Characteristic varieties of quasi-projective manifolds and orbifolds

Artal Bartolo, Enrique¹ ; Cogolludo-Agustín, José Ignacio¹ ; Matei, Daniel²

¹ Departamento de Matemáticas, IUMA, Facultad de Ciencias, Universidad de Zaragoza, c/ Pedro Cerbuna 12, E-50009 Zaragoza, Spain
² Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania

Geometry & topology, Tome 17 (2013) no. 1, pp. 273-309.

Résumé

The present paper considers the structure of the space of characters of quasi-projective manifolds. Such a space is stratified by the cohomology support loci of rank one local systems called characteristic varieties. The classical structure theorem of characteristic varieties is due to Arapura and it exhibits the positive-dimensional irreducible components as pull-backs obtained from morphisms onto complex curves.

In this paper a different approach is provided, using morphisms onto orbicurves, which accounts also for zero-dimensional components and gives more precise information on the positive-dimensional characteristic varieties. In the course of proving this orbifold version of Arapura’s structure theorem, a gap in his proof is completed. As an illustration of the benefits of the orbifold approach, new obstructions for a group to be the fundamental group of a quasi-projective manifold are obtained.

Détail

DOI : 10.2140/gt.2013.17.273

Classification : 32S20, 32S50, 58K65, 14B05, 14H30, 14H50
Keywords: characteristic varieties, local systems, cohomology jumping loci, cohomology with twisted coefficients, quasi-projective groups, orbicurves, orbifolds

Bibliographie

[1] D Arapura, Geometry of cohomology support loci for local systems, I, J. Algebraic Geom. 6 (1997) 563

[2] E Artal Bartolo, J Carmona Ruber, J I Cogolludo-Agustín, Essential coordinate components of characteristic varieties, Math. Proc. Cambridge Philos. Soc. 136 (2004) 287

[3] E Artal Bartolo, J I Cogolludo-Agustín, On the connection between fundamental groups and pencils with multiple fibers, J. Singul. 2 (2010) 1

[4] E Artal Bartolo, J I Cogolludo-Agustín, A Libgober, Depth of cohomology support loci for quasi-projective varieties via orbifold pencils

[5] E Artal Bartolo, J I Cogolludo-Agustín, D Matei, Quasi-projectivity, Artin–Tits groups, and pencil maps, from: "Topology of algebraic varieties and singularities" (editors J I Cogolludo-Agustín, E Hironaka), Contemp. Math. 538, Amer. Math. Soc. (2011) 113

[6] G Barthel, F Hirzebruch, T Höfer, Geradenkonfigurationen und Algebraische Flächen, 4, Friedr. Vieweg Sohn (1987)

[7] I Bauer, Irrational pencils on non-compact algebraic manifolds, Internat. J. Math. 8 (1997) 441

[8] A Beauville, Annulation du H1 pour les fibrés en droites plats, from: "Complex algebraic varieties" (editors K Hulek, T Peternell, M Schneider, F O Schreyer), Lecture Notes in Math. 1507, Springer (1992) 1

[9] R Bieri, W D Neumann, R Strebel, A geometric invariant of discrete groups, Invent. Math. 90 (1987) 451

[10] F Bruhat, J Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math. (1972) 5

[11] N Budur, Unitary local systems, multiplier ideals, and polynomial periodicity of Hodge numbers, Adv. Math. 221 (2009) 217

[12] F Campana, Ensembles de Green–Lazarsfeld et quotients résolubles des groupes de Kähler, J. Algebraic Geom. 10 (2001) 599

[13] F Campana, Orbifoldes géométriques spéciales et classification biméromorphe des variétés kählériennes compactes, J. Inst. Math. Jussieu 10 (2011) 809

[14] F Campana, Quotients résolubles ou nilpotents des groupes de Kähler orbifoldes, Manuscripta Math. 135 (2011) 117

[15] F Campana, Special orbifolds and birational classification: a survey, from: "Classification of algebraic varieties" (editors C Faber, G van der Geer, E Looijenga), EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich (2011) 123

[16] F Catanese, Moduli and classification of irregular Kaehler manifolds (and algebraic varieties) with Albanese general type fibrations, Invent. Math. 104 (1991) 263

[17] F Catanese, J Keum, K Oguiso, Some remarks on the universal cover of an open K3 surface, Math. Ann. 325 (2003) 279

[18] J I Cogolludo-Agustín, Topological invariants of the complement to arrangements of rational plane curves, Mem. Amer. Math. Soc. 159 (2002)

[19] J I Cogolludo-Agustín, A Libgober, Mordell–Weil groups of elliptic threefolds and the Alexander module of plane curves, to appear in Crelle’s Journal

[20] D C Cohen, Triples of arrangements and local systems, Proc. Amer. Math. Soc. 130 (2002) 3025

[21] D C Cohen, G Denham, A I Suciu, Torsion in Milnor fiber homology, Algebr. Geom. Topol. 3 (2003) 511

[22] K Corlette, C Simpson, On the classification of rank-two representations of quasiprojective fundamental groups, Compos. Math. 144 (2008) 1271

[23] P Deligne, Équations différentielles à points singuliers réguliers, 163, Springer (1970)

[24] T Delzant, Trees, valuations and the Green–Lazarsfeld set, Geom. Funct. Anal. 18 (2008) 1236

[25] A Dimca, Characteristic varieties and constructible sheaves, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl. 18 (2007) 365

[26] A Dimca, On the irreducible components of characteristic varieties, An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat. 15 (2007) 67

[27] A Dimca, On admissible rank one local systems, J. Algebra 321 (2009) 3145

[28] A Dimca, L Maxim, Multivariable Alexander invariants of hypersurface complements, Trans. Amer. Math. Soc. 359 (2007) 3505

[29] A Dimca, Ş Papadima, A I Suciu, Alexander polynomials: essential variables and multiplicities, Int. Math. Res. Not. 2008 (2008) 36

[30] A Dimca, Ş Papadima, A I Suciu, Topology and geometry of cohomology jump loci, Duke Math. J. 148 (2009) 405

[31] R H Fox, Free differential calculus, V : the Alexander matrices re-examined, Ann. of Math. 71 (1960) 408

[32] M Green, R Lazarsfeld, Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville, Invent. Math. 90 (1987) 389

[33] H A Hamm, Lefschetz theorems for singular varieties, from: "Singularities, Part 1" (editor P Orlik), Proc. Sympos. Pure Math. 40, Amer. Math. Soc. (1983) 547

[34] E Hironaka, Alexander stratifications of character varieties, Ann. Inst. Fourier (Grenoble) 47 (1997) 555

[35] G Levitt, R-trees and the Bieri–Neumann–Strebel invariant, Publ. Mat. 38 (1994) 195

[36] A Libgober, Characteristic varieties of algebraic curves, from: "Applications of algebraic geometry to coding theory, physics and computation" (editors C Ciliberto, F Hirzebruch, R Miranda, M Teicher), NATO Sci. Ser. II Math. Phys. Chem. 36, Kluwer Acad. Publ. (2001) 215

[37] A Libgober, Non vanishing loci of Hodge numbers of local systems, Manuscripta Math. 128 (2009) 1

[38] M V Nori, Zariski’s conjecture and related problems, Ann. Sci. École Norm. Sup. 16 (1983) 305

[39] F Serrano, Multiple fibres of a morphism, Comment. Math. Helv. 65 (1990) 287

[40] C Simpson, Lefschetz theorems for the integral leaves of a holomorphic one-form, Compositio Math. 87 (1993) 99

[41] C Simpson, Subspaces of moduli spaces of rank one local systems, Ann. Sci. École Norm. Sup. 26 (1993) 361

[42] Y T Siu, Strong rigidity for Kähler manifolds and the construction of bounded holomorphic functions, from: "Discrete groups in geometry and analysis" (editor R Howe), Progr. Math. 67, Birkhäuser (1987) 124

[43] K Timmerscheidt, Mixed Hodge theory for unitary local systems, J. Reine Angew. Math. 379 (1987) 152

[44] O Zariski, The topological discriminant group of a Riemann surface of genus p, Amer. J. Math. 59 (1937) 335