Characteristic classes of complex hypersurfaces

Sylvain E. Cappell, Laurentiu Maxim, J. Schürmann Jörg, Julius L. Shaneson

Research output: Contribution to journalArticle

Abstract

The Milnor-Hirzebruch class of a locally complete intersection X in an algebraic manifold M measures the difference between the (Poincaré dual of the) Hirzebruch class of the virtual tangent bundle of X and, respectively, the Brasselet-Schürmann-Yokura (homology) Hirzebruch class of X. In this note, we calculate the Milnor-Hirzebruch class of a globally defined algebraic hypersurface X in terms of the corresponding Hirzebruch invariants of vanishing cycles and singular strata in a Whitney stratification of X. Our approach is based on Schürmann's specialization property for the motivic Hirzebruch class transformation of Brasselet-Schürmann-Yokura. The present results also yield calculations of Todd, Chern and L-type characteristic classes of hypersurfaces.

Original languageEnglish (US)
Pages (from-to)2616-2647
Number of pages32
JournalAdvances in Mathematics
Volume225
Issue number5
DOIs
StatePublished - Dec 2010

Keywords

  • Characteristic classes
  • Hodge theory
  • Hypersurfaces
  • Intersection homology
  • Knot theory
  • Milnor fiber
  • Singularities
  • Vanishing cycles

ASJC Scopus subject areas

  • Mathematics(all)

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    Cappell, S. E., Maxim, L., Schürmann Jörg, J., & Shaneson, J. L. (2010). Characteristic classes of complex hypersurfaces. Advances in Mathematics, 225(5), 2616-2647. https://doi.org/10.1016/j.aim.2010.05.007