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A criterion for the logarithmic differential operators to be generated by vector fields

Author: Mathias Schulze
Journal: Proc. Amer. Math. Soc. 135 (2007), 3631-3640
MSC (2000): Primary 32C38, 13A30
Published electronically: August 7, 2007
MathSciNet review: 2336579
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Abstract: We study divisors in a complex manifold in view of the property that the algebra of logarithmic differential operators along the divisor is generated by logarithmic vector fields. We give

  • a sufficient criterion for the property,
  • a simple proof of F.J. Calderón-Moreno's theorem that free divisors have the property,
  • a proof that divisors in dimension $ 3$ with only isolated quasi-homogeneous singularities have the property,
  • an example of a nonfree divisor with nonisolated singularity having the property,
  • an example of a divisor not having the property, and
  • an algorithm to compute the V-filtration along a divisor up to a given order.

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Additional Information

Mathias Schulze
Affiliation: Department of Mathematics, Oklahoma State University, 401 MSCS, Stillwater, Oklahoma 74078

Keywords: Free divisor, hyperplane arrangement, logarithmic differential operator, symmetric algebra, V-filtration
Received by editor(s): September 16, 2005
Received by editor(s) in revised form: September 2, 2006
Published electronically: August 7, 2007
Additional Notes: The author is grateful to M. Granger for many valuable discussions and comments and to F.J. Castro-Jiménez, L. Narváez-Macarro, and J.M. Ucha-Enríquez for explaining their results and ideas.
Communicated by: Michael Stillman
Article copyright: © Copyright 2007 American Mathematical Society