Sheaf of logarithmic differential forms

In algebraic geometry, the sheaf of logarithmic differential p-forms $\Omega^p_X(\log D)$ on a smooth projective variety X along a smooth divisor $D = \sum D_j$ is defined and fits into the exact sequence of locally free sheaves:

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

where $i_j: D_j \to X$ 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 $D_j, 1 \le j \le k$ and not on $D_j, j > k$, then

${du_1 \over u_1}, \dots, {du_k \over u_k}, \, du_k, \dots, du_n$

form a basis of $\Omega^1_X(\log D)$ at x, where $u_j$ are local coordinates around x such that $u_j, 1 \le j \le k$ are local parameters for $D_j, 1 \le j \le k$.

See also

• Poincaré residue

References

• de Jong, Algebraic de Rham cohomology.
• P. Deligne, Equations Différentielles à Points Singuliers Réguliers. Lecture Notes in Math. 163.