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In algebraic geometry, the sheaf of logarithmic differential p-forms Ω X p {\displaystyle \Omega _{X}^{p}} on a smooth projective variety X along a smooth divisor D = ∑ D j {\displaystyle D=\sum D_{j}} is defined and fits into the exact sequence of locally free sheaves:
where i j : D j → X {\displaystyle i_{j}\colon D_{j}\to X} are the inclusions of irreducible divisors , and β {\displaystyle \beta } is called the residue map when p is 1.
For example, if x is a closed point on D j , 1 ≤ j ≤ k {\displaystyle D_{j},1\leq j\leq k} and not on D j , j > k {\displaystyle D_{j},j>k} , then
form a basis of Ω X 1 {\displaystyle \Omega _{X}^{1}} at x, where u j {\displaystyle u_{j}} are local coordinates around x such that u j , 1 ≤ j ≤ k {\displaystyle u_{j},1\leq j\leq k} are local parameters for D j , 1 ≤ j ≤ k {\displaystyle D_{j},1\leq j\leq k}.