Linear free divisors and the global logarithmic comparison theorem

[Diviseurs linéairement libres et le théorème de comparaison logarithmique global]

Granger, Michel ; Mond, David ; Nieto-Reyes, Alicia ; Schulze, Mathias

Annales de l'Institut Fourier, Tome 59 (2009), p. 811-850 / Harvested from Numdam

Résumé

Une hypersurface complexe de $\mathbb{C}^n$ est appelée un diviseur linéairement libre (ou DLL) si son module de champs de vecteur logarithmiques a une base globale formée de champs de vecteurs linéaires. Nous classifions tous les DLL pour $n$ au plus égal à $4$.

Par analogie avec le théorème de comparaison de Grothendieck, on dit que le théorème de comparaison logarithmique global (ou TCLG) est vrai pour $D$ si le complexe des formes différentielles logarithmiques globales permet de calculer la cohomologie de $\mathbb{C}^{n}\setminus D$ à coefficients complexes. Nous mettons en évidence un critère général pour qu’un DLL ait la propriété TCLG, et nous démontrons que ce critère s’applique lorsque l’algèbre de Lie des champs de vecteurs logarithmiques linéaires est réductive. Pour $n$ inférieur ou égal à $4$, nous montrons que le TCLG est vrai pour tous les DLL.

Nous montrons que les DLL qui apparaissent naturellement comme discriminants dans les espaces de représentations de carquois pour des racines de Schur réelles satisfont au TCLG. Comme corollaire nous obtenons une démonstration topologique d’un résultat de V. Kac sur le nombre de composantes irréductibles de tels discriminants.

Abstract

A complex hypersurface $D$ in $\mathbb{C}^n$ is a linear free divisor (LFD) if its module of logarithmic vector fields has a global basis of linear vector fields. We classify all LFDs for $n$ at most $4$.

By analogy with Grothendieck’s comparison theorem, we say that the global logarithmic comparison theorem (GLCT) holds for $D$ if the complex of global logarithmic differential forms computes the complex cohomology of $\mathbb{C}^{n}\setminus D$. We develop a general criterion for the GLCT for LFDs and prove that it is fulfilled whenever the Lie algebra of linear logarithmic vector fields is reductive. For $n$ at most $4$, we show that the GLCT holds for all LFDs.

We show that LFDs arising naturally as discriminants in quiver representation spaces (of real Schur roots) fulfill the GLCT. As a by-product we obtain a topological proof of a theorem of V. Kac on the number of irreducible components of such discriminants.

Détail

Publié le : 2009-01-01

MR 2521436 | Zbl 1163.32014 | 5 citations dans Numdam

DOI : https://doi.org/10.5802/aif.2448
Classification:  32S20,  14F40,  20G10,  17B66
Mots clés: diviseur linéairement libre, espace vectorielle préhomogène, cohomologie de De Rham, théorème de comparaison logarithmique, cohomologie des algèbres de Lie, représentation des quivers

Bibliographie

[1] Anosov, D. V.; Aranson, S. Kh.; Arnold, V. I.; Bronshtein, I. U.; Grines, V. Z.; Il’Yashenko, Yu. S. Ordinary differential equations and smooth dynamical systems, Springer-Verlag, Berlin (1997) (Translated from the 1985 Russian original by E. R. Dawson and D. O’Shea, Third printing of the 1988 translation [Dynamical systems. I, Encyclopaedia Math. Sci., 1, Springer, Berlin, 1988; MR0970793 (89g:58060)]) | MR 1633529 | MR 970793 | Zbl 0858.34001

[2] Artin, M. On the solutions of analytic equations, Invent. Math., Tome 5 (1968), pp. 277-291 | Article | MR 232018 | Zbl 0172.05301

[3] Buchweitz, Ragnar-Olaf; Mond, David Linear free divisors and quiver representations, Singularities and computer algebra, Cambridge Univ. Press, Cambridge (London Math. Soc. Lecture Note Ser.) Tome 324 (2006), pp. 41-77 | MR 2228227 | Zbl 1101.14013

[4] Calderón-Moreno, Francisco; Narváez-Macarro, Luis The module $\mathscr{D}f^s$ for locally quasi-homogeneous free divisors, Compositio Math., Tome 134 (2002) no. 1, pp. 59-74 | Article | MR 1931962 | Zbl 1017.32023

[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, F. J.; Ucha-Enríquez, J. M. Logarithmic comparison theorem and some Euler homogeneous free divisors, Proc. Amer. Math. Soc., Tome 133 (2005) no. 5, p. 1417-1422 (electronic) | Article | MR 2111967 | Zbl 1077.32012

[7] 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

[8] Fehér, L.M.; Patakfalvi, Zs. The incidence class and the hierarchy of orbits (2007) (http://arxiv.org/abs/0705.3834)

[9] Gabriel, Peter Unzerlegbare Darstellungen. I, Manuscripta Math., Tome 6 (1972), p. 71-103; correction, ibid. 6 (1972), 309 | Article | MR 332887 | Zbl 0232.08001

[10] Granger, Michel; Mond, David; Nieto, Alicia; Schulze, Mathias Linear free divisors and the global logarithmic comparison theorem (2006) (http://arxiv.org/abs/math/0607045)

[11] 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

[12] Grothendieck, A. On the de Rham cohomology of algebraic varieties, Inst. Hautes Études Sci. Publ. Math. (1966) no. 29, pp. 95-103 | Article | Numdam | MR 199194 | Zbl 0145.17602

[13] Hauser, Herwig; Müller, Gerd The cancellation property for direct products of analytic space germs, Math. Ann., Tome 286 (1990) no. 1-3, pp. 209-223 | Article | MR 1032931 | Zbl 0702.32008

[14] Humphreys, James E. Linear algebraic groups, Springer-Verlag, New York (1975) (Graduate Texts in Mathematics, No. 21) | MR 396773 | Zbl 0325.20039

[15] Jacobson, Nathan Lie algebras, Dover Publications Inc., New York (1979) (Republication of the 1962 original) | MR 559927

[16] Kac, V. G. Infinite root systems, representations of graphs and invariant theory. II, J. Algebra, Tome 78 (1982) no. 1, pp. 141-162 | Article | MR 677715 | Zbl 0497.17007

[17] Kraft, H.; Riedtmann, Ch. Geometry of representations of quivers, Representations of algebras (Durham, 1985), Cambridge Univ. Press, Cambridge (London Math. Soc. Lecture Note Ser.) Tome 116 (1986), pp. 109-145 | MR 897322 | Zbl 0632.16019

[18] Nieto-Reyes, Alicia M.Phil Thesis, University of Warwick, Coventry, England (2005) (Masters thesis)

[19] Onishchik, A. L.; Vinberg, È. B. Lie groups and algebraic groups, Springer-Verlag, Berlin, Springer Series in Soviet Mathematics (1990) (Translated from the Russian and with a preface by D. A. Leites) | MR 1064110 | Zbl 0722.22004

[20] 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

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

[22] 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

[23] Sato, M.; Kimura, T. A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J., Tome 65 (1977), pp. 1-155 | MR 430336 | Zbl 0321.14030

[24] Schofield, Aidan Semi-invariants of quivers, J. London Math. Soc. (2), Tome 43 (1991) no. 3, pp. 385-395 | Article | MR 1113382 | Zbl 0779.16005

[25] Serre, Jean-Pierre Complex semisimple Lie algebras, Springer-Verlag, Berlin, Springer Monographs in Mathematics (2001) (Translated from the French by G. A. Jones, Reprint of the 1987 edition) | MR 1808366 | Zbl 1058.17005

[26] Terao, Hiroaki Forms with logarithmic pole and the filtration by the order of the pole, Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977), Kinokuniya Book Store, Tokyo (1978), pp. 673-685 | MR 578880 | Zbl 0429.32015

[27] Torrelli, Tristan On meromorphic functions defined by a differential system of order 1, Bull. Soc. Math. France, Tome 132 (2004) no. 4, pp. 591-612 | Numdam | MR 2131905 | Zbl 1080.32011

[28] Walther, Uli Bernstein-Sato polynomial versus cohomology of the Milnor fiber for generic hyperplane arrangements, Compos. Math., Tome 141 (2005) no. 1, pp. 121-145 | Article | MR 2099772 | Zbl 1070.32021

[29] 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