Local cohomology of logarithmic forms  [ Cohomologie locale des formes logarithmiques ]
Annales de l'Institut Fourier, Tome 63 (2013) no. 3, pp. 1177-1203.

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.

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.

DOI : https://doi.org/10.5802/aif.2787
Classification : 32S22,  52C35,  16W25
Mots clés : arrangements d’hyperplans, forme logarithmique différentielle, diviseur libre
@article{AIF_2013__63_3_1177_0,
     author = {Denham, G. and Schenck, H. and Schulze, M. and Wakefield, M. and Walther, U.},
     title = {Local cohomology of logarithmic forms},
     journal = {Annales de l'Institut Fourier},
     pages = {1177--1203},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {63},
     number = {3},
     year = {2013},
     doi = {10.5802/aif.2787},
     mrnumber = {3137483},
     zbl = {1277.32030},
     language = {en},
     url = {www.numdam.org/item/AIF_2013__63_3_1177_0/}
}
Denham, G.; Schenck, H.; Schulze, M.; Wakefield, M.; Walther, U. Local cohomology of logarithmic forms. Annales de l'Institut Fourier, Tome 63 (2013) no. 3, pp. 1177-1203. doi : 10.5802/aif.2787. http://www.numdam.org/item/AIF_2013__63_3_1177_0/

[1] Borel, A.; Grivel, P.-P.; Kaup, B.; Haefliger, A.; Malgrange, B.; Ehlers, F. Algebraic D-modules, Perspectives in Mathematics, Volume 2, Academic Press Inc., Boston, MA, 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), Volume 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), Volume 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., Volume 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., Volume 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., Volume 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, Volume 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., Volume 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., Volume 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), Volume 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), Volume 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., Volume 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., Volume 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., Volume 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), Volume 26 (1975) no. 6, pp. 595-601 | Article | MR 396534 | Zbl 0335.13007

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

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

[19] Mangeney, Marguerite; Peskine, Christian; Szpiro, Lucien Anneaux de Gorenstein, et torsion en algèbre commutative, 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, Secrétariat mathématique, Paris, 1967 | MR 225844

[20] Matsumura, Hideyuki Commutative ring theory, Cambridge Studies in Advanced Mathematics, Volume 8, Cambridge University Press, Cambridge, 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, Volume 241 (2001) no. 2, pp. 699-719 | Article | MR 1843320 | Zbl 1047.14007

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

[24] Roos, Jan-Erik Bidualité et structure des foncteurs dérivés de lim dans la catégorie des modules sur un anneau régulier, C. R. Acad. Sci. Paris, Volume 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, Volume 136 (1991) no. 2, pp. 376-400 | Article | MR 1089305 | Zbl 0732.13010

[26] Saito, Kyoji Quasihomogene isolierte Singularitäten von Hyperflächen, Invent. Math., Volume 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., Volume 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), Volume 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 (Lect. Notes Pure Appl. Math.) Volume 244, Chapman & Hall/CRC, Boca Raton, FL, 2006, pp. 211-226 | MR 2184799 | Zbl 1099.13030

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

[31] Solomon, L.; Terao, H. A formula for the characteristic polynomial of an arrangement, Adv. in Math., Volume 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., Volume 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., Volume 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., Volume 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., Volume 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., Volume 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, Volume 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 (Itogi Nauki i Tekhniki), Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1983, pp. 56-93 | MR 735440 | Zbl 0554.58011

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