Sheaf of logarithmic differential forms

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

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

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

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

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

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

See also

• Poincaré residue

Notes

1. Deligne, Part II, Lemma 3.2.1. harvnb error: no target: CITEREFDeligne (help)

References

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