In algebraic geometry, the sheaf of logarithmic differential p-forms
Ω
X
p
(
log
D
)
on a smooth projective variety X along a smooth divisor
D
=
∑
D
j
is defined and fits into the exact sequence of locally free sheaves:
0
→
Ω
X
p
→
Ω
X
p
(
log
D
)
→
β
⊕
j
i
j
∗
Ω
D
j
p
−
1
→
0
,
p
≥
1
where
i
j
:
D
j
→
X
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.
For example, if x is a closed point on
D
j
,
1
≤
j
≤
k
and not on
D
j
,
j
>
k
, then
d
u
1
u
1
,
…
,
d
u
k
u
k
,
d
u
k
,
…
,
d
u
n
form a basis of
Ω
X
1
(
log
D
)
at x, where
u
j
are local coordinates around x such that
u
j
,
1
≤
j
≤
k
are local parameters for
D
j
,
1
≤
j
≤
k
.
