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Vanishing and bases for cohomology of partially trivial local systems on hyperplane arrangements

Author(s): Yukihito Kawahara
Journal: Proc. Amer. Math. Soc. 133 (2005), 1907-1915.
MSC (2000): Primary 14F40; Secondary 32S22, 55N25
Posted: January 21, 2005
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Abstract | References | Similar articles | Additional information

Abstract: In this paper we prove a vanishing theorem and construct bases for the cohomology of partially trivial local systems on complements of hyperplane arrangements. As a result, we obtain a non-resonance condition for partially trivial local systems.

References:

[Ar]
Arapura, D., Geometry of cohomology support loci for local systems I, J. Alg. Geom. 6 (1997), 563-597. MR 1487227 (98m:32043)

[AK]
Aomoto, K. and Kita, M., Hypergeometric functions (in Japanese), Tokyo Springer, (1994).

[Co1]
Cohen, D., Morse inequalities for arrangements, Adv. Math. 134 (1998), 43-45. MR 1612383 (98m:52017)

[Co2]
Cohen, D., Triples of arrangements and local systems, Proc. Amer. Math. Soc. 130 (2002), no. 10, 3025-3031. MR 1908926 (2003g:32044)

[CO1]
Cohen, D. and Orlik, P., Arrangements and local systems, Math. Res. Lett. 7 (2000), no.2-3, 299-316. MR 1764324 (2001i:57040)

[CO2]
Cohen, D. and Orlik, P., Gauss-Manin connections for arrangements, I Eigenvalues, Compositio Math. 136 (2003), no. 3, 299-316. MR 1977008 (2004a:32042)

[CS]
Cohen, D. and Suciu, A., Characteristic varieties of arrangements, Math. Proc. Cambridge Philos. Soc. 127 (1999), 33-54. MR 1692519 (2000m:32036)

[De]
Deligne, P., Équations différentielles à points singuliers réguliers, Lecture Notes in Math. 163 (1970). MR 0417174 (54:5232)

[ESV]
Esnault, H., Schechtman, V., and Viehweg, E., Cohomology of local systems on the complement of hyperplanes, Invent. Math. 109 (1992), 557-561; Erratum 112 (1993), 447. MR 1176205 (93g:32051) MR 1213111 (94b:32061)

[EFK]
Etingof, P., Frenkel, I., and Kirillov, A. Jr., Lectures on representation theory and Knizhnik-Zamolodchikov equations, Math. Surveys and Monographs 58, Amer. Math. Soc., 1998. MR 1629472 (2001b:32028)

[Fa]
Falk, M., Combinatorial and algebraic structure in Orlik-Solomon algebras, European J. Combin. 22 (2001), no. 5, 687-698. MR 1845492 (2002f:52023)

[FT]
Falk, M. and Terao, H., $\beta$NBC-bases for cohomology of local systems on hyperplane complements, Trans. Amer. Math. Soc. 349 (1997), no. 1, 189-202. MR 1401770 (97g:52029)

[Gel]
Gelfand, I., General theory of hypergeometric functions, Soviet Math. Dokl. 33 (1986), 573-577.

[Ka]
Kawahara, Y., The twisted De Rham cohomology for Basic Constructions of Hyperplane arrangements and its applications, preprint.

[Li]
Libgober, A., Characteristic varieties of algebraic curves, Applications of algebraic geometry to coding theory, physics and computation (Eilat, 2001), 215-254, NATO Sci. Ser. II Math. Phys. Chem., 36, Kluwer Acad. Publ., Dordrecht, 2001. MR 1866902 (2003e:14008)

[LY]
Libgober, A. and Yuzvinsky, S., Cohomology of the Orlik-Solomon algebras and local systems, Compositio Math. 121 (2000), no. 3, 337-361. MR 1761630 (2001j:52032)

[OT]
Orlik, P. and Terao, H., Arrangements of Hyperplanes, Grundlehren der mathematischen Wissenschaften 300, Springer-Verlag, (1992). MR 1217488 (94e:52014)

[OT2]
-, Arrangements and Hypergeometric integrals, MSJ, Mem. vol.9, Math. Soc. Japan, (2001). MR 1814008 (2003a:32048)

[STV]
Schechtman, V., Terao, H., and Varchenko, A., Local systems over complements of hyperplanes and the Kac-Kazhdan conditions for singular vectors, J. Pure Appl. Algebra 100 (1995), 93-102. MR 1344845 (96j:32047)

[Te]
Terao, H., Moduli space of combinatorially equivalent arrangements of hyperplanes and logarithmic Gauss-Manin connections, Topology Appl. 118 (2002), 255-274. MR 1877728 (2003e:32049)

[Va]
Varchenko, A., Multidimensional Hypergeometric Functions and Representation Theory of Lie Algebras and Quantum Groups, Adv. Ser. Math. Phys., vol. 21, World Scientific, River Edge, NJ, 1995. MR 1384760 (99i:32029)

[Yu1]
Yuzvinsky, S., Cohomology of the Brieskorn-Orlik-Solomon algebras, Comm. Algebra 23 (1995), 5339-5354. MR 1363606 (97a:52023)

[Yu2]
Yuzvinsky, S., Orlik-Solomon algebras in algebra and topology, (Russian) Uspekhi Mat. Nauk 56 (2001), no. 2(338), 87-166; translation in Russian Math. Surveys 56 (2001), no. 2, 293-364. MR 1859708 (2002i:14047)

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Additional Information:

Yukihito Kawahara
Affiliation: Department of Mathematics, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji-shi, Tokyo 192-0397, Japan
Email: ykawa@comp.metro-u.ac.jp

DOI: 10.1090/S0002-9939-05-07745-2
PII: S 0002-9939(05)07745-2
Keywords: Hyperplane arrangement, local system, twisted de Rham cohomology.
Received by editor(s): November 7, 2003
Received by editor(s) in revised form: February 26, 2004
Posted: January 21, 2005
Communicated by: Paul Goerss
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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