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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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 | References | Similar Articles | Additional Information

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
Email: mschulze@math.okstate.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08969-1
PII: S 0002-9939(07)08969-1
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