Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-27T02:47:07.907Z Has data issue: false hasContentIssue false
  • English
  • Français

Characteristic varieties and logarithmic differential 1-forms

Part of: Cycles and subschemes Singularities Curves (Co)homology theory

Published online by Cambridge University Press:  22 January 2010

Alexandru Dimca
Alexandru Dimca*
Affiliation:
Laboratoire J.A. Dieudonné, UMR du CNRS 6621, Université de Nice Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 02, France (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We introduce in this paper a hypercohomology version of the resonance varieties and obtain some relations to the characteristic varieties of rank one local systems on a smooth quasi-projective complex variety M. A logarithmic resonance variety is also considered and, as an application, we determine the first characteristic variety of the configuration space of n distinct labeled points on an elliptic curve. Finally, for a logarithmic 1-form α on M we investigate the relation between the resonance degree of α and the codimension of the zero set of α on a good compactification of M. This question was inspired by the recent work by Cohen, Denham, Falk and Varchenko.

Keywords

twisted cohomologylogarithmic de Rham complexzeroes of 1-formsHodge theory

MSC classification

Secondary: 14C30: Transcendental methods, Hodge theory , Hodge conjecture 14F40: de Rham cohomology 14H52: Elliptic curves 32S22: Relations with arrangements of hyperplanes
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 1 , January 2010 , pp. 129 - 144
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Arapura, D., Geometry of cohomology support loci for local systems I, J. Algebraic Geom. 6 (1997), 563–597.Google Scholar
[2]Bauer, I., Irrational pencils on non-compact algebraic manifolds, Internat. J. Math. 8 (1997), 441–450.CrossRefGoogle Scholar
[3]Beauville, A., Annulation du H1 pour les fibrés en droites plats, in Complex algebraic varieties (Bayreuth, 1990), Lecture Notes in Mathematics, vol. 1507 (Springer, Berlin, 1992), 1–15.Google Scholar
[4]Catanese, F., Moduli and classification of irregular Kaehler manifolds (and algebraic varieties) with Albanese general type fibrations, Invent. Math. 104 (1991), 263–289 (with an appendix by A. Beauville).CrossRefGoogle Scholar
[5]Catanese, F., Fibred surfaces, varieties isogenous to a product and related moduli spaces, Amer. J. Math. 122 (2000), 1–44.CrossRefGoogle Scholar
[6]Cohen, D. and Suciu, A., Characteristic varieties of arrangements, Math. Proc. Cambridge Philos. Soc. 127 (1999), 33–53.CrossRefGoogle Scholar
[7]Deligne, P., Equations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, vol. 163 (Springer, Berlin, 1970).CrossRefGoogle Scholar
[8]Deligne, P., Théorie de Hodge II, Publ. Math. Inst. Hautes Études Sci. 40 (1972), 5–57.CrossRefGoogle Scholar
[9]Denham, G., Zeroes of 1-forms and resonance of free arrangements, Oberwolfach Report 40/2007.Google Scholar
[10]Dimca, A., Characteristic varieties and constructible sheaves, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 18 (2007), 365–389.CrossRefGoogle Scholar
[11]Dimca, A., On the isotropic subspace theorems, Bull. Math. Soc. Sci. Math. Roumanie 51 (2008), 307–324.Google Scholar
[12]Dimca, A. and Maxim, L., Multivariable Alexander invariants of hypersurface complements, Trans. Amer. Math. Soc. 359 (2007), 3505–3528.CrossRefGoogle Scholar
[13]Dimca, A., Papadima, S. and Suciu, A., Topology and geometry of cohomology jump loci, Duke Math. J. 148 (2009), 405–457.CrossRefGoogle Scholar
[14]Eisenbud, D., Commutative algebra, Graduate Texts in Mathematics, vol. 150, third edition (Springer, Berlin, 1999).Google Scholar
[15]Esnault, H., Schechtman, V. and Viehweg, E., Cohomology of local systems on the complement of hyperplanes, Invent. Math. 109 (1992), 557–561 Erratum, ibid 112, (1993) 447.CrossRefGoogle Scholar
[16]Falk, M. J., Resonance and zeros of logarithmic one-forms with hyperplane poles, Oberwolfach Report 40/2007.Google Scholar
[17]Green, M. and Lazarsfeld, R., Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville, Invent. Math. 90 (1987), 389–407.CrossRefGoogle Scholar
[18]Harris, J., Algebraic geometry, Graduate Texts in Mathematics, vol. 133 (Springer, New York, 1992).CrossRefGoogle Scholar
[19]Libgober, A., First order deformations for rank one local systems with a non-vanishing cohomology, Topology Appl. 118 (2002), 159–168.CrossRefGoogle Scholar
[20]Peters, C. and Steenbrink, J., Mixed hodge structures, Ergebnisse der Mathematik und ihrer Grenzgebiete 3, Folge 52 (Springer, Berlin, 2008).Google Scholar
[21]Schechtman, V. V. and Varchenko, A. N., Arrangements of hyperplanes and Lie algebra homology, Invent. Math. 106 (1991), 139–194.CrossRefGoogle Scholar
[22]Scherbak, I. and Varchenko, A. N., Critical points of functions, sℓ 2 representations, and Fuchsian differential equations with only univalued solutions, Mosc. Math. J. 3 (2003), 621–645.CrossRefGoogle Scholar
[23]Simpson, C., The Hodge filtration on nonabelian cohomology, Proceedings of Symposia in Pure Mathematics, vol. 62/2 (American Mathematical Society, Providence, RI, 1997), 217–281.Google Scholar
[24]Simpson, C., A weight two phenomenon for the moduli of rank one local systems on open varieties, Proceedings of Symposia in Pure Mathematics, vol. 78 (American Mathematical Society, Providence, RI, 2008), 175–214.Google Scholar
[25]Suciu, A., Translated tori in the characteristic varieties of complex hyperplane arrangements. Arrangements in Boston: a conference on hyperplane arrangements (1999), Topology Appl. 118 (2002), 209–223.CrossRefGoogle Scholar
[26]Voisin, C., On the homotopy types of compact Kähler and complex projective manifolds, Invent. Math. 157 (2004), 329–343.CrossRefGoogle Scholar
[27]Voisin, C., Hodge structures on cohomology algebras and geometry, Math. Ann. 341 (2008), 39–69.CrossRefGoogle Scholar