi-isomorphism. Thus

$H^k(X;\mathbf{C}) = \mathbb{H}^k(Y, \Omega^{\bullet}_Y(\log D))$

where $\mathbb{H}^{\bullet}$ denotes hypercohomology of a complex of abelian sheaves. There is^[2] a decreasing filtration $W_{\bullet} \Omega^p_Y(\log D)$ given by

$W_{m}\Omega^p_Y(\log D) = \begin{cases} 0 & m < 0\\ \Omega^p_Y(\log D) & m\geq p \\ \Omega^{p-m}_Y\wedge \Omega^m_Y(\log D) & 0\leq m \leq p \end{cases}$

which, along with the trivial increasing filtration $F^{\bullet}\Omega^p_Y(\log D)$ on logarithmic p-forms, produces filtrations on cohomology

$W_mH^k(X; \mathbf{C}) = \text{Im}(\mathbb{H}^k(Y, W_{m-k}\Omega^{\bullet}_Y(\log D))\rightarrow H^k(X; \mathbf{C}))$

$F^pH^k(X; \mathbf{C}) = \text{Im}(\mathbb{H}^k(Y, F^p\Omega^{\bullet}_Y(\log D))\rightarrow H^k(X; \mathbf{C}))$.

One shows^[2] that $W_mH^k(X; \mathbf{C})$ can actually be defined over Q. Then the filtrations $W_{\bullet}, F^{\bullet}$ on cohomology give rise to a mixed Hodge structure on $H^k(X; \mathbf{Z})$.

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 H¹(S), when by the Dolbeault isomorphism it is interpreted as the sheaf cohomology group H⁰(S,Ω); this is tautologous considering their definition. The H^1,0 direct summand in H¹(S), as well as being interpreted as H¹(S,O) where O is the sheaf of holomorphic functions on S, can be identified more concretely with a vector space of logarithmic differentials.

See also

• Algebraic Geometry
• Adjunction formula
• Differential of the first kind
• Residue Theorem

References

1. Deligne, Pierre. Equations différentielles à points singuliers réguliers. Lecture Notes in Mathematics. 163. Berlin-Heidelberg-New York: Springer-Verlag.
2. Chris A.M. Peters; Joseph H.M. Steenbrink (2007). Mixed Hodge Structures. Springer. ISBN 978-3-540-77017-6
3. Phillip A. Griffiths; Joseph Harris (1979). Principles of Algebraic Geometry. Wiley-Interscience. ISBN 0-471-05059-8.