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A criterion for the logarithmic differential operators to be generated by vector fields
Author(s):
Mathias
Schulze
Journal:
Proc. Amer. Math. Soc.
135
(2007),
3631-3640.
MSC (2000):
Primary 32C38, 13A30
Posted:
August 7, 2007
MathSciNet review:
2336579
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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
 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.
References:
-
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- Francisco J. Calderón-Moreno, Logarithmic differential operators and logarithmic de Rham complexes relative to a free divisor, Ann. Sci. École Norm. Sup. (4) 32 (1999), no. 5, 701-714. MR 1710757 (2000g:32010)
- [CMNM05]
- Francisco Javier Calderón-Moreno and Luis Narváez-Macarro, Dualité et comparaison sur les complexes de de Rham logarithmiques par rapport aux diviseurs libres, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 1, 47-75. MR 2141288 (2006d:32008)
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- Gert-Martin Greuel and Gerhard Pfister, A Singular introduction to commutative algebra, Springer-Verlag, Berlin, 2002, With contributions by Olaf Bachmann, Christoph Lossen and Hans Schönemann, With 1 CD-ROM (Windows, Macintosh, and UNIX). MR 1930604 (2003k:13001)
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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:
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
Posted:
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
Copyright of article:
Copyright
2007,
American Mathematical Society
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