Logarithmic form

In algebraic geometry and the theory of complex manifolds, a logarithmic differential form is a differential form with poles of a certain kind. The concept was introduced by Pierre Deligne. 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 (a sum of distinct codimension-1 complex subspaces), and ω a holomorphic p-form on X−D. If both ω and dω 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 subsheaf of the meromorphic p-forms on X, denoted

Ω

X

p

(

log

⁡

D

)

.

{\displaystyle \Omega _{X}^{p}(\log D).}

The name comes from the fact that in complex analysis,

d

(

log

⁡

z

)

=

d

z

/

z

{\displaystyle d(\log z)=dz/z}

; here

d

z

/

z

{\displaystyle dz/z}

is a typical example of a 1-form on the complex numbers C with a logarithmic pole at the origin. Differential forms such as

d

z

/

z

{\displaystyle dz/z}

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

Ω

X

p

(

log

⁡

D

)

{\displaystyle \Omega _{X}^{p}(\log D)}

and the fact that the exterior derivative d satisfies d2 = 0, one has

d

Ω

X

p

(

log

⁡

D

)

(

U

)

⊂

Ω

X

p

+

1

(

log

⁡

D

)

(

U

)

{\displaystyle d\Omega _{X}^{p}(\log D)(U)\subset \Omega _{X}^{p+1}(\log D)(U)}

for every open subset U of X. Thus the logarithmic differentials form a complex of sheaves

(

Ω

X

∙

(

log

⁡

D

)

,

d

)

{\displaystyle (\Omega _{X}^{\bullet }(\log D),d)}

, known as the logarithmic de Rham complex associated to the divisor D. This is a subcomplex of the direct image

j

∗

(

Ω

X

−

D

∙

)

{\displaystyle j_{*}(\Omega _{X-D}^{\bullet })}

, where

j

:

X

−

D

→

X

{\displaystyle j:X-D\rightarrow X}

is the inclusion and

Ω

X

−

D

∙

{\displaystyle \Omega _{X-D}^{\bullet }}

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

j

∗

(

Ω

X

−

D

∙

)

{\displaystyle j_{*}(\Omega _{X-D}^{\bullet })}

generated by the holomorphic differential forms

Ω

X

∙

{\displaystyle \Omega _{X}^{\bullet }}

together with the 1-forms

d

f

/

f

{\displaystyle df/f}

for holomorphic functions

f

{\displaystyle f}

that are nonzero outside D. Note that

d

(

f

g

)

f

g

=

d

f

f

+

d

g

g

.

{\displaystyle {\frac {d(fg)}{fg}}={\frac {df}{f}}+{\frac {dg}{g}}.}

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

z

1

,

…

,

z

n

{\displaystyle z_{1},\ldots ,z_{n}}

such that x is the origin and D is defined by the equation

z

1

⋯

z

k

=

0

{\displaystyle z_{1}\cdots z_{k}=0}

for some

0

≤

k

≤

n

{\displaystyle 0\leq k\leq n}

.