Sheaf Of Logarithmic Differential Forms

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

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

where ${\displaystyle i_j:D_j\to X}$ 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 ${\displaystyle D_j,1\le j\le k}$ and not on ${\displaystyle D_j,j>k}$, then

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

form a basis of ${\displaystyle {\mathrm{\Omega }}_X^1(\log D)}$ at x, where ${\displaystyle u_j}$ are local coordinates around x such that ${\displaystyle u_j,1\le j\le k}$ are local parameters for ${\displaystyle D_j,1\le j\le k}$.

• Poincaré residue

• 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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