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Logarithmic Comparison Theorem and some Euler homogeneous free divisors
Author(s):
F.
J.
Castro-Jiménez;
J.
M.
Ucha-Enríquez
Journal:
Proc. Amer. Math. Soc.
133
(2005),
1417-1422.
MSC (2000):
Primary 32S20;
Secondary 14F10, 32S40
Posted:
November 1, 2004
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Abstract:
Let  be a free divisor germ in a complex manifold  of dimension  . It is an open problem to find out which are the properties required for  to satisfy the so-called Logarithmic Comparison Theorem (LCT), that is, when the complex of logarithmic differential forms computes the cohomology of the complement of  . We give a family of Euler homogeneous free divisors which, somewhat unexpectedly, does not satisfy the LCT.
References:
-
- 1.
- Calderón-Moreno F.J., D. Mond, L. Narváez-Macarro and F.J. Castro-Jiménez. Logarithmic Cohomology of the Complement of a Plane Curve. Comment. Math. Helv. 77 (2002), no. 1, 24-38. MR 1898392 (2003e:32047)
- 2.
- Castro-Jiménez, F.J., Mond, D. and Narváez-Macarro, L. Cohomology of the complement of a free divisor. Trans. Amer. Math. Soc. 348 (1996), no. 8, 3037-3049. MR 1363009 (96k:32072)
- 3.
- Castro-Jiménez, F.J., Ucha-Enríquez, J.M. Testing the Logarithmic Comparison Theorem In preparation.
- 4.
- Castro-Jiménez, F.J., Ucha-Enríquez, J.M. Explicit comparison theorems for
 -modules. J. Symbolic Comput., Special Issue on Effective Methods in Rings of Differential Operators, 32 (2001) no. 6, 677-685. MR 1866710 (2003m:16034) - 5.
- Castro-Jiménez, F. J. and J. M. Ucha-Enríquez. Free Divisors and Duality for
 -Modules Proc. Steklov Inst. of Math., volume 238, pages 88-96. 2002. MR 1969307 (2004a:32017) - 6.
- Grothendieck, A. On the de Rham cohomology of algebraic varieties. Publ. Math. de l'I.H.E.S. 29 (1966), 95-103. MR 0199194 (33:7343)
- 7.
- Saito, K. Theory of logarithmic differential forms and logarithmic vector fields. J. Fac. Sci. Univ. Tokyo 27:256-291, 1980. MR 0586450 (83h:32023)
- 8.
- Torrelli, T. Equations fonctionnelles pour une fonction sur un espace singulier. Ph. D. Thesis, 1998.
- 9.
- Torrelli, T. Sur les germes de fonctions méromorphes définis par un système differentiel d'ordre 1. Preprint, 2002
- 10.
- Ucha-Enríquez, J.M. Métodos constructivos en álgebras de operadores diferenciales. Ph. D. Thesis, Universidad de Sevilla, 1999.
- 11.
- Walther, U. Bernstein-Sato polynomial versus cohomology of the Milnor fiber for generic arrangements of hyperplanes Preprint math.AG/0204080.
- 12.
- Wiens, J. and Yuzvinsky, S. De Rham Cohomology of logarithmic forms on arrangements of hyperplanes. Trans. Amer. Math. Soc. 349 (1997) no. 4, 1653-1662. MR 1407505 (97h:52013)
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Additional Information:
F.
J.
Castro-Jiménez
Affiliation:
Departamento de Álgebra, Facultad de Matemáticas, Universidad de Sevilla, Apdo 1160, E-41080 Sevilla, Spain
Email:
castro@us.es
J.
M.
Ucha-Enríquez
Affiliation:
Departamento de Álgebra, Facultad de Matemáticas, Universidad de Sevilla, Apdo 1160, E-41080 Sevilla, Spain
Email:
ucha@us.es
DOI:
10.1090/S0002-9939-04-07678-6
PII:
S 0002-9939(04)07678-6
Keywords:
Free divisor,
Logarithmic Comparison Theorem,
$D$-modules,
Euler-homogeneous divisor
Received by editor(s):
July 21, 2003
Received by editor(s) in revised form:
January 8, 2004
Posted:
November 1, 2004
Additional Notes:
This work was partially supported by DGESIC BFM-2001-3164 and FQM-333.
Communicated by:
Michael Stillman
Copyright of article:
Copyright
2004,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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