Non-vanishing of the twisted cohomology on the complement of hypersurfaces

Kawahara, Yukihito

doi:10.1090/S0002-9939-08-09224-1

Non-vanishing of the twisted cohomology on the complement of hypersurfaces
HTML articles powered by AMS MathViewer

by Yukihito Kawahara PDF

Proc. Amer. Math. Soc. 136 (2008), 1967-1975 Request permission

Abstract:

Generically, the cohomology with coefficients in a local system of rank one on the complement in $\mathbb {P}^n$ of the union of a finite number of hypersurfaces vanishes except in the highest dimension. We study the non-generic case, in which the cohomology in other dimensions does not vanish. When the hypersurfaces are hyperplanes, many examples of this kind are known. In this paper, we consider the case in which the hypersurfaces need not be hyperplanes. We prove that the hypersurfaces given by some particular linear systems have non-vanishing local system cohomologies.

References

Kazuhiko Aomoto, Un théorème du type de Matsushima-Murakami concernant l’intégrale des fonctions multiformes, J. Math. Pures Appl. (9) 52 (1973), 1–11. MR 396563
Koji Cho, A generalization of Kita and Noumi’s vanishing theorems of cohomology groups of local system, Nagoya Math. J. 147 (1997), 63–69. MR 1475166, DOI 10.1017/S0027763000006310
D. C. Cohen, A. Dimca, and P. Orlik, Nonresonance conditions for arrangements, Ann. Inst. Fourier (Grenoble) 53 (2003), no. 6, 1883–1896 (English, with English and French summaries). MR 2038782, DOI 10.5802/aif.1994
Daniel C. Cohen and Alexander I. Suciu, Characteristic varieties of arrangements, Math. Proc. Cambridge Philos. Soc. 127 (1999), no. 1, 33–53. MR 1692519, DOI 10.1017/S0305004199003576
Pierre Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, Vol. 163, Springer-Verlag, Berlin-New York, 1970 (French). MR 0417174, DOI 10.1007/BFb0061194
Alexandru Dimca, Sheaves in topology, Universitext, Springer-Verlag, Berlin, 2004. MR 2050072, DOI 10.1007/978-3-642-18868-8
Hélène Esnault, Vadim Schechtman, and Eckart Viehweg, Cohomology of local systems on the complement of hyperplanes, Invent. Math. 109 (1992), no. 3, 557–561. MR 1176205, DOI 10.1007/BF01232040
Michael Falk, Arrangements and cohomology, Ann. Comb. 1 (1997), no. 2, 135–157. MR 1629681, DOI 10.1007/BF02558471
Michael Falk and Hiroaki Terao, $\beta$nbc-bases for cohomology of local systems on hyperplane complements, Trans. Amer. Math. Soc. 349 (1997), no. 1, 189–202. MR 1401770, DOI 10.1090/S0002-9947-97-01844-8
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
Akio Hattori, Topology of $C^{n}$ minus a finite number of affine hyperplanes in general position, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), no. 2, 205–219. MR 379883
Yukihito Kawahara, The twisted de Rham cohomology for basic constructions of hyperplane arrangements and its applications, Hokkaido Math. J. 34 (2005), no. 2, 489–505. MR 2159008, DOI 10.14492/hokmj/1285766233
Yukihito Kawahara, Vanishing and bases for cohomology of partially trivial local systems on hyperplane arrangements, Proc. Amer. Math. Soc. 133 (2005), no. 7, 1907–1915. MR 2137854, DOI 10.1090/S0002-9939-05-07745-2
Yukihito Kawahara, The non-vanishing cohomology of Orlik-Solomon algebras, Tokyo J. Math. 30 (2007), no. 1, 223–238. MR 2328065, DOI 10.3836/tjm/1184963658
Michitake Kita, On vanishing of the twisted rational de Rham cohomology associated with hypergeometric functions, Nagoya Math. J. 135 (1994), 55–85. MR 1295817, DOI 10.1017/S0027763000004955
Michitake Kita and Masatoshi Noumi, On the structure of cohomology groups attached to the integral of certain many-valued analytic functions, Japan. J. Math. (N.S.) 9 (1983), no. 1, 113–157. MR 722538, DOI 10.4099/math1924.9.113
Toshitake Kohno, Homology of a local system on the complement of hyperplanes, Proc. Japan Acad. Ser. A Math. Sci. 62 (1986), no. 4, 144–147. MR 846350
A. Libgober, Characteristic varieties of algebraic curves, Applications of algebraic geometry to coding theory, physics and computation (Eilat, 2001) NATO Sci. Ser. II Math. Phys. Chem., vol. 36, Kluwer Acad. Publ., Dordrecht, 2001, pp. 215–254. MR 1866902
A. Libgober, Homotopy groups of complements to ample divisors, math.AG/0404341.
Anatoly Libgober and Sergey Yuzvinsky, Cohomology of the Orlik-Solomon algebras and local systems, Compositio Math. 121 (2000), no. 3, 337–361. MR 1761630, DOI 10.1023/A:1001826010964
Peter Orlik and Hiroaki Terao, Arrangements of hyperplanes, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 300, Springer-Verlag, Berlin, 1992. MR 1217488, DOI 10.1007/978-3-662-02772-1
Peter Orlik and Hiroaki Terao, Arrangements and hypergeometric integrals, MSJ Memoirs, vol. 9, Mathematical Society of Japan, Tokyo, 2001. MR 1814008
Vadim Schechtman, Hiroaki Terao, and Alexander Varchenko, Local systems over complements of hyperplanes and the Kac-Kazhdan conditions for singular vectors, J. Pure Appl. Algebra 100 (1995), no. 1-3, 93–102. MR 1344845, DOI 10.1016/0022-4049(95)00014-N
Sergey Yuzvinsky, Cohomology of the Brieskorn-Orlik-Solomon algebras, Comm. Algebra 23 (1995), no. 14, 5339–5354. MR 1363606, DOI 10.1080/00927879508825535
Sergey Yuzvinsky, Realization of finite abelian groups by nets in $\Bbb P^2$, Compos. Math. 140 (2004), no. 6, 1614–1624. MR 2098405, DOI 10.1112/S0010437X04000600