Hodge genera of algebraic varieties, II

Sylvain Cappell, Anatoly Libgober, Laurentiu G. Maxim, Julius L. Shaneson

Research output: Contribution to journalArticle

Abstract

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.

Original languageEnglish (US)
Pages (from-to)925-972
Number of pages48
JournalMathematische Annalen
Volume345
Issue number4
DOIs
StatePublished - Sep 2009

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Algebraic Variety
Genus
Michael Francis Atiyah
Characteristic Classes
Complex Manifolds
Monodromy
Morphism
Morphisms
Compact Manifold
Multiplicative
Signature
Express
Deviation
Singularity
Module
Curve
Invariant
Term

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Cappell, S., Libgober, A., Maxim, L. G., & Shaneson, J. L. (2009). Hodge genera of algebraic varieties, II. 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.

In: Mathematische Annalen, Vol. 345, No. 4, 09.2009, p. 925-972.

Research output: Contribution to journalArticle

Cappell, S, Libgober, A, Maxim, LG & Shaneson, JL 2009, 'Hodge genera of algebraic varieties, II', Mathematische Annalen, vol. 345, no. 4, pp. 925-972. https://doi.org/10.1007/s00208-009-0389-6
Cappell S, Libgober A, Maxim LG, Shaneson JL. Hodge genera of algebraic varieties, II. Mathematische Annalen. 2009 Sep;345(4):925-972. https://doi.org/10.1007/s00208-009-0389-6
Cappell, Sylvain ; Libgober, Anatoly ; Maxim, Laurentiu G. ; Shaneson, Julius L. / Hodge genera of algebraic varieties, II. In: Mathematische Annalen. 2009 ; Vol. 345, No. 4. pp. 925-972.
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