Local cohomology of logarithmic forms

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

doi:10.5802/aif.2787

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) no. 3, pp. 1177-1203.

Résumé
Abstract

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.

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

@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/

Bibliographie

[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 $lim_{\to}$ 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