Deformations of free and linear free divisors

Michele Torielli[1]

  • [1] University of Warwick Department of Mathematics Coventry CV4 7AL (U.K.)

Annales de l’institut Fourier (2013)

  • Volume: 63, Issue: 6, page 2097-2136
  • ISSN: 0373-0956

Abstract

top
We study deformations of free and linear free divisors. We introduce a complex similar to the de Rham complex whose cohomology calculates the deformation spaces. This cohomology turns out to be zero for all reductive linear free divisors and to be constructible for Koszul free divisors and weighted homogeneous free divisors.

How to cite

top

Torielli, Michele. "Deformations of free and linear free divisors." Annales de l’institut Fourier 63.6 (2013): 2097-2136. <http://eudml.org/doc/275425>.

@article{Torielli2013,
abstract = {We study deformations of free and linear free divisors. We introduce a complex similar to the de Rham complex whose cohomology calculates the deformation spaces. This cohomology turns out to be zero for all reductive linear free divisors and to be constructible for Koszul free divisors and weighted homogeneous free divisors.},
affiliation = {University of Warwick Department of Mathematics Coventry CV4 7AL (U.K.)},
author = {Torielli, Michele},
journal = {Annales de l’institut Fourier},
keywords = {Free divisor; linear free divisor; non-isolated singularity; deformation theory; logarithmic de Rham cohomology; free divisor},
language = {eng},
number = {6},
pages = {2097-2136},
publisher = {Association des Annales de l’institut Fourier},
title = {Deformations of free and linear free divisors},
url = {http://eudml.org/doc/275425},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Torielli, Michele
TI - Deformations of free and linear free divisors
JO - Annales de l’institut Fourier
PY - 2013
PB - Association des Annales de l’institut Fourier
VL - 63
IS - 6
SP - 2097
EP - 2136
AB - We study deformations of free and linear free divisors. We introduce a complex similar to the de Rham complex whose cohomology calculates the deformation spaces. This cohomology turns out to be zero for all reductive linear free divisors and to be constructible for Koszul free divisors and weighted homogeneous free divisors.
LA - eng
KW - Free divisor; linear free divisor; non-isolated singularity; deformation theory; logarithmic de Rham cohomology; free divisor
UR - http://eudml.org/doc/275425
ER -

References

top
  1. Camilo Arias Abad, Marius Crainic, Representations up to homotopy of Lie algebroids, J. Reine Angew. Math. 663 (2012), 91-126 Zbl1238.58010MR2889707
  2. R.O. Buchweitz, D. Mond, Linear free divisors and quiver representations, Singularities and computer algebra 324 (2006) Zbl1101.14013MR2228227
  3. F. J. Calderón Moreno, Logarithmic differential operators and logarithmic de Rham complexes relative to a free divisor, Ann. Sci. École Norm. Sup. 4e série 32 (1999), 701-714 Zbl0955.14013MR1710757
  4. F. J. Calderón Moreno, L. Narváez Macarro, Locally quasi-homogeneous free divisors are Koszul free, Proceedings of the Steklov Institute of Mathematics-Interperiodica Translation 238 (2002), 72-76 Zbl1031.32006MR1969305
  5. F. J. Calderón Moreno, L. Narváez Macarro, Dualité et comparaison sur les complexes de de Rham logarithmiques par rapport aux diviseurs libres, Ann. Inst. Fourier (Grenoble) 55 (2005), 47-75 Zbl1089.32003MR2141288
  6. I. De Gregorio, D. Mond, C. Sevenheck, Linear free divisors and Frobenius manifolds, Compos. Math 145 (2009), 1305-1350 Zbl1238.32022MR2551998
  7. Jacques Dixmier, Enveloping algebras, (1977), North-Holland Publishing Co., Amsterdam Zbl0346.17010MR498740
  8. M. Granger, D. Mond, A. Nieto-Reyes, M. Schulze, Linear free divisors and the global logarithmic comparison theorem, Ann. Inst. Fourier (Grenoble) 59 (2009), 811-850 Zbl1163.32014MR2521436
  9. G.-M. Greuel, C. Lossen, E. Shustin, Introduction to singularities and deformations, (2007), Springer, Berlin Zbl1125.32013MR2290112
  10. Robin Hartshorne, Deformation theory, 257 (2010), Springer, New York Zbl1186.14004MR2583634
  11. G. Hochschild, J.-P. Serre, Cohomology of Lie algebras, Annals of Mathematics 57 (1953), 591-603 Zbl0053.01402MR54581
  12. P. Orlik, H. Terao, Arrangements of hyperplanes, 300 (1992), Springer-Verlag, Berlin Zbl0757.55001MR1217488
  13. G. S. Rinehart, Differential forms on general commutative algebras, Trans. Amer. Math. Soc. 108 (1963), 195-222 Zbl0113.26204MR154906
  14. K. Saito, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), 265-291 Zbl0496.32007MR586450
  15. M. Schlessinger, Functors of Artin rings, Trans. Amer. Math. Soc. 130 (1968), 208-222 Zbl0167.49503MR217093
  16. C. Sevenheck, Lagrange-singularitäten und ihre deformationen, (1999), Heinrich-Heine Universität, Düsseldorf 
  17. C. Sevenheck, Lagrangian singularities, (2003), Cuvillier Verlag, Göttingen Zbl1059.14006MR2015318
  18. C. Sevenheck, D. van Straten, Deformation of singular Lagrangian subvarieties, Math. Ann. 327 (2003), 79-102 Zbl1051.14006MR2005122
  19. M. Torielli, Free divisors and their deformations, (2012) Zbl1301.14004
  20. C. A. Weibel, An introduction to homological algebra, 38 (1994), Cambridge University Press, Cambridge Zbl0797.18001MR1269324

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.