Sheaf of logarithmic differential forms

In algebraic geometry, the sheaf of logarithmic differential p-forms $\Omega _{X}^{p}(\log D)$ 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:

$0\to \Omega _{X}^{p}\to \Omega _{X}^{p}(\log D){\overset {\beta }{\to }}\bigoplus _{j}{i_{j}}_{*}\Omega _{D_{j}}^{p-1}\to 0,\,p\geq 1$

where $i_{j}\colon D_{j}\to X$ are the inclusions of irreducible divisors (and the pushforwards along them are extension by zero), and $\beta$ is called the residue map when p is 1.

For example, if x is a closed point on $D_{j},1\leq j\leq k$ and not on $D_{j},j>k$, then

${du_{1} \over u_{1}},\dots ,{du_{k} \over u_{k}},\,du_{k+1},\dots ,du_{n}$

form a basis of $\Omega _{X}^{1}(\log D)$ at x, where $u_{j}$ are local coordinates around x such that $u_{j},1\leq j\leq k$ are local parameters for $D_{j},1\leq j\leq k$.

See also

• Poincaré residue

References

• Aise Johan de Jong, Algebraic de Rham cohomology.
• Deligne, Pierre (2008). Equations Differentielles a Points Singuliers Reguliers (in French). Springer. p. 163. ISBN 978-3-540-05190-9. OCLC 466097729.