Topology and geometry of cohomology jump loci
Alexandru Dimca, Ştefan Papadima, and Alexander I. Suciu
Source: Duke Math. J. Volume 148, Number 3 (2009), 405-457.
Abstract
We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We define relative characteristic and resonance varieties, $\mathscr{V}_k$ and $\mathscr{R}_k$, related to twisted group cohomology with coefficients of arbitrary rank. We show that the germs at the origin of $\mathscr{V}_k$ and $\mathscr{R}_k$ are analytically isomorphic if the group is $1$-formal; in particular, the tangent cone to $\mathscr{V}_k$ at $1$ equals $\mathscr{R}_k$. 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
Primary Subjects: 14F35, 20F14, 55N25
Secondary Subjects: 14M12, 20F36, 55P62
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
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.dmj/1245350753
Digital Object Identifier: doi:10.1215/00127094-2009-030
Zentralblatt MATH identifier:
05578928
Mathematical Reviews number (MathSciNet):
MR2527322
References
L. A. Alaniya, Cohomology with local coefficients of some nilmanifolds, Russian Math. Surveys 54, no. 5 (1999), 1019--1020.
Mathematical Reviews (MathSciNet):
MR1741665
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.
Mathematical Reviews (MathSciNet):
MR1379330
D. Arapura, Geometry of cohomology support loci for local systems, I, J. Algebraic Geom. 6 (1997), 563--597.
Mathematical Reviews (MathSciNet):
MR1487227
Zentralblatt MATH:
0923.14010
D. Arapura, P. Bressler, and M. Ramachandran, On the fundamental group of a compact Kähler manifold, Duke Math. J. 68 (1992), 477--488.
Mathematical Reviews (MathSciNet):
MR1194951
Digital Object Identifier: doi:10.1215/S0012-7094-92-06819-0
Project Euclid: euclid.dmj/1077294146
I. Bauer, Irrational pencils on non-compact algebraic manifolds, Internat. J. Math. 8 (1997), 441--450.
Mathematical Reviews (MathSciNet):
MR1460895
Digital Object Identifier: doi:10.1142/S0129167X97000226
Zentralblatt MATH:
0896.14008
A. Beauville, Annulation du $H^1$ et systèmes paracanoniques sur les surfaces, J. Reine Angew. Math. 388 (1988), 149--157.
Mathematical Reviews (MathSciNet):
MR0944188
—, ``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.
Mathematical Reviews (MathSciNet):
MR1178716
Digital Object Identifier: doi:10.1007/BFb0094507
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.
Mathematical Reviews (MathSciNet):
MR1465330
Digital Object Identifier: doi:10.1007/s002220050168
Zentralblatt MATH:
0888.20021
R. Bezrukavnikov, Koszul DG-algebras arising from configuration spaces, Geom. Funct. Anal. 4 (1994), 119--135.
Mathematical Reviews (MathSciNet):
MR1262702
Digital Object Identifier: doi:10.1007/BF01895836
Zentralblatt MATH:
0829.55009
E. Brieskorn, Die Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe, Invent. Math. 12 (1971), 57--61.
Mathematical Reviews (MathSciNet):
MR0293615
Digital Object Identifier: doi:10.1007/BF01389827
F. Campana, Ensembles de Green-Lazarsfeld et quotients résolubles des groupes de Kähler, J. Algebraic Geom. 10 (2001), 599--622.
Mathematical Reviews (MathSciNet):
MR1838973
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.
Mathematical Reviews (MathSciNet):
MR1098610
Digital Object Identifier: doi:10.1007/BF01245076
—, Fibred Kähler and quasi-projective groups, Adv. Geom. 2003, suppl., S13--S27.
Mathematical Reviews (MathSciNet):
MR2028385
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.
Mathematical Reviews (MathSciNet):
MR1368655
Zentralblatt MATH:
0930.55006
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.
Mathematical Reviews (MathSciNet):
MR0458461
Digital Object Identifier: doi:10.1016/0001-8708(77)90120-7
Zentralblatt MATH:
0345.58003
D. C. Cohen and A. I. Suciu, Characteristic varieties of arrangements, Math. Proc. Cambridge Philos. Soc. 127 (1999), 33--53.
Mathematical Reviews (MathSciNet):
MR1692519
Digital Object Identifier: doi:10.1017/S0305004199003576
Zentralblatt MATH:
0963.32018
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. $\!$;
Mathematical Reviews (MathSciNet):
MR0498551
Mathematical Reviews (MathSciNet):
MR0498552
Digital Object Identifier: doi:10.1007/BF02684692
P. Deligne, P. Griffiths, J. Morgan, and D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), 245--274.
Mathematical Reviews (MathSciNet):
MR0382702
Digital Object Identifier: doi:10.1007/BF01389853
A. Dimca, Singularities and Topology of Hypersurfaces, Universitext, Springer, New York, 1992.
Mathematical Reviews (MathSciNet):
MR1194180
Zentralblatt MATH:
0753.57001
—, Characteristic varieties and constructible sheaves, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Nature. Rend. Lincei (9) Mat. Appl. 18 (2007), 365--389.
Mathematical Reviews (MathSciNet):
MR2349994
—, 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.
Mathematical Reviews (MathSciNet):
MR2299465
Digital Object Identifier: doi:10.1090/S0002-9947-07-04241-9
Zentralblatt MATH:
1119.32012
A. Dimca, S. Papadima, and A. Suciu, Alexander polynomials: Essential variables and multiplicities, Int. Math. Res. Not. IMRN 2008, Art. ID rnm119.
Mathematical Reviews (MathSciNet):
MR2416998
Zentralblatt MATH:
1156.32018
—, Quasi-Kähler Bestvina-Brady groups, J. Algebraic Geom. 17 (2008), 185--197.
Mathematical Reviews (MathSciNet):
MR2357684
—, 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.
Mathematical Reviews (MathSciNet):
MR1322960
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. $\!$;
Mathematical Reviews (MathSciNet):
MR1176205
Mathematical Reviews (MathSciNet):
MR1213111
Digital Object Identifier: doi:10.1007/BF01232040
M. Falk, Arrangements and cohomology, Ann. Comb. 1 (1997), 135--157.
Mathematical Reviews (MathSciNet):
MR1629681
Digital Object Identifier: doi:10.1007/BF02558471
Zentralblatt MATH:
0941.52022
M. Falk and S. Yuzvinsky, Multinets, resonance varieties, and pencils of plane curves, Compos. Math. 143 (2007), 1069--1088.
Mathematical Reviews (MathSciNet):
MR2339840
Zentralblatt MATH:
1122.52009
R. H. Fox, Free differential calculus, II: The isomorphism problem of groups, Ann. of Math. (2) 59 (1954), 196--210.
Mathematical Reviews (MathSciNet):
MR0062125
Digital Object Identifier: doi:10.2307/1969686
JSTOR: links.jstor.org
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.
Mathematical Reviews (MathSciNet):
MR0972343
Digital Object Identifier: doi:10.1007/BF02699127
M. Green and R. Lazarsfeld, Higher obstructions to deforming cohomology groups of line bundles, J. Amer. Math. Soc. 4 (1991), 87--103.
Mathematical Reviews (MathSciNet):
MR1076513
Digital Object Identifier: doi:10.2307/2939255
JSTOR: links.jstor.org
Zentralblatt MATH:
0735.14004
M. Gromov, Sur le groupe fondamental d'une variété kählérienne, C. R. Acad. Sci. Paris Sér. I Math. 308, no. 3 (1989), 67--70.
Mathematical Reviews (MathSciNet):
MR0983460
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.
Mathematical Reviews (MathSciNet):
MR1234261
Zentralblatt MATH:
0831.57005
—, Infinitesimal presentations of the Torelli groups, J. Amer. Math. Soc. 10 (1997), 597--651.
Mathematical Reviews (MathSciNet):
MR1431828
Digital Object Identifier: doi:10.1090/S0894-0347-97-00235-X
JSTOR: links.jstor.org
Zentralblatt MATH:
0915.57001
J. Harris, Algebraic Geometry: A First Course, Grad. Texts in Math. 133, Springer, New York, 1995.
Mathematical Reviews (MathSciNet):
MR1416564
P. J. Hilton and U. Stammbach, A Course in Homological Algebra, Grad. Texts in Math. 4, Springer, New York, 1971.
Mathematical Reviews (MathSciNet):
MR0346025
J. E. Humphreys, Linear Algebraic Groups, Grad. Texts in Math. 21, Springer, New York, 1975.
Mathematical Reviews (MathSciNet):
MR0396773
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.
Mathematical Reviews (MathSciNet):
MR1733326
Digital Object Identifier: doi:10.1007/BF02701766
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.
Mathematical Reviews (MathSciNet):
MR0726138
Project Euclid: euclid.nmj/1118787354
Zentralblatt MATH:
0503.57001
J. KolláR, Shafarevich Maps and Automorphic Forms, M. B. Porter Lectures, Princeton Univ. Press, Princeton, 1995.
Mathematical Reviews (MathSciNet):
MR1341589
M. Laurent, Équations diophantiennes exponentielles, Invent. Math. 78 (1984), 299--327.
Mathematical Reviews (MathSciNet):
MR0767195
Digital Object Identifier: doi:10.1007/BF01388597
M. Lazard, Sur les groupes nilpotents et les anneaux de Lie, Ann. Sci. École Norm. Sup. (3) 71 (1954), 101--190.
Mathematical Reviews (MathSciNet):
MR0088496
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.
Mathematical Reviews (MathSciNet):
MR1272031
Zentralblatt MATH:
0833.14012
—, ``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.
Mathematical Reviews (MathSciNet):
MR1866902
Zentralblatt MATH:
1045.14016
—, First order deformations for rank one local systems with a non-vanishing cohomology, Topology Appl. 118 (2002), 159--168.
Mathematical Reviews (MathSciNet):
MR1877722
Digital Object Identifier: doi:10.1016/S0166-8641(01)00048-7
Zentralblatt MATH:
1010.52016
A. Libgober and S. Yuzvinsky, Cohomology of the Orlik-Solomon algebras and local systems, Compositio Math. 121 (2000), 337--361.
Mathematical Reviews (MathSciNet):
MR1761630
Digital Object Identifier: doi:10.1023/A:1001826010964
Zentralblatt MATH:
0952.52020
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.
Mathematical Reviews (MathSciNet):
MR0422434
M. Markl and ş. Papadima, Homotopy Lie algebras and fundamental groups via deformation theory, Ann. Inst. Fourier (Grenoble) 42 (1992), 905--935.
Mathematical Reviews (MathSciNet):
MR1196099
W. S. Massey, Completion of link modules, Duke Math. J. 47 (1980), 399--420.
Mathematical Reviews (MathSciNet):
MR0575904
Digital Object Identifier: doi:10.1215/S0012-7094-80-04724-9
Project Euclid: euclid.dmj/1077314042
Zentralblatt MATH:
0464.57001
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.
Mathematical Reviews (MathSciNet):
MR1796900
Zentralblatt MATH:
0974.32020
—, Homotopy types of complements of $2$-arrangements in $\RR\sp 4$, Topology 39 (2000), 61--88.
Mathematical Reviews (MathSciNet):
MR1710992
Digital Object Identifier: doi:10.1016/S0040-9383(98)00058-5
—, Hall invariants, homology of subgroups, and characteristic varieties, Int. Math. Res. Not. 2002, no. 9, 465--503.
Mathematical Reviews (MathSciNet):
MR1884468
Digital Object Identifier: doi:10.1155/S107379280210907X
Zentralblatt MATH:
1061.20040
J. Meier and L. Vanwyk, The Bieri-Neumann-Strebel invariants for graph groups, Proc. London Math. Soc. (3) 71 (1995), 263--280.
Mathematical Reviews (MathSciNet):
MR1337468
Digital Object Identifier: doi:10.1112/plms/s3-71.2.263
Zentralblatt MATH:
0835.20037
J. Milnor, Singular Points of Complex Hypersurfaces, Ann. of Math. Stud. 61, Princeton Univ. Press, Princeton, 1968.
Mathematical Reviews (MathSciNet):
MR0239612
J. W. Morgan, The algebraic topology of smooth algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 48 (1978), 137--204.
Mathematical Reviews (MathSciNet):
MR0516917
Digital Object Identifier: doi:10.1007/BF02684316
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.
Mathematical Reviews (MathSciNet):
MR0838822
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.
Mathematical Reviews (MathSciNet):
MR1352516
Zentralblatt MATH:
0852.20023
—, Campbell-Hausdorff invariants of links, Proc. London Math. Soc. (3) 75 (1997), 641--670.
Mathematical Reviews (MathSciNet):
MR1466664
Digital Object Identifier: doi:10.1112/S0024611597000464
Zentralblatt MATH:
0901.57009
—, ``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.
Mathematical Reviews (MathSciNet):
MR2432117
S. Papadima and A. I. Suciu, Chen Lie algebras, Int. Math. Res. Not. 2004, no. 21, 1057--1086.
Mathematical Reviews (MathSciNet):
MR2037049
Digital Object Identifier: doi:10.1155/S1073792804132017
Zentralblatt MATH:
1076.17007
—, Algebraic invariants for right-angled Artin groups, Math. Ann. 334 (2006), 533--555.
Mathematical Reviews (MathSciNet):
MR2207874
Digital Object Identifier: doi:10.1007/s00208-005-0704-9
Zentralblatt MATH:
05013674
—, When does the associated graded Lie algebra of an arrangement group decompose?, Comment. Math. Helv. 81 (2006), 859--875.
Mathematical Reviews (MathSciNet):
MR2271225
—, Toric complexes and Artin kernels, Adv. Math. 220 (2009), 441--477.
Mathematical Reviews (MathSciNet):
MR2466422
Digital Object Identifier: doi:10.1016/j.aim.2008.09.008
Zentralblatt MATH:
05480732
D. Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205--295.
Mathematical Reviews (MathSciNet):
MR0258031
Digital Object Identifier: doi:10.2307/1970725
JSTOR: links.jstor.org
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.
Mathematical Reviews (MathSciNet):
MR1978716
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.
Mathematical Reviews (MathSciNet):
MR1344845
Digital Object Identifier: doi:10.1016/0022-4049(95)00014-N
Zentralblatt MATH:
0849.32025
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.
Mathematical Reviews (MathSciNet):
MR0098097
C. I. Simpson, Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 5--95.
Mathematical Reviews (MathSciNet):
MR1179076
Digital Object Identifier: doi:10.1007/BF02699491
— Subspaces of moduli spaces of rank one local systems, Ann. Sci. École Norm. Sup. (4) 26 (1993), 361--401.
Mathematical Reviews (MathSciNet):
MR1222278
Zentralblatt MATH:
0798.14005
D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269--331.
Mathematical Reviews (MathSciNet):
MR0646078
Digital Object Identifier: doi:10.1007/BF02684341
C. H. Taubes, The existence of anti-self-dual conformal structures, J. Differential Geom. 36 (1992), 163--253.
Mathematical Reviews (MathSciNet):
MR1168984
Project Euclid: euclid.jdg/1214448445
Zentralblatt MATH:
0822.53006
B. Totaro, Configuration spaces of algebraic varieties, Topology 35 (1996), 1057--1067.
Mathematical Reviews (MathSciNet):
MR1404924
Digital Object Identifier: doi:10.1016/0040-9383(95)00058-5
C. Voisin, On the homotopy types of compact Kähler and complex projective manifolds, Invent. Math. 157 (2004), 329--343.
Mathematical Reviews (MathSciNet):
MR2076925
Digital Object Identifier: doi:10.1007/s00222-003-0352-1
H. Whitney, Complex Analytic Varieties, Addison-Wesley, Reading, Mass., 1972.
Mathematical Reviews (MathSciNet):
MR0387634
Zentralblatt MATH:
0265.32008
G. M. Ziegler, On the difference between real and complex arrangements, Math. Z. 212 (1993), 1--11.
Mathematical Reviews (MathSciNet):
MR1200161
Digital Object Identifier: doi:10.1007/BF02571638
Zentralblatt MATH:
0789.57018
Duke Mathematical Journal
