Let be a divisor on a smooth algebraic variety . We investigate the geometry of the Jacobian scheme of , homological invariants derived from logarithmic differential forms along , and their relationship with the property that 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.
Soit une variété algébrique lisse et un diviseur sur . Nous étudions la géométrie du schéma Jacobien de , les invariants homologiques provenant des formes différentielles logarithmiques le long de , et leur relation avec la propriété que 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.
Accepted:
DOI: 10.5802/aif.2787
Keywords: hyperplane arrangement, logarithmic, differential form, free divisor
Mot clés : arrangements d’hyperplans, forme logarithmique différentielle, diviseur libre
Denham, G. 1; Schenck, H. 2; Schulze, M. 3; Wakefield, M. 4; Walther, U. 5
@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{\textquoteright}institut Fourier}, volume = {63}, number = {3}, year = {2013}, doi = {10.5802/aif.2787}, mrnumber = {3137483}, zbl = {1277.32030}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2787/} }
TY - JOUR AU - Denham, G. AU - Schenck, H. AU - Schulze, M. AU - Wakefield, M. AU - Walther, U. TI - Local cohomology of logarithmic forms JO - Annales de l'Institut Fourier PY - 2013 SP - 1177 EP - 1203 VL - 63 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2787/ DO - 10.5802/aif.2787 LA - en ID - AIF_2013__63_3_1177_0 ER -
%0 Journal Article %A Denham, G. %A Schenck, H. %A Schulze, M. %A Wakefield, M. %A Walther, U. %T Local cohomology of logarithmic forms %J Annales de l'Institut Fourier %D 2013 %P 1177-1203 %V 63 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2787/ %R 10.5802/aif.2787 %G en %F 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, Volume 63 (2013) no. 3, pp. 1177-1203. doi : 10.5802/aif.2787. https://aif.centre-mersenne.org/articles/10.5802/aif.2787/
[1] Algebraic -modules, Perspectives in Mathematics, 2, Academic Press Inc., Boston, MA, 1987, xii+355 pages | MR
[2] 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 | Zbl
[3] Functions on discriminants, J. London Math. Soc. (2), Volume 30 (1984) no. 3, pp. 551-567 | DOI | MR | Zbl
[4] 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 | DOI | Numdam | MR | Zbl
[5] Logarithmic cohomology of the complement of a plane curve, Comment. Math. Helv., Volume 77 (2002) no. 1, pp. 24-38 | DOI | MR | Zbl
[6] Cohomology of the complement of a free divisor, Trans. Amer. Math. Soc., Volume 348 (1996) no. 8, pp. 3037-3049 | DOI | MR | Zbl
[7] Critical points and resonance of hyperplane arrangements, Canad. J. Math., Volume 63 (2011) no. 5, pp. 1038-1057 | DOI | MR | Zbl
[8] 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
[9] A counterexample to Orlik’s conjecture, Proc. Amer. Math. Soc., Volume 118 (1993) no. 3, pp. 927-929 | DOI | MR | Zbl
[10] Direct methods for primary decomposition, Invent. Math., Volume 110 (1992) no. 2, pp. 207-235 | DOI | MR | Zbl
[11] Linear free divisors and the global logarithmic comparison theorem, Ann. Inst. Fourier (Grenoble), Volume 59 (2009) no. 2, pp. 811-850 | DOI | Numdam | MR | Zbl
[12] Free divisors in prehomogeneous vector spaces, Proc. Lond. Math. Soc. (3), Volume 102 (2011) no. 5, pp. 923-950 | DOI | MR | Zbl
[13] On the formal structure of logarithmic vector fields, Compos. Math., Volume 142 (2006) no. 3, pp. 765-778 | DOI | MR | Zbl
[14] On the symmetry of -functions of linear free divisors, Publ. Res. Inst. Math. Sci., Volume 46 (2010) no. 3, pp. 479-506 | DOI | MR | Zbl
[15] Linear free divisors and Frobenius manifolds, Compos. Math., Volume 145 (2009) no. 5, pp. 1305-1350 | DOI | MR | Zbl
[16] Zur homologischen Dimension äusserer Potenzen von Moduln, Arch. Math. (Basel), Volume 26 (1975) no. 6, pp. 595-601 | DOI | MR | Zbl
[17] Freie Auflösungen äusserer Potenzen, Manuscripta Math., Volume 21 (1977) no. 4, pp. 341-355 | DOI | MR | Zbl
[18] Isolated singular points on complete intersections, London Mathematical Society Lecture Note Series, 77, Cambridge University Press, Cambridge, 1984, xi+200 pages | MR | Zbl
[19] 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, 69 pages | MR
[20] Commutative ring theory, Cambridge Studies in Advanced Mathematics, 8, Cambridge University Press, Cambridge, 1989, xiv+320 pages (Translated from the Japanese by M. Reid) | MR | Zbl
[21] Adjoint divisors and free divisors, arXiv.org, math.AG, 2010 (1001.1095, Submitted)
[22] The module of logarithmic -forms of a locally free arrangement, J. Algebra, Volume 241 (2001) no. 2, pp. 699-719 | DOI | MR | Zbl
[23] Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 300, Springer-Verlag, Berlin, 1992, xviii+325 pages | MR | Zbl
[24] Bidualité et structure des foncteurs dérivés de dans la catégorie des modules sur un anneau régulier, C. R. Acad. Sci. Paris, Volume 254 (1962), pp. 1556-1558 | MR | Zbl
[25] A free resolution of the module of logarithmic forms of a generic arrangement, J. Algebra, Volume 136 (1991) no. 2, pp. 376-400 | DOI | MR | Zbl
[26] Quasihomogene isolierte Singularitäten von Hyperflächen, Invent. Math., Volume 14 (1971), pp. 123-142 | DOI | MR | Zbl
[27] 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 | Zbl
[28] Bernstein polynomials and spectral numbers for linear free divisors, Ann. Inst. Fourier (Grenoble), Volume 61 (2011) no. 1, pp. 379-400 | arXiv | DOI | Numdam | MR | Zbl
[29] 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 | Zbl
[30] Simple singularities and simple algebraic groups, Lecture Notes in Mathematics, 815, Springer, Berlin, 1980, x+175 pages | MR | Zbl
[31] A formula for the characteristic polynomial of an arrangement, Adv. in Math., Volume 64 (1987) no. 3, pp. 305-325 | DOI | MR | Zbl
[32] A note on the discriminant of a space curve, Manuscripta Math., Volume 87 (1995) no. 2, pp. 167-177 | DOI | MR | Zbl
[33] 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 | DOI | MR | Zbl
[34] 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 | Zbl
[35] The module of derivations for an arrangement of subspaces, Pacific J. Math., Volume 198 (2001) no. 2, pp. 501-512 | DOI | MR | Zbl
[36] De Rham cohomology of logarithmic forms on arrangements of hyperplanes, Trans. Amer. Math. Soc., Volume 349 (1997) no. 4, pp. 1653-1662 | DOI | MR | Zbl
[37] A free resolution of the module of derivations for generic arrangements, J. Algebra, Volume 136 (1991) no. 2, pp. 432-438 | DOI | MR | Zbl
[38] 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 | Zbl
[39] Combinatorial construction of logarithmic differential forms, Adv. Math., Volume 76 (1989) no. 1, pp. 116-154 | DOI | MR | Zbl
Cited by Sources: