Duke Mathematical Journal

Topology and geometry of cohomology jump loci

Alexandru Dimca, Ştefan Papadima, and Alexander I. Suciu

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We define relative characteristic and resonance varieties, Vk and Rk, related to twisted group cohomology with coefficients of arbitrary rank. We show that the germs at the origin of Vk and Rk are analytically isomorphic if the group is 1-formal; in particular, the tangent cone to Vk at 1 equals Rk. These new obstructions to 1-formality lead to a striking rationality property of the usual resonance varieties. A detailed analysis of the irreducible components of the tangent cone at 1 to the first characteristic variety yields powerful obstructions to realizing a finitely presented group as the fundamental group of a smooth, complex quasi-projective algebraic variety. This sheds new light on a classical problem of J.-P. Serre. Applications to arrangements, configuration spaces, coproducts of groups, and Artin groups are given

Article information

Duke Math. J., Volume 148, Number 3 (2009), 405-457.

First available in Project Euclid: 18 June 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14F35: Homotopy theory; fundamental groups [See also 14H30] 20F14: Derived series, central series, and generalizations 55N25: Homology with local coefficients, equivariant cohomology
Secondary: 14M12: Determinantal varieties [See also 13C40] 20F36: Braid groups; Artin groups 55P62: Rational homotopy theory


Dimca, Alexandru; Papadima, Ştefan; Suciu, Alexander I. Topology and geometry of cohomology jump loci. Duke Math. J. 148 (2009), no. 3, 405--457. doi:10.1215/00127094-2009-030. https://projecteuclid.org/euclid.dmj/1245350753

Export citation


  • L. A. Alaniya, Cohomology with local coefficients of some nilmanifolds, Russian Math. Surveys 54, no. 5 (1999), 1019--1020.
  • J. Amorós, m. burger, k. corlette, d. kotschick, and D. Toledo, Fundamental Groups of Compact Kähler Manifolds, Math. Surveys Monogr. 44, Amer. Math. Soc. Providence, 1996.
  • D. Arapura, Geometry of cohomology support loci for local systems, I, J. Algebraic Geom. 6 (1997), 563--597.
  • D. Arapura, P. Bressler, and M. Ramachandran, On the fundamental group of a compact Kähler manifold, Duke Math. J. 68 (1992), 477--488.
  • I. Bauer, Irrational pencils on non-compact algebraic manifolds, Internat. J. Math. 8 (1997), 441--450.
  • A. Beauville, Annulation du $H^1$ et systèmes paracanoniques sur les surfaces, J. Reine Angew. Math. 388 (1988), 149--157.
  • —, ``Annulation du $H\sp 1$ pour les fibrés en droites plats'' in Complex Algebraic Varieties (Bayreuth, Germany, 1990), Lecture Notes in Math. 1507, Springer, Berlin, 1992, 1--15.
  • B. Berceanu and S. Papadima, Universal representations of braid and braid-permutation groups, preprint,\arxiv0708.0634v1[math.GR], to appear in J. Knot Theory Ramifications.
  • M. Bestvina and N. Brady, Morse theory and finiteness properties of groups, Invent. Math. 129 (1997), 445--470.
  • R. Bezrukavnikov, Koszul DG-algebras arising from configuration spaces, Geom. Funct. Anal. 4 (1994), 119--135.
  • E. Brieskorn, Die Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe, Invent. Math. 12 (1971), 57--61.
  • F. Campana, Ensembles de Green-Lazarsfeld et quotients résolubles des groupes de Kähler, J. Algebraic Geom. 10 (2001), 599--622.
  • G. Castelnuovo, Sulle superficie aventi il genere aritmetico negativo, Palermo Rend. 20 (1905), 55--60.
  • F. Catanese, Moduli and classification of irregular Kaehler manifolds (and algebraic varieties) with Albanese general type fibrations, Invent. Math. 104 (1991), 263--289.
  • —, Fibred Kähler and quasi-projective groups, Adv. Geom. 2003, suppl., S13--S27.
  • R. Charney and M. W. Davis, ``Finite $K(\pi, 1)$s for Artin groups'' in Prospects in Topology (Princeton, 1994), Ann. of Math. Stud. 138, Princeton Univ. Press, Princeton, 1995, 110--124.
  • K.-T. Chen, Extension of $C\sp\infty$ function algebra by integrals and Malcev completion of $\pi\sb1$, Advances in Math. 23 (1977), 181--210.
  • D. C. Cohen and A. I. Suciu, Characteristic varieties of arrangements, Math. Proc. Cambridge Philos. Soc. 127 (1999), 33--53.
  • M. De Franchis, Sulle superficie algebriche le quali contengono un fascio irrazionale di curve, Palermo Rend. 20 (1905), 49--54.
  • P. Deligne, Théorie de Hodge, II, Inst. Hautes Études Sci. Publ. Math. 40 (1972), 5--57.; Théorie de Hodge, III, Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5--77. $\!$;
  • P. Deligne, P. Griffiths, J. Morgan, and D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), 245--274.
  • A. Dimca, Singularities and Topology of Hypersurfaces, Universitext, Springer, New York, 1992.
  • —, Characteristic varieties and constructible sheaves, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Nature. Rend. Lincei (9) Mat. Appl. 18 (2007), 365--389.
  • —, On the isotropic subspace theorems, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 51 (2008), 307--324.
  • A. Dimca and L. Maxim, Multivariable Alexander invariants of hypersurface complements, Trans. Amer. Math. Soc. 359 (2007), 3505--3528.
  • A. Dimca, S. Papadima, and A. Suciu, Alexander polynomials: Essential variables and multiplicities, Int. Math. Res. Not. IMRN 2008, Art. ID rnm119.
  • —, Quasi-Kähler Bestvina-Brady groups, J. Algebraic Geom. 17 (2008), 185--197.
  • —, Formality, Alexander invariants, and a question of Serre, preprint,\arxivmath/0512480v3[math.AT]
  • —, Non-finiteness properties of fundamental groups of smooth projective varieties, J. Reine Angew. Math. 629 (2009), 89--105.
  • A. Dimca and A. I. Suciu, Which $3$-manifold groups are Kähler groups?, preprint,\arxiv0709.4350v2[math.AG], to appear in J. Eur. Math. Soc. (JEMS).
  • D. Eisenbud, Commutative Algebra: With a View Towards Algebraic Geometry, Grad. Texts in Math. 150, Springer, New York, 1995.
  • H. Esnault, V. Schechtman, and E. Viehweg, Cohomology of local systems on the complement of hyperplanes, Invent. Math. 109 (1992), 557--561.; Erratum, Invent. Math. 112 (1993), 447. $\!$;
  • M. Falk, Arrangements and cohomology, Ann. Comb. 1 (1997), 135--157.
  • M. Falk and S. Yuzvinsky, Multinets, resonance varieties, and pencils of plane curves, Compos. Math. 143 (2007), 1069--1088.
  • R. H. Fox, Free differential calculus, II: The isomorphism problem of groups, Ann. of Math. (2) 59 (1954), 196--210.
  • W. M. Goldman and J. J. Millson, The deformation theory of representations of fundamental groups of compact Kähler manifolds, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 43--96.
  • M. Green and R. Lazarsfeld, Higher obstructions to deforming cohomology groups of line bundles, J. Amer. Math. Soc. 4 (1991), 87--103.
  • M. Gromov, Sur le groupe fondamental d'une variété kählérienne, C. R. Acad. Sci. Paris Sér. I Math. 308, no. 3 (1989), 67--70.
  • R. M. Hain, ``Completions of mapping class groups and the cycle $C-C^-$'' in Mapping Class Groups and Moduli Spaces of Riemann Surfaces (Göttingen/Seattle, 1991), Contemp. Math. 150, Amer. Math. Soc., Providence, 1993, 75--105.
  • —, Infinitesimal presentations of the Torelli groups, J. Amer. Math. Soc. 10 (1997), 597--651.
  • J. Harris, Algebraic Geometry: A First Course, Grad. Texts in Math. 133, Springer, New York, 1995.
  • P. J. Hilton and U. Stammbach, A Course in Homological Algebra, Grad. Texts in Math. 4, Springer, New York, 1971.
  • J. E. Humphreys, Linear Algebraic Groups, Grad. Texts in Math. 21, Springer, New York, 1975.
  • M. Kapovich and J. J. Millson, On representation varieties of Artin groups, projective arrangements and the fundamental groups of smooth complex algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 88 (1998), 5--95.
  • T. Kohno, On the holonomy Lie algebra and the nilpotent completion of the fundamental group of the complement of hypersurfaces, Nagoya Math. J. 92 (1983), 21--37.
  • J. KolláR, Shafarevich Maps and Automorphic Forms, M. B. Porter Lectures, Princeton Univ. Press, Princeton, 1995.
  • M. Laurent, Équations diophantiennes exponentielles, Invent. Math. 78 (1984), 299--327.
  • M. Lazard, Sur les groupes nilpotents et les anneaux de Lie, Ann. Sci. École Norm. Sup. (3) 71 (1954), 101--190.
  • A. S. Libgober, ``Groups which cannot be realized as fundamental groups of the complements to hypersurfaces in $C\sp N$'' in Algebraic geometry and Its Applications (West Lafayette, Ind., 1990), Springer, New York, 1994, 203--207.
  • —, ``Characteristic varieties of algebraic curves'' in Applications of Algebraic Geometry to Coding Theory, Physics and Computation (Eilat, Israel, 2001), NATO Sci. Ser. II Math. Phys. Chem. 36, Kluwer, Dordrecht, 2001, 215--254.
  • —, First order deformations for rank one local systems with a non-vanishing cohomology, Topology Appl. 118 (2002), 159--168.
  • A. Libgober and S. Yuzvinsky, Cohomology of the Orlik-Solomon algebras and local systems, Compositio Math. 121 (2000), 337--361.
  • A. Macinic and ş. Papadima, Characteristic varieties of nilpotent groups and applications, preprint,\arxiv0710.5398v1[math.AT], to appear in the Proceedings of the 6th Congress of Romanian Mathematicians, Romanian Academy, Bucharest
  • W. Magnus, A. Karrass, and D. Solitar, Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations, 2nd ed., Dover, New York, 1976.
  • M. Markl and ş. Papadima, Homotopy Lie algebras and fundamental groups via deformation theory, Ann. Inst. Fourier (Grenoble) 42 (1992), 905--935.
  • W. S. Massey, Completion of link modules, Duke Math. J. 47 (1980), 399--420.
  • D. Matei and A. Suciu, ``Cohomology rings and nilpotent quotients of real and complex arrangements'' in Arrangements (Tokyo, 1998), Adv. Stud. Pure Math. 27, Math. Soc. Japan, Tokyo, 2000, 185--215.
  • —, Homotopy types of complements of $2$-arrangements in $\RR\sp 4$, Topology 39 (2000), 61--88.
  • —, Hall invariants, homology of subgroups, and characteristic varieties, Int. Math. Res. Not. 2002, no. 9, 465--503.
  • J. Meier and L. Vanwyk, The Bieri-Neumann-Strebel invariants for graph groups, Proc. London Math. Soc. (3) 71 (1995), 263--280.
  • J. Milnor, Singular Points of Complex Hypersurfaces, Ann. of Math. Stud. 61, Princeton Univ. Press, Princeton, 1968.
  • J. W. Morgan, The algebraic topology of smooth algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 48 (1978), 137--204.
  • A. J. Narkawicz, Cohomology jumping loci and the relative Malcev completion, Ph.D. dissertation, Duke University, Durham, N. C., 2007, http://hdl.handle.net/10161/441
  • S. P. Novikov, Bloch homology: Critical points of functions and closed $1$-forms, Soviet Math. Dokl. 33, no. 2 (1986), 551--555.
  • ş. Papadima, ``Finite determinacy phenomena for finitely presented groups'' in Proceedings of the 2nd Gauss Symposium: Conference A: Mathematics and Theoretical Physics (Munich, 1993), Sympos. Gaussiana, de Gruyter, Berlin, 1995, 507--528.
  • —, Campbell-Hausdorff invariants of links, Proc. London Math. Soc. (3) 75 (1997), 641--670.
  • —, ``Global versus local algebraic fundamental groups'' in Mini-Workshop: Topology of Closed One-Forms and Cohomology Jumping Loci (Oberwolfach, Germany, 2007), Oberwolfach Rep. 4, Eur. Math. Soc., Zürich, 2007, 2340--2341.
  • S. Papadima and A. I. Suciu, Chen Lie algebras, Int. Math. Res. Not. 2004, no. 21, 1057--1086.
  • —, Algebraic invariants for right-angled Artin groups, Math. Ann. 334 (2006), 533--555.
  • —, When does the associated graded Lie algebra of an arrangement group decompose?, Comment. Math. Helv. 81 (2006), 859--875.
  • —, Toric complexes and Artin kernels, Adv. Math. 220 (2009), 441--477.
  • D. Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205--295.
  • A. Reznikov, ``The structure of Kähler groups, I: Second cohomology'' in Motives, Polylogarithms and Hodge Theory, Part II (Irvine, Calif., 1998), Int. Press Lect. Ser. 3, Int. Press, Somerville, Mass., 2002, 717--730.
  • V. Schechtman, H. Terao, and A. Varchenko, Local systems over complements of hyperplanes and the Kac-Kazhdan condition for singular vectors, J. Pure Appl. Algebra 100 (1995), 93--102.
  • J.-P. Serre, ``Sur la topologie des variétés algébriques en charactéristique $p$'' in Symposium internacional de topología algebraica (Mexico City, 1958), Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958, 24--53.
  • C. I. Simpson, Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 5--95.
  • Subspaces of moduli spaces of rank one local systems, Ann. Sci. École Norm. Sup. (4) 26 (1993), 361--401.
  • D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269--331.
  • C. H. Taubes, The existence of anti-self-dual conformal structures, J. Differential Geom. 36 (1992), 163--253.
  • B. Totaro, Configuration spaces of algebraic varieties, Topology 35 (1996), 1057--1067.
  • C. Voisin, On the homotopy types of compact Kähler and complex projective manifolds, Invent. Math. 157 (2004), 329--343.
  • H. Whitney, Complex Analytic Varieties, Addison-Wesley, Reading, Mass., 1972.
  • G. M. Ziegler, On the difference between real and complex arrangements, Math. Z. 212 (1993), 1--11.