Tohoku Mathematical Journal

Regular functions transversal at infinity

Alexandru Dimca and Anatoly Libgober

Full-text: Open access

Abstract

We generalize and complete some of Maxim's recent results on Alexander invariants of a polynomial transversal to the hyperplane at infinity. Roughly speaking, and surprisingly, such a polynomial behaves, both topologically and algebraically (e.g., in terms of the variation of MHS on the cohomology of its smooth fibers), like a homogeneous polynomial.

Article information

Source
Tohoku Math. J. (2), Volume 58, Number 4 (2006), 549-564.

Dates
First available in Project Euclid: 1 February 2007

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1170347689

Digital Object Identifier
doi:10.2748/tmj/1170347689

Mathematical Reviews number (MathSciNet)
MR2297199

Zentralblatt MATH identifier
1149.32016

Subjects
Primary: 32S20: Global theory of singularities; cohomological properties [See also 14E15]
Secondary: 32S22: Relations with arrangements of hyperplanes [See also 52C35] 32S35: Mixed Hodge theory of singular varieties [See also 14C30, 14D07] 32S40: Monodromy; relations with differential equations and D-modules 32S55: Milnor fibration; relations with knot theory [See also 57M25, 57Q45] 32S60: Stratifications; constructible sheaves; intersection cohomology [See also 58Kxx] 14D05: Structure of families (Picard-Lefschetz, monodromy, etc.) 14J70: Hypersurfaces 14F17: Vanishing theorems [See also 32L20] 14F45: Topological properties

Keywords
Hypersurface complement Alexander polynomials local system Milnor fiber perverse sheaves mixed Hodge structure

Citation

Dimca, Alexandru; Libgober, Anatoly. Regular functions transversal at infinity. Tohoku Math. J. (2) 58 (2006), no. 4, 549--564. doi:10.2748/tmj/1170347689. https://projecteuclid.org/euclid.tmj/1170347689


Export citation

References

  • J. Briançon, Ph. Maisonobe and M. Merle, Localisation de systèmes différentiels, stratifications de Whitney et condition de Thom, Invent. Math. 117 (1994), 531--550.
  • S. A. Broughton, Milnor numbers and the topology of polynomial hypersurfaces, Invent. Math. 92 (1988), 217--241.
  • D. C. Cohen, A. Dimca and P. Orlik, Nonresonance conditions for arrangements, Ann. Inst. Fourier (Grenoble) 53 (2003), 1883--1896.
  • P. Deligne, Theorie de Hodge. II, Publ. Math. Inst. Hautes Études Sci. 40 (1971), 5--58.
  • A. Dimca, Singularities and topology of hypersurfaces, Universitext, Springer-Verlag, New York, 1992.
  • A. Dimca, Sheaves in topology, Universitext, Springer-Verlag, Berlin, 2004.
  • A. Dimca, Hyperplane arrangements, $M$-tame polynomials and twisted cohomology, Commutative Algebra, Singularities and Computer Algebra (Sinaia, 2002), 113--126, NATO Sci. Ser. II Math. Phys. Chem. 115, Kluwer Acad. Publ., Dordrecht, 2003.
  • A. Dimca, Monodromy at infinity for polynomials in two variables, J. Algebraic Geom. 7 (1998), 771--779.
  • A. Dimca and A. Némethi, Hypersurface complements, Alexander modules and monodromy, Real and complex singularities,19--43, Contemp. Math. 354, Amer. Math. Soc., Providence, R.I., 2004.
  • A. Dimca and S. Papadima, Hypersurface complements, Milnor fibers and higher homotopy groups of arrangements, Ann. of Math. (2) 158 (2003), 473--507.
  • A. Dimca and A. Libgober, Local topology of reducible divisors, to appear in Proc. Sao Carlos conference on Singularities, P. Brasselet and M. Ruas editors.
  • H. Esnault, Fibre de Milnor d'un cône sur une courbe plane singulière, Invent. Math. 68 (1982), 477--496.
  • R. Garcí a López and A. Némethi, On the monodromy at infinity of a polynomial map I, Compositio Math. 100 (1996), 205--231; II, Compositio Math. 115 (1999), 1--20.
  • R. Garcí a López and A. Némethi, Hodge numbers attached to a polynomial map, Ann. Inst. Fourier (Grenoble) 49 (1999), 1547--1579.
  • M. Goresky and R. MacPherson, Stratified Morse theory, Singularities, Part 1 (Arcata, Calif., 1981), 517--533, Proc. Sympos. Pure Math. 40, Amer. Math. Soc., Providence, R.I., 1983.
  • H. A. Hamm, Lefschetz theorems for singular varieties, Singularities, Part 1 (Arcata, Calif., 1981), 547--557, Proc. Sympos. Pure Math. 40, Amer. Math. Soc., Providence, R.I., 1983.
  • V. S. Kulikov and Vik. S. Kulikov, On the monodromy and mixed Hodge structure on the cohomology of the infinite cyclic covering of the complement to a plane algebraic curve, Izv. Math. 59 (1995), 367--386.
  • A. Libgober, Homotopy groups of the complements to singular hypersurfaces II, Ann. of Math. (2) 139 (1994), 117--144.
  • A. Libgober, Position of singularities of hypersurfaces and the topology of their complements. Algebraic geometry 5, J. Math. Sci. 82 (1996), 3194--3210.
  • A. Libgober, Eigenvalues for the monodromy of the Milnor fibers of arrangements, Trends in singularities, Trends in Math., Birkhäuser, Basel, 2002.
  • A. Libgober, Isolated non-normal crossings, Real and complex singularities, 145--160, Contemp. Math. 354, Amer. Math. Soc., Providence, R.I., 2004.
  • F. Loeser and M. Vaquié, Le polynôme d'Alexander d'une courbe plane projective, Topology 29 (1990), 163--173.
  • L. Maxim, Intersection homology and Alexander modules of hypersurface complements, arXiv.math.AT/0409412.
  • J. Milnor, Singular points of of complex hypersurfaces, Ann. of Math. Stud. 61. Princeton Univ. Press, Princeton, N.J., 1968.
  • J. Morgan, The algebraic topology of smooth algebraic varieties, Publ. Math. Inst. Haute Etudes Sci. 48 (1978), 137--204.
  • A. Némethi and A. Zaharia, Milnor fibration at infinity, Indag. Math. (N.S.) 3 (1992), 323--335.
  • W. Neumann and P. Norbury, Unfolding polynomial maps at infinity, Math Ann. 318 (2000), 149--180.
  • M. Oka, Alexander polynomial of sextics, J. Knot Theory Ramifications 12 (2003), 619--636.
  • L. Păunescu and A. Zaharia, Remarks on the Milnor fibration at infinity, Manuscripta Math. 103 (2000), 351--361.
  • C. Sabbah, Hypergeometric periods for a tame polynomial, C. R. Acad. Sci. Paris, Sér. I Math. 328 (1999), 603--608.
  • M. Saito, Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci. 24 (1988), 849--995.
  • M. Saito, Mixed Hodge modules, Publ. Res. Inst. Math. Sci. 26 (1990), 221--333.
  • M. Saito, Mixed Hodge complexes on algabraic varieties, Math. Ann. 316 (2000), 283--331.
  • D. Siersma and M. Tibăr, Singularities at infinity and their vanishing cycles, Duke Math. J. 80 (1995), 771--783.
  • J. Steenbrink and S. Zucker, Variations of mixed Hodge structures I, Invent. Math. 80 (1985), 489--542.
  • C. A. Weibel, An introduction to homological algebra, Cambridge Stud. Adv. Math. 38, Cambridge Univ. Press, Cambridge, 1994.