Sheaf of logarithmic differential forms

In algebraic geometry, the sheaf of logarithmic differential p-forms ΩXp(log⁡D) on a smooth projective variety X along a smooth divisor D=∑Dj is defined and fits into the exact sequence of locally free sheaves:

0->ΩXp->ΩXp(log⁡D)->β⨁jij*ΩDjp−1->0,p≥1

where ij:Dj->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 Dj,1≤j≤k and not on Dj,j>k, then

du1u1,…,dukuk,duk+1,…,dun

form a basis of ΩX1(log⁡D) at x, where uj are local coordinates around x such that uj,1≤j≤k are local parameters for Dj,1≤j≤k.

## See also

* Poincaré residue

## Notes

1. ↑ Deligne 2008, Part II, Lemma 3.2.1.

## References

* Aise Johan de Jong, Algebraic de Rham cohomology.
* Deligne, Pierre (2008) (in fr). Equations Differentielles a Points Singuliers Reguliers. Springer. p. 163. ISBN 978-3-540-05190-9. OCLC 466097729. http://www.springerlink.com/openurl.asp?genre=book&isbn=978-3-540-05190-9.

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