Sheaf of logarithmic differential forms

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

$0\to {\mathrm{\Omega }}_X^p\to {\mathrm{\Omega }}_X^p(\log D)\stackrel{\beta }{\to }\underset{j}{⨁}{i_j}_{*}{\mathrm{\Omega }}_{D_j}^{p-1}\to 0,p\ge 1$

where $i_j:D_j\to X$ are the inclusions of irreducible divisors (and the pushforwards along them are extension by zero), and $\beta$ 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

$\frac{d{u}_1}{u_1},\dots ,\frac{d{u}_k}{u_k},d{u}_{k+1},\dots ,d{u}_n$

form a basis of ${\mathrm{\Omega }}_X^1(\log D)$ 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

• 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.