Logarithmic form: Difference between revisions

Revision as of 02:36, 28 April 2023

Revision as of 02:37, 28 April 2023

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==Logarithmic differentials in algebraic geometry==

In algebraic geometry, the vector bundle of '''logarithmic differential ''p''-forms''' <math>\Omega^p_X(\log D)</math> on a [[smooth scheme]] ''X'' over a field, with respect to a [[divisor (algebraic geometry)|divisor]] <math>D = \sum D_j</math> with simple normal crossings, is defined as above: sections of <math>\Omega^p_X(\log D)</math> are (algebraic) differential forms ω on <math>X-D</math> such that both ω and ''d''ω have a pole of order at most one along ''D''.<ref>Deligne (1970), Lemma II.3.2.1.</ref> Explicitly, for a closed point ''x'' that lies in <math>D_j</math> for <math>1 \le j \le k</math> and not in <math>D_j</math> for <math>j > k</math>, let <math>u_j</math> be regular functions on some open neighborhood ''U'' of ''Xx'' such that <math>D_j</math> is the closed subscheme defined by <math>u_j=0</math> inside ''U'' for <math>1 \le j \le k</math>, and ''x'' is the closed subscheme of ''U'' defined by <math>u_1=\cdots=u_n=0</math>. Then a basis of sections of <math>\Omega^1_X(\log D)</math> on ''U'' is given by:

:<math>{du_1 \over u_1}, \dots, {du_k \over u_k}, \, du_{k+1}, \dots, du_n.</math>

This describes the vector bundle <math>\Omega^1_X(\log D)</math> on ''X'', and then <math>\Omega^p_X(\log D)</math> is the ''p''th exterior power of <math>\Omega^1_X(\log D)</math>.