Hodge genera of algebraic varieties, II

Abstract

We study the behavior of Hodge-genera under algebraic maps. We prove that the motivic χ c y -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 χ 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 χ 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.A.

Euler characteristics of algebraic varieties

“…[1,2,9]). These are self-dual with respect to Verdier duality (and become important in the context of perverse sheaves and mixed Hodge modules, as in our forthcoming papers [3,4]). The normalization axiom for I CV (in the conventions of [1]) yields that I CV | V = Q V [dim(V )], with dim(V ) := dim R (V )/2 (the complex dimension in the complex algebraic/analytic context).…”

“…In this paper, we only make use of equation (3.9), and this could be proven directly by working in F V (Y ) by following the same arguments as above. However, the formula of equation (3.8) is particularly important since in the complex algebraic context it extends to the framework of Grothendieck groups of algebraic mixed Hodge modules that will be used in our forthcoming paper [4]. Of course, the technical condition used in proving formula (3.8) is not generally satisfied, but it holds under the assumption of trivial monodromy along all strata V ∈ V (e.g., if all strata V are simply connected).…”

“…This note is a first step in an ongoing project that deals with the study of genera of complex algebraic (respectively, analytic) varieties. In forthcoming papers [3,4] we will discuss the behavior of Hodge-theoretic genera under proper morphisms and provide explicit formulae for the pushforward of various characteristic classes. The functorial approach and the language of Grothendieck groups of constructible sheaves used in this paper allow a simple translation of the underlying ideas to the forthcoming papers, where Grothendieck groups of Saito's algebraic mixed Hodge modules will be employed.…”

Hodge genera of algebraic varieties I

“…In a future paper, we will consider the behavior under proper algebraic maps of χ y -genera that are defined by using the Hodge-Deligne numbers of (compactly supported) cohomology groups of a possibly singular algebraic variety and deal with nontrivial monodromy considerations (cf. [10]). We point out that preliminary results on the Euler characteristics χ −1 and I χ −1 , and on the homology MacPherson-Chern classes [25] T −1 = c * ⊗ Q and I T −1 = I c * ⊗ Q of complex algebraic (respectively, compact complex analytic) varieties, have been already obtained by the authors in [11].…”

“…Contributions of nontrivial monodromy to such formulae will be the subject of further studies; e.g., see our forthcoming paper[10]; see also[2,3] for some results on monodromy contributions for signatures and related characteristic classes in topological settings.…”