Local cohomology of logarithmic forms

[Cohomologie locale des formes logarithmiques]

Denham, G. ; Schenck, H. ; Schulze, M. ; Wakefield, M. ; Walther, U.

Annales de l'Institut Fourier, Tome 63 (2013), p. 1177-1203 / Harvested from Numdam

Résumé

Soit $X$ une variété algébrique lisse et $Y$ un diviseur sur $X$. Nous étudions la géométrie du schéma Jacobien de $Y$, les invariants homologiques provenant des formes différentielles logarithmiques le long de $Y$, et leur relation avec la propriété que $Y$ soit un diviseur libre. Nous considérons les arrangements d’hyperplans comme source d’exemples et de contre-exemples. En particulier, nous faisons un calcul complet de la cohomologie locale des formes logarithmiques d’arrangements d’hyperplans génériques.

Abstract

Let $Y$ be a divisor on a smooth algebraic variety $X$. We investigate the geometry of the Jacobian scheme of $Y$, homological invariants derived from logarithmic differential forms along $Y$, and their relationship with the property that $Y$ be a free divisor. We consider arrangements of hyperplanes as a source of examples and counterexamples. In particular, we make a complete calculation of the local cohomology of logarithmic forms of generic hyperplane arrangements.

Détail

Publié le : 2013-01-01

MR 3137483 | Zbl 1277.32030

DOI : https://doi.org/10.5802/aif.2787
Classification:  32S22,  52C35,  16W25
Mots clés: arrangements d’hyperplans, forme logarithmique différentielle, diviseur libre

Bibliographie

[1] Borel, A.; Grivel, P.-P.; Kaup, B.; Haefliger, A.; Malgrange, B.; Ehlers, F. Algebraic $D$-modules, Academic Press Inc., Boston, MA, Perspectives in Mathematics, Tome 2 (1987) | MR 882000

[2] Brieskorn, E. Singular elements of semi-simple algebraic groups, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, Gauthier-Villars, Paris (1971), pp. 279-284 | MR 437798 | Zbl 0223.22012

[3] Bruce, J. W. Functions on discriminants, J. London Math. Soc. (2), Tome 30 (1984) no. 3, pp. 551-567 | Article | MR 810963 | Zbl 0605.58011

[4] Calderón-Moreno, Francisco J. Logarithmic differential operators and logarithmic de Rham complexes relative to a free divisor, Ann. Sci. École Norm. Sup. (4), Tome 32 (1999) no. 5, pp. 701-714 | Article | Numdam | MR 1710757 | Zbl 0955.14013

[5] Calderón Moreno, Francisco J.; Mond, David; Narváez Macarro, Luis; Castro Jiménez, Francisco J. Logarithmic cohomology of the complement of a plane curve, Comment. Math. Helv., Tome 77 (2002) no. 1, pp. 24-38 | Article | MR 1898392 | Zbl 1010.32016

[6] Castro-Jiménez, Francisco J.; Narváez-Macarro, Luis; Mond, David Cohomology of the complement of a free divisor, Trans. Amer. Math. Soc., Tome 348 (1996) no. 8, pp. 3037-3049 | Article | MR 1363009 | Zbl 0862.32021

[7] Cohen, D.; Denham, G.; Falk, M.; Varchenko, A. Critical points and resonance of hyperplane arrangements, Canad. J. Math., Tome 63 (2011) no. 5, pp. 1038-1057 | Article | MR 2866070 | Zbl 1228.32028

[8] Denham, Graham; Schulze, Mathias Complexes, duality and Chern classes of logarithmic forms along hyperplane arrangements, Advanced Studies in Pure Mathematics, Tome 62 (2011) http://xxx.lanl.gov/abs/1004.4237 (in press) | MR 2933791

[9] Edelman, Paul H.; Reiner, Victor A counterexample to Orlik’s conjecture, Proc. Amer. Math. Soc., Tome 118 (1993) no. 3, pp. 927-929 | Article | MR 1134624 | Zbl 0791.52013

[10] Eisenbud, David; Huneke, Craig; Vasconcelos, Wolmer Direct methods for primary decomposition, Invent. Math., Tome 110 (1992) no. 2, pp. 207-235 | Article | MR 1185582 | Zbl 0770.13018

[11] Granger, Michel; Mond, David; Nieto-Reyes, Alicia; Schulze, Mathias Linear free divisors and the global logarithmic comparison theorem, Ann. Inst. Fourier (Grenoble), Tome 59 (2009) no. 2, pp. 811-850 | Article | Numdam | MR 2521436 | Zbl 1163.32014

[12] Granger, Michel; Mond, David; Schulze, Mathias Free divisors in prehomogeneous vector spaces, Proc. Lond. Math. Soc. (3), Tome 102 (2011) no. 5, pp. 923-950 | Article | MR 2795728 | Zbl 1231.14042

[13] Granger, Michel; Schulze, Mathias On the formal structure of logarithmic vector fields, Compos. Math., Tome 142 (2006) no. 3, pp. 765-778 | Article | MR 2231201 | Zbl 1096.32016

[14] Granger, Michel; Schulze, Mathias On the symmetry of $b$-functions of linear free divisors, Publ. Res. Inst. Math. Sci., Tome 46 (2010) no. 3, pp. 479-506 | Article | MR 2760735 | Zbl 1202.14046

[15] De Gregorio, Ignacio; Mond, David; Sevenheck, Christian Linear free divisors and Frobenius manifolds, Compos. Math., Tome 145 (2009) no. 5, pp. 1305-1350 | Article | MR 2551998 | Zbl 1238.32022

[16] Lebelt, Karsten Zur homologischen Dimension äusserer Potenzen von Moduln, Arch. Math. (Basel), Tome 26 (1975) no. 6, pp. 595-601 | Article | MR 396534 | Zbl 0335.13007

[17] Lebelt, Karsten Freie Auflösungen äusserer Potenzen, Manuscripta Math., Tome 21 (1977) no. 4, pp. 341-355 | Article | MR 450253 | Zbl 0365.13004

[18] Looijenga, E. J. N. Isolated singular points on complete intersections, Cambridge University Press, Cambridge, London Mathematical Society Lecture Note Series, Tome 77 (1984) | MR 747303 | Zbl 0552.14002

[19] Mangeney, Marguerite; Peskine, Christian; Szpiro, Lucien Anneaux de Gorenstein, et torsion en algèbre commutative, Secrétariat mathématique, Paris, Séminaire d’Algèbre Commutative dirigé par Pierre Samuel, 1966/67. Texte rédigé, d’après des exposés de Maurice Auslander, Marquerite Mangeney, Christian Peskine et Lucien Szpiro. École Normale Supérieure de Jeunes Filles (1967) | MR 225844

[20] Matsumura, Hideyuki Commutative ring theory, Cambridge University Press, Cambridge, Cambridge Studies in Advanced Mathematics, Tome 8 (1989) (Translated from the Japanese by M. Reid) | MR 1011461 | Zbl 0666.13002

[21] Mond, David; Schulze, Mathias Adjoint divisors and free divisors, arXiv.org, math.AG (2010) (1001.1095, Submitted)

[22] Mustaţǎ, Mircea; Schenck, Henry K. The module of logarithmic $p$-forms of a locally free arrangement, J. Algebra, Tome 241 (2001) no. 2, pp. 699-719 | Article | MR 1843320 | Zbl 1047.14007

[23] Orlik, Peter; Terao, Hiroaki Arrangements of hyperplanes, Springer-Verlag, Berlin, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Tome 300 (1992) | MR 1217488 | Zbl 0757.55001

[24] Roos, Jan-Erik Bidualité et structure des foncteurs dérivés de $\varinjlim $ dans la catégorie des modules sur un anneau régulier, C. R. Acad. Sci. Paris, Tome 254 (1962), pp. 1556-1558 | MR 136639 | Zbl 0105.01303

[25] Rose, Lauren L.; Terao, Hiroaki A free resolution of the module of logarithmic forms of a generic arrangement, J. Algebra, Tome 136 (1991) no. 2, pp. 376-400 | Article | MR 1089305 | Zbl 0732.13010

[26] Saito, Kyoji Quasihomogene isolierte Singularitäten von Hyperflächen, Invent. Math., Tome 14 (1971), pp. 123-142 | Article | MR 294699 | Zbl 0224.32011

[27] Saito, Kyoji Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math., Tome 27 (1980) no. 2, pp. 265-291 | MR 586450 | Zbl 0496.32007

[28] Sevenheck, Christian Bernstein polynomials and spectral numbers for linear free divisors, Ann. Inst. Fourier (Grenoble), Tome 61 (2011) no. 1, pp. 379-400 http://arxiv.org/abs/0905.0971 | Article | Numdam | MR 2828135 | Zbl 1221.34237

[29] Simis, Aron Differential idealizers and algebraic free divisors, Commutative algebra, Chapman & Hall/CRC, Boca Raton, FL (Lect. Notes Pure Appl. Math.) Tome 244 (2006), pp. 211-226 | MR 2184799 | Zbl 1099.13030

[30] Slodowy, Peter Simple singularities and simple algebraic groups, Springer, Berlin, Lecture Notes in Mathematics, Tome 815 (1980) | MR 584445 | Zbl 0441.14002

[31] Solomon, L.; Terao, H. A formula for the characteristic polynomial of an arrangement, Adv. in Math., Tome 64 (1987) no. 3, pp. 305-325 | Article | MR 888631 | Zbl 0625.05001

[32] Van Straten, D. A note on the discriminant of a space curve, Manuscripta Math., Tome 87 (1995) no. 2, pp. 167-177 | Article | MR 1334939 | Zbl 0858.32031

[33] Terao, Hiroaki Free arrangements of hyperplanes and unitary reflection groups, Proc. Japan Acad. Ser. A Math. Sci., Tome 56 (1980) no. 8, pp. 389-392 http://projecteuclid.org/getRecord?id=euclid.pja/1195516722 | Article | MR 596011 | Zbl 0476.14016

[34] Terao, Hiroaki Discriminant of a holomorphic map and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math., Tome 30 (1983) no. 2, pp. 379-391 | MR 722502 | Zbl 0535.32003

[35] Wiens, Jonathan The module of derivations for an arrangement of subspaces, Pacific J. Math., Tome 198 (2001) no. 2, pp. 501-512 | Article | MR 1835521 | Zbl 1062.14068

[36] Wiens, Jonathan; Yuzvinsky, Sergey De Rham cohomology of logarithmic forms on arrangements of hyperplanes, Trans. Amer. Math. Soc., Tome 349 (1997) no. 4, pp. 1653-1662 | Article | MR 1407505 | Zbl 0948.52014

[37] Yuzvinsky, Sergey A free resolution of the module of derivations for generic arrangements, J. Algebra, Tome 136 (1991) no. 2, pp. 432-438 | Article | MR 1089307 | Zbl 0732.13009

[38] Zakalyukin, V. M. Reconstructions of fronts and caustics depending on a parameter, and versality of mappings, Current problems in mathematics, Vol. 22, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow (Itogi Nauki i Tekhniki) (1983), pp. 56-93 | MR 735440 | Zbl 0554.58011

[39] Ziegler, Günter M. Combinatorial construction of logarithmic differential forms, Adv. Math., Tome 76 (1989) no. 1, pp. 116-154 | Article | MR 1004488 | Zbl 0725.05032