Sheaf of logarithmic differential forms
In algebraic geometry, the sheaf of logarithmic differential p-forms [math]\displaystyle{ \Omega^p_X(\log D) }[/math] on a smooth projective variety X along a smooth divisor [math]\displaystyle{ D = \sum D_j }[/math] is defined and fits into the exact sequence of locally free sheaves:
- [math]\displaystyle{ 0 \to \Omega^p_X \to \Omega^p_X(\log D) \overset{\beta}\to \bigoplus_j {i_j}_*\Omega^{p-1}_{D_j} \to 0, \, p \ge 1 }[/math]
where [math]\displaystyle{ i_j\colon D_j \to X }[/math] are the inclusions of irreducible divisors (and the pushforwards along them are extension by zero), and [math]\displaystyle{ \beta }[/math] is called the residue map when p is 1.
For example,[1] if x is a closed point on [math]\displaystyle{ D_j, 1 \le j \le k }[/math] and not on [math]\displaystyle{ D_j, j \gt k }[/math], then
- [math]\displaystyle{ {du_1 \over u_1}, \dots, {du_k \over u_k}, \, du_{k+1}, \dots, du_n }[/math]
form a basis of [math]\displaystyle{ \Omega^1_X(\log D) }[/math] at x, where [math]\displaystyle{ u_j }[/math] are local coordinates around x such that [math]\displaystyle{ u_j, 1 \le j \le k }[/math] are local parameters for [math]\displaystyle{ D_j, 1 \le j \le k }[/math].
See also
Notes
- ↑ 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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