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,[1] 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

Deligne, Part II, Lemma 3.2.1.

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

Mathematics Encyclopedia