### 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) |
---|---|

Pages (from-to) | 2616-2647 |

Number of pages | 32 |

Journal | Advances in Mathematics |

Volume | 225 |

Issue number | 5 |

DOIs | |

State | Published - Dec 2010 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Advances in Mathematics*,

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

**Characteristic classes of complex hypersurfaces.** / Cappell, Sylvain; Maxim, Laurentiu; Schürmann Jörg, J.; Shaneson, Julius L.

Research output: Contribution to journal › Article

*Advances in Mathematics*, vol. 225, no. 5, pp. 2616-2647. https://doi.org/10.1016/j.aim.2010.05.007

}

TY - JOUR

T1 - Characteristic classes of complex hypersurfaces

AU - Cappell, Sylvain

AU - Maxim, Laurentiu

AU - Schürmann Jörg, J.

AU - Shaneson, Julius L.

PY - 2010/12

Y1 - 2010/12

N2 - 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.

AB - 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.

KW - Characteristic classes

KW - Hodge theory

KW - Hypersurfaces

KW - Intersection homology

KW - Knot theory

KW - Milnor fiber

KW - Singularities

KW - Vanishing cycles

UR - http://www.scopus.com/inward/record.url?scp=77956186096&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77956186096&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2010.05.007

DO - 10.1016/j.aim.2010.05.007

M3 - Article

VL - 225

SP - 2616

EP - 2647

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 5

ER -