### 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 language | English (US) |
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Pages (from-to) | 2616-2647 |

Number of pages | 32 |

Journal | Advances in Mathematics |

Volume | 225 |

Issue number | 5 |

DOIs | |

State | Published - 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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## Cite this

*Advances in Mathematics*,

*225*(5), 2616-2647. https://doi.org/10.1016/j.aim.2010.05.007