Linear free divisors and the global logarithmic comparison theorem

Michel Granger[1]; David Mond[2]; Alicia Nieto-Reyes[3]; Mathias Schulze[4]

  • [1] Université d’Angers Département de Mathématiques 2 bd. Lavoisier 49045 Angers (France)
  • [2] University of Warwick Mathematics Institute Coventry CV47AL (England)
  • [3] Universidad de Cantabria Departamento de Matematicas, Estadistica y Computacion (Spain)
  • [4] Oklahoma State University Department of Mathematics Stillwater, OK 74078 (United States)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 2, page 811-850
  • ISSN: 0373-0956

Abstract

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A complex hypersurface D in 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 ℂ n ∖ 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.

How to cite

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Granger, Michel, et al. "Linear free divisors and the global logarithmic comparison theorem." Annales de l’institut Fourier 59.2 (2009): 811-850. <http://eudml.org/doc/10412>.

@article{Granger2009,
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.},
affiliation = {Université d’Angers Département de Mathématiques 2 bd. Lavoisier 49045 Angers (France); University of Warwick Mathematics Institute Coventry CV47AL (England); Universidad de Cantabria Departamento de Matematicas, Estadistica y Computacion (Spain); Oklahoma State University Department of Mathematics Stillwater, OK 74078 (United States)},
author = {Granger, Michel, Mond, David, Nieto-Reyes, Alicia, Schulze, Mathias},
journal = {Annales de l’institut Fourier},
keywords = {Free divisor; prehomogeneous vector space; De Rham cohomology; logarithmic comparison theorem; Lie algebra cohomology; quiver representation; free divisor; de Rham cohomology},
language = {eng},
number = {2},
pages = {811-850},
publisher = {Association des Annales de l’institut Fourier},
title = {Linear free divisors and the global logarithmic comparison theorem},
url = {http://eudml.org/doc/10412},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Granger, Michel
AU - Mond, David
AU - Nieto-Reyes, Alicia
AU - Schulze, Mathias
TI - Linear free divisors and the global logarithmic comparison theorem
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 2
SP - 811
EP - 850
AB - 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.
LA - eng
KW - Free divisor; prehomogeneous vector space; De Rham cohomology; logarithmic comparison theorem; Lie algebra cohomology; quiver representation; free divisor; de Rham cohomology
UR - http://eudml.org/doc/10412
ER -

References

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  1. D. V. Anosov, S. Kh. Aranson, V. I. Arnold, I. U. Bronshtein, V. Z. Grines, Yu. S. Il’yashenko, Ordinary differential equations and smooth dynamical systems, (1997), Springer-Verlag, Berlin Zbl0858.34001MR970793
  2. M. Artin, On the solutions of analytic equations, Invent. Math. 5 (1968), 277-291 Zbl0172.05301MR232018
  3. Ragnar-Olaf Buchweitz, David Mond, Linear free divisors and quiver representations, Singularities and computer algebra 324 (2006), 41-77, Cambridge Univ. Press, Cambridge Zbl1101.14013MR2228227
  4. Francisco Calderón-Moreno, Luis Narváez-Macarro, The module 𝒟 f s for locally quasi-homogeneous free divisors, Compositio Math. 134 (2002), 59-74 Zbl1017.32023MR1931962
  5. Francisco J. Calderón Moreno, David Mond, Luis Narváez Macarro, Francisco J. Castro Jiménez, Logarithmic cohomology of the complement of a plane curve, Comment. Math. Helv. 77 (2002), 24-38 Zbl1010.32016MR1898392
  6. F. J. Castro-Jiménez, J. M. Ucha-Enríquez, Logarithmic comparison theorem and some Euler homogeneous free divisors, Proc. Amer. Math. Soc. 133 (2005), 1417-1422 (electronic) Zbl1077.32012MR2111967
  7. Francisco J. Castro-Jiménez, Luis Narváez-Macarro, David Mond, Cohomology of the complement of a free divisor, Trans. Amer. Math. Soc. 348 (1996), 3037-3049 Zbl0862.32021MR1363009
  8. L.M. Fehér, Zs. Patakfalvi, The incidence class and the hierarchy of orbits, (2007) 
  9. Peter Gabriel, Unzerlegbare Darstellungen. I, Manuscripta Math. 6 (1972), 71-103; correction, ibid. 6 (1972), 309 Zbl0232.08001MR332887
  10. Michel Granger, David Mond, Alicia Nieto, Mathias Schulze, Linear free divisors and the global logarithmic comparison theorem, (2006) Zbl1163.32014
  11. Michel Granger, Mathias Schulze, On the formal structure of logarithmic vector fields, Compos. Math. 142 (2006), 765-778 Zbl1096.32016MR2231201
  12. A. Grothendieck, On the de Rham cohomology of algebraic varieties, Inst. Hautes Études Sci. Publ. Math. (1966), 95-103 Zbl0145.17602MR199194
  13. Herwig Hauser, Gerd Müller, The cancellation property for direct products of analytic space germs, Math. Ann. 286 (1990), 209-223 Zbl0702.32008MR1032931
  14. James E. Humphreys, Linear algebraic groups, (1975), Springer-Verlag, New York Zbl0325.20039MR396773
  15. Nathan Jacobson, Lie algebras, (1979), Dover Publications Inc., New York Zbl0121.27504MR559927
  16. V. G. Kac, Infinite root systems, representations of graphs and invariant theory. II, J. Algebra 78 (1982), 141-162 Zbl0497.17007MR677715
  17. H. Kraft, Ch. Riedtmann, Geometry of representations of quivers, Representations of algebras (Durham, 1985) 116 (1986), 109-145, Cambridge Univ. Press, Cambridge Zbl0632.16019MR897322
  18. Alicia Nieto-Reyes, M.Phil Thesis, (2005) 
  19. A. L. Onishchik, È. B. Vinberg, Lie groups and algebraic groups, (1990), Springer-Verlag, Berlin Zbl0722.22004MR1064110
  20. Peter Orlik, Hiroaki Terao, Arrangements of hyperplanes, 300 (1992), Springer-Verlag, Berlin Zbl0757.55001MR1217488
  21. Kyoji Saito, Quasihomogene isolierte Singularitäten von Hyperflächen, Invent. Math. 14 (1971), 123-142 Zbl0224.32011MR294699
  22. Kyoji Saito, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), 265-291 Zbl0496.32007MR586450
  23. M. Sato, T. Kimura, A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J. 65 (1977), 1-155 Zbl0321.14030MR430336
  24. Aidan Schofield, Semi-invariants of quivers, J. London Math. Soc. (2) 43 (1991), 385-395 Zbl0779.16005MR1113382
  25. Jean-Pierre Serre, Complex semisimple Lie algebras, (2001), Springer-Verlag, Berlin Zbl1058.17005MR1808366
  26. Hiroaki Terao, 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) (1978), 673-685, Kinokuniya Book Store, Tokyo Zbl0429.32015MR578880
  27. Tristan Torrelli, On meromorphic functions defined by a differential system of order 1, Bull. Soc. Math. France 132 (2004), 591-612 Zbl1080.32011MR2131905
  28. Uli Walther, Bernstein-Sato polynomial versus cohomology of the Milnor fiber for generic hyperplane arrangements, Compos. Math. 141 (2005), 121-145 Zbl1070.32021MR2099772
  29. Jonathan Wiens, Sergey Yuzvinsky, De Rham cohomology of logarithmic forms on arrangements of hyperplanes, Trans. Amer. Math. Soc. 349 (1997), 1653-1662 Zbl0948.52014MR1407505

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