### Abstract

We study the behavior of Hodge-genera under algebraic maps. We prove that the motivic X_{y}^{c} -genus satisfies the "stratified multiplicative property", which shows how to compute the invariant of the source of a morphism from its values on varieties arising from the singularities of the map. By considering morphisms to a curve, we obtain a Hodge-theoretic version of the Riemann-Hurwitz formula. We also study the monodromy contributions to the X_{y}-genus of a family of compact complex manifolds, and prove an Atiyah-Meyer type formula in the algebraic and analytic contexts. This formula measures the deviation from multiplicativity of the X_{y}-genus, and expresses the correction terms as higher-genera associated to the period map; these higher-genera are Hodge-theoretic extensions of Novikov higher-signatures to analytic and algebraic settings. Characteristic class formulae of Atiyah-Meyer type are also obtained by making use of Saito's theory of mixed Hodge modules.

Original language | English (US) |
---|---|

Pages (from-to) | 925-972 |

Number of pages | 48 |

Journal | Mathematische Annalen |

Volume | 345 |

Issue number | 4 |

DOIs | |

State | Published - Sep 2009 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Mathematische Annalen*,

*345*(4), 925-972. https://doi.org/10.1007/s00208-009-0389-6

**Hodge genera of algebraic varieties, II.** / Cappell, Sylvain; Libgober, Anatoly; Maxim, Laurentiu G.; Shaneson, Julius L.

Research output: Contribution to journal › Article

*Mathematische Annalen*, vol. 345, no. 4, pp. 925-972. https://doi.org/10.1007/s00208-009-0389-6

}

TY - JOUR

T1 - Hodge genera of algebraic varieties, II

AU - Cappell, Sylvain

AU - Libgober, Anatoly

AU - Maxim, Laurentiu G.

AU - Shaneson, Julius L.

PY - 2009/9

Y1 - 2009/9

N2 - We study the behavior of Hodge-genera under algebraic maps. We prove that the motivic Xyc -genus satisfies the "stratified multiplicative property", which shows how to compute the invariant of the source of a morphism from its values on varieties arising from the singularities of the map. By considering morphisms to a curve, we obtain a Hodge-theoretic version of the Riemann-Hurwitz formula. We also study the monodromy contributions to the Xy-genus of a family of compact complex manifolds, and prove an Atiyah-Meyer type formula in the algebraic and analytic contexts. This formula measures the deviation from multiplicativity of the Xy-genus, and expresses the correction terms as higher-genera associated to the period map; these higher-genera are Hodge-theoretic extensions of Novikov higher-signatures to analytic and algebraic settings. Characteristic class formulae of Atiyah-Meyer type are also obtained by making use of Saito's theory of mixed Hodge modules.

AB - We study the behavior of Hodge-genera under algebraic maps. We prove that the motivic Xyc -genus satisfies the "stratified multiplicative property", which shows how to compute the invariant of the source of a morphism from its values on varieties arising from the singularities of the map. By considering morphisms to a curve, we obtain a Hodge-theoretic version of the Riemann-Hurwitz formula. We also study the monodromy contributions to the Xy-genus of a family of compact complex manifolds, and prove an Atiyah-Meyer type formula in the algebraic and analytic contexts. This formula measures the deviation from multiplicativity of the Xy-genus, and expresses the correction terms as higher-genera associated to the period map; these higher-genera are Hodge-theoretic extensions of Novikov higher-signatures to analytic and algebraic settings. Characteristic class formulae of Atiyah-Meyer type are also obtained by making use of Saito's theory of mixed Hodge modules.

UR - http://www.scopus.com/inward/record.url?scp=70350166409&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70350166409&partnerID=8YFLogxK

U2 - 10.1007/s00208-009-0389-6

DO - 10.1007/s00208-009-0389-6

M3 - Article

VL - 345

SP - 925

EP - 972

JO - Mathematische Annalen

JF - Mathematische Annalen

SN - 0025-5831

IS - 4

ER -