Alexander stratifications of character varieties
Annales de l'Institut Fourier, Volume 47 (1997) no. 2, p. 555-583

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.

Dans cet article, nous donnons une exposition élémentaire sur la stratification d’Alexander qui vient de deux points de vue : la cohomologie des groupes et les revêtements finis et abéliens des CW-complexes. Nous appliquons ces méthodes à l’étude des groupes qui sont isomorphes à un groupe fondamental d’une variété compacte kählérienne.

@article{AIF_1997__47_2_555_0,
     author = {Hironaka, Eriko},
     title = {Alexander stratifications of character varieties},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {47},
     number = {2},
     year = {1997},
     pages = {555-583},
     doi = {10.5802/aif.1573},
     zbl = {0870.57003},
     mrnumber = {98e:14020},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1997__47_2_555_0}
}
Hironaka, Eriko. Alexander stratifications of character varieties. Annales de l'Institut Fourier, Volume 47 (1997) no. 2, pp. 555-583. doi : 10.5802/aif.1573. http://www.numdam.org/item/AIF_1997__47_2_555_0/

[AS] S. Adams and P. Sarnak, Betti numbers of congruence groups, Israel J. of Math., 88 (1994), 31-72. | Zbl 0843.11027

[ABCKT] J. Amoròs, M. Burger, K. Corlette, D. Kotschick and D. Toledo, Fundamental groups of compact Kähler manifolds, Mathematical surveys and Monographs, 44, Am. Math. Soc. (1996). | MR 97d:32037 | Zbl 0849.32006

[Ar1] D. Arapura, Higgs line bundles, Green-Lazarsfeld sets, and maps of Kähler manifolds to curves, Bull. Am. Math. Soc., 26 (1992), 310-314. | Zbl 0759.14016

[Ar2] D. Arapura, Survey on the fundamental group of smooth complex projective varieties, MSRI Series, 28 (1995). | MR 97d:32038 | Zbl 0873.14021

[Be] A. Beauville, Annulation du H1 et systèmes paraconiques sur les surfaces, J. Reine Angew. Math., 388 (1988), 149-157. | MR 89i:14032 | Zbl 0639.14017

[Br] K. Brown, Cohomology of Groups, GTM 87, Springer-Verlag, New York, 1982. | MR 83k:20002 | Zbl 0584.20036

[Cat] F. Catanese, Moduli and classification of irregular Kaehler manifolds (and algebraic varieties) with Albanese general type fibrations, Invent. Math., 104 (1991), 263-389. | MR 92f:32049 | Zbl 0743.32025

[Fox] R. Fox, Free Differential Calculus I, Annals of Math., 57 (1953), 547-560. | MR 14,843d | Zbl 0050.25602

[Gro] M. Gromov, Sur le groupe fondamental d'une variété kählérienne, C. R. Acad. Sci. Paris., 308, I (1989), 67-70. | MR 90i:53090 | Zbl 0661.53049

[GL] M. Green and R. Lazarsfeld, Higher obstructions to deforming cohomology groups of line bundles, J. Am. Math. Soc., 4 (1991), 87-103. | MR 92i:32021 | Zbl 0735.14004

[Ha] R. Hartshorne, Algebraic Geometry, GTM 52, Springer-Verlag, New York, 1977. | MR 57 #3116 | Zbl 0367.14001

[La] M. Laurent, Équations diophantines exponentielles, Invent. Math., 78 (1982), 833-851.

[Sim] C. Simpson, Subspaces of moduli spaces of rank one local systems, Ann. scient. École Norm. Sup., 4 (1993). | Numdam | MR 94f:14008 | Zbl 0798.14005

[Siu] Y.-T. Siu, Strong rigidity for Kähler manifolds and the construction of bounded holomorphic functions, Discrete Groups in Analysis, Birkhäuser, 1987, 123-151. | Zbl 0647.53052