Alexander stratifications of character varieties

Eriko Hironaka

Annales de l'institut Fourier (1997)

  • Volume: 47, Issue: 2, page 555-583
  • ISSN: 0373-0956

Abstract

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Equations defining the jumping loci for the first cohomology group of one-dimensional representations of a finitely presented group Γ can be effectively computed using Fox calculus. In this paper, we give an exposition of Fox calculus in the language of group cohomology and in the language of finite abelian coverings of CW complexes. Work of Arapura and Simpson imply that if Γ is the fundamental group of a compact Kähler manifold, then the strata are finite unions of translated affine subtori. It follows that for Kähler groups the jumping loci must be defined by binomial ideals. As we will show, this is not the case for general finitely presented groups. Thus, the “binomial condition” can be used as a criterion for proving certain finitely presented groups are not Kähler.

How to cite

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Hironaka, Eriko. "Alexander stratifications of character varieties." Annales de l'institut Fourier 47.2 (1997): 555-583. <http://eudml.org/doc/75237>.

@article{Hironaka1997,
abstract = {Equations defining the jumping loci for the first cohomology group of one-dimensional representations of a finitely presented group $\Gamma $ can be effectively computed using Fox calculus. In this paper, we give an exposition of Fox calculus in the language of group cohomology and in the language of finite abelian coverings of CW complexes. Work of Arapura and Simpson imply that if $\Gamma $ is the fundamental group of a compact Kähler manifold, then the strata are finite unions of translated affine subtori. It follows that for Kähler groups the jumping loci must be defined by binomial ideals. As we will show, this is not the case for general finitely presented groups. Thus, the “binomial condition” can be used as a criterion for proving certain finitely presented groups are not Kähler.},
author = {Hironaka, Eriko},
journal = {Annales de l'institut Fourier},
keywords = {Alexander invariants; Betti numbers; binomial ideals; character varieties; complex projective varieties; unbranched coverings; CW-complexes; fundamental groups; Kähler groups},
language = {eng},
number = {2},
pages = {555-583},
publisher = {Association des Annales de l'Institut Fourier},
title = {Alexander stratifications of character varieties},
url = {http://eudml.org/doc/75237},
volume = {47},
year = {1997},
}

TY - JOUR
AU - Hironaka, Eriko
TI - Alexander stratifications of character varieties
JO - Annales de l'institut Fourier
PY - 1997
PB - Association des Annales de l'Institut Fourier
VL - 47
IS - 2
SP - 555
EP - 583
AB - Equations defining the jumping loci for the first cohomology group of one-dimensional representations of a finitely presented group $\Gamma $ can be effectively computed using Fox calculus. In this paper, we give an exposition of Fox calculus in the language of group cohomology and in the language of finite abelian coverings of CW complexes. Work of Arapura and Simpson imply that if $\Gamma $ is the fundamental group of a compact Kähler manifold, then the strata are finite unions of translated affine subtori. It follows that for Kähler groups the jumping loci must be defined by binomial ideals. As we will show, this is not the case for general finitely presented groups. Thus, the “binomial condition” can be used as a criterion for proving certain finitely presented groups are not Kähler.
LA - eng
KW - Alexander invariants; Betti numbers; binomial ideals; character varieties; complex projective varieties; unbranched coverings; CW-complexes; fundamental groups; Kähler groups
UR - http://eudml.org/doc/75237
ER -

References

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  10. [GL] M. GREEN and R. LAZARSFELD, Higher obstructions to deforming cohomology groups of line bundles, J. Am. Math. Soc., 4 (1991), 87-103. Zbl0735.14004MR92i:32021
  11. [Ha] R. HARTSHORNE, Algebraic Geometry, GTM 52, Springer-Verlag, New York, 1977. Zbl0367.14001MR57 #3116
  12. [La] M. LAURENT, Équations diophantines exponentielles, Invent. Math., 78 (1982), 833-851. 
  13. [Sim] C. SIMPSON, Subspaces of moduli spaces of rank one local systems, Ann. scient. École Norm. Sup., 4 (1993). Zbl0798.14005MR94f:14008
  14. [Siu] Y.-T. SIU, Strong rigidity for Kähler manifolds and the construction of bounded holomorphic functions, Discrete Groups in Analysis, Birkhäuser, 1987, 123-151. Zbl0647.53052

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