Freeness of hyperplane arrangements and related topics
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 23 (2014) no. 2, pp. 483-512.

Cet article est un développement des notes de l’exposé donnée par l’auteur à la conférence « Arrangements en Pyrénées », en juin 2012. Nous discutons les relations entre les problèmes de liberté et ceux de décomposabilité pour les fibrés vectoriels, plusieurs techniques qui prouvent la liberté pour des arrangements d’hyperplans, la théorie de K. Saito des dérivations primitives pour les arrangements de Coxeter, leur application à des problèmes combinatoires et quelques conjectures liées.

These are the expanded notes of the lecture by the author in “Arrangements in Pyrénées”, June 2012. We are discussing relations of freeness and splitting problems of vector bundles, several techniques proving freeness of hyperplane arrangements, K. Saito’s theory of primitive derivations for Coxeter arrangements, their application to combinatorial problems and related conjectures.

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Yoshinaga, Masahiko. Freeness of hyperplane arrangements and related topics. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 23 (2014) no. 2, pp. 483-512. doi : 10.5802/afst.1413. http://www.numdam.org/articles/10.5802/afst.1413/

[1] Abe (T.).— Exponents of 2-multiarrangements and freeness of 3-arrangements. to appear in J. Alg. Combin.

[2] Abe (T.), Numata (Y.).— Exponents of 2-multiarrangements and multiplicity lattices. J. of Alg. Comb. 35, no. 1, p. 1-17 (2012). | MR | Zbl

[3] Abe (T.), Terao (H.), Wakefield (M.).— The characteristic polynomial of a multiarrangement, Adv. in Math. 215, p. 825-838 (2007). | MR | Zbl

[4] Abe (T.), Yoshinaga (M.).— Splitting criterion for reflexive sheaves. Proc. Amer. Math. Soc. 136, no. 6, p. 1887-1891 (2008). | MR | Zbl

[5] Abe (T.), Yoshinaga (M.).— Coxeter multiarrangements with quasi-constant multiplicities. J. Algebra 322, no. 8, p. 2839-2847 (2009). | MR | Zbl

[6] Abe (T.), Yoshinaga (M.).— Free arrangements and coefficients of characteristic polynomials. arXiv:1109.0668, Preprint | MR

[7] Athanasiadis (C.A.).— Deformations of Coxeter hyperplane arrangements and their characteristic polynomials. Arrangements–Tokyo (1998), p. 1-26, Adv. Stud. Pure Math., 27, Kinokuniya, Tokyo (2000). | MR | Zbl

[8] Athanasiadis (C.A.).— Extended Linial hyperplane arrangements for root systems and a conjecture of Postnikov and Stanley. J. Algebraic Combin. 10, no. 3, p. 207-225 (1999). | MR | Zbl

[9] Athanasiadis (C.A.).— Generalized Catalan numbers, Weyl groups and arrangements of hyperplanes. Bull. London Math. Soc. 36, no. 3, p. 294-302 (2004). | MR | Zbl

[10] Bertone (C.), M. Roggero (M.).— Splitting type, global sections and Chern classes for torsion free sheaves on P n . J. Korean Math. Soc. 47, no. 6, p. 1147-1165 (2010). | MR | Zbl

[11] Chevalley (C.).— Invariants of finite groups generated by reflections. Amer. J. Math. 77, p. 778-782 (1955). | MR | Zbl

[12] Edelman (P. H.), Reiner (V.).— Free arrangements and rhombic tilings. Discrete Comput. Geom. 15, no. 3, p. 307-340 (1996). | MR | Zbl

[13] Edelman (P. H.), Reiner (V.).— Not all free arrangements are K(π,1). Bull. Amer. Math. Soc. (N.S.) 32 p. 61-65 (1995). | MR | Zbl

[14] Elencwajg (G.), Forster (O.).— Bounding cohomology groups of vector bundles on Pn. Math. Ann. 246, no. 3, p. 251-270 (1979/80). | EuDML | MR | Zbl

[15] Gao (R.), Pei (D.), Terao (H.).— The Shi arrangement of the type D . Proc. Japan Acad. Ser. A Math. Sci., 88, p. 41-45 (2012). | MR | Zbl

[16] Hartshorne (R.).— Stable reflexive sheaves. Math. Ann. 254, p. 121-176 (1980). | EuDML | MR | Zbl

[17] Hartshorne (R.).— Algebraic Geometry. Springer GTM 52. | MR | Zbl

[18] Headley (P.).— On a family of hyperplane arrangements related to the affine Weyl groups. J. Alg. Comb. 6 p. 331-338 (1997). | MR | Zbl

[19] Mustaţǎ (M.), Schenck (H.).— The module of logarithmic p-forms of a locally free arrangement, J. Algebra 241, p. 699-719 (2001). | MR | Zbl

[20] Okonek (C.), Schneider (M.), Spindler (H.).— Vector bundles on complex projective spaces. Progress in Mathematics, 3. Birkhäuser, Boston, Mass. (1980). (Revised version). | MR | Zbl

[21] Orlik (P.), Solomon (L.).— Combinatorics and topology of complements of hyperplanes. Invent. Math. 56, no. 2, p. 167-189 (1980). | EuDML | MR | Zbl

[22] Orlik (P.) and Terao (H.).— Arrangements of hyperplanes. Grundlehren der Mathematischen Wissenschaften, 300. Springer-Verlag, Berlin, 1992. xviii+325 pp. | MR | Zbl

[23] Postnikov (A.), Stanley (R.).— Deformations of Coxeter hyperplane arrangements. J. Combin. Theory Ser. A 91, no. 1-2, p. 544-597 (2000). | MR | Zbl

[24] Saito (K.).— Theory of logarithmic differential forms and logarithmic vector fields. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27, no. 2, p. 265-291 (1980). | MR | Zbl

[25] Saito (K.).— Period mapping associated to a primitive form. Publ. Res. Inst. Math. Sci. 19, no. 3, p. 1231-1264 (1983). | MR | Zbl

[26] Saito (K.).— On a linear structure of the quotient variety by a finite reflexion group. Publ. Res. Inst. Math. Sci. 29, no. 4, p. 535-579 (1993). | MR | Zbl

[27] Saito (K.).— Uniformization of the orbifold of a finite reflection group. Frobenius manifolds, p. 265-320, Aspects Math., E36, Vieweg, Wiesbaden (2004). | MR | Zbl

[28] Schenck (H.).— A rank two vector bundle associated to a three arrangement, and its Chern polynomial. Adv. Math. 149, no. 2, p. 214-229 (2000). | MR | Zbl

[29] Schenck (H.).— Elementary modifications and line configurations in 2 . Comment. Math. Helv. 78, no. 3, p. 447-462 (2003). | MR | Zbl

[30] Schenck (H.).— S. Tohǎneanu, Freeness of conic-line arrangements in 2 . Comment. Math. Helv. 84, no. 2, p. 235-258 (2009). | MR | Zbl

[31] Schulze (M.).— Freeness and multirestriction of hyperplane arrangements. Compositio Math. 148, p. 799-806 (2012). | MR | Zbl

[32] Silvotti (R.).— On the Poincaré polynomial of a complement of hyperplanes. Math. Res. Lett. 4, no. 5, p. 645-661 (1997). | MR | Zbl

[33] Solomon (L.).— Invariants of finite reflection groups. Nagoya Math. J. 22 p. 57-64 (1963). | MR | Zbl

[34] Solomon (L.), Terao (H.).— A formula for the characteristic polynomial of an arrangement. Adv. in Math. 64, no. 3, p. 305-325 (1987). | MR | Zbl

[35] Solomon (L.), Terao (H.).— The double Coxeter arrangement. Comm. Math. Helv. 73, p. 237-258 (1998). | MR | Zbl

[36] Suyama (D.).— A basis construction for the Shi arrangement of the type B or C . arXiv:1205.6294

[37] Suyama (D.), Terao (H.).— The Shi arrangements and the Bernoulli polynomials. To appear in Bull. London Math. Soc. | MR | Zbl

[38] Terao (H.).— Arrangements of hyperplanes and their freeness. I, II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27, no. 2, p. 293-320 (1980). | MR | Zbl

[39] Terao (H.).— Generalized exponents of a free arrangement of hyperplanes and Shepherd-Todd-Brieskorn formula. Invent. Math. 63, no. 1, p. 159-179 (1981). | EuDML | MR | Zbl

[40] Terao (H.).— The exponents of a free hypersurface. Singularities, Part 2 (Arcata, Calif., 561-566 1981), Proc. Sympos. Pure Math., 40, Amer. Math. Soc., Providence, RI (1983). | MR | Zbl

[41] Terao (H.).— Multiderivations of Coxeter arrangements. Invent. Math. 148, no. 3, p. 659-674 (2002). | MR | Zbl

[42] Terao (H.).— The Hodge filtration and the contact-order filtration of derivations of Coxeter arrangements. Manuscripta Math. 118, no. 1, p. 1-9 (2005). | MR | Zbl

[43] Wakamiko (A.).— On the Exponents of 2-Multiarrangements. Tokyo J. Math. 30, p. 99-116 (2007). | MR | Zbl

[44] Wakefield (M.), Yuzvinsky (S.).— Derivations of an effective divisor on the complex projective line. Trans. A. M. S. 359, p. 4389-4403 (2007). | MR | Zbl

[45] Yamada (H.).— Lie group theoretical construction of period mapping. Math. Z. 220, no. 2, p. 231-255 (1995). | EuDML | MR | Zbl

[46] Yoshinaga (M.).— The primitive derivation and freeness of multi-Coxeter arrangements. Proc. Japan Acad., 78, Ser. A, p. 116-119 (2002). | MR | Zbl

[47] Yoshinaga (M.).— Characterization of a free arrangement and conjecture of Edelman and Reiner. Invent. Math. 157, no. 2, p. 449-454 (2004). | MR | Zbl

[48] Yoshinaga (M.).— On the freeness of 3-arrangements. Bull. London Math. Soc. 37, no. 1, p. 126-134 (2005). | MR | Zbl

[49] Yoshinaga (M.).— On the extendability of free multiarrangements. Arrangements, Local Systems and Singularities: CIMPA Summer School, Galatasaray University, Istanbul (2007), p. 273-281, Progress in Mathematics, 283, Birkhäuser, Basel (2009). | MR

[50] Yoshinaga (M.).— Arrangements, multiderivations, and adjoint quotient map of type ADE. Arrangements of Hyperplanes–Sapporo 2009, Advanced Studies in Pure Math., vol. 62. | Zbl

[51] Yuzvinsky (S.).— Cohomology of local sheaves on arrangement lattices. Proc. Amer. Math. Soc. 112, no. 4, p. 1207-1217 (1991). | MR | Zbl

[52] Yuzvinsky (S.).— The first two obstructions to the freeness of arrangements. Trans. Amer. Math. Soc. 335, no. 1, p. 231-244 (1993). | MR | Zbl

[53] Yuzvinsky (S.).— Free and locally free arrangements with a given intersection lattice. Proc. Amer. Math. Soc. 118, no. 3, p. 745-752 (1993). | MR | Zbl

[54] Ziegler (G.).— Multiarrangements of hyperplanes and their freeness. Singularities (Iowa City, IA, 1986), p. 345-359, Contemp. Math., 90, Amer. Math. Soc., Providence, RI (1989). | MR | Zbl

[55] Ziegler (G.).— Matroid representations and free arrangements. Trans. Amer. Math. Soc. 320, no. 2, p. 525-541 (1990). | MR | Zbl

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