Abstract. The aim of this note is to study the behavior of intersection homology Euler characteristic under morphisms of algebraic varieties. The main result is a direct application of the BBDG decomposition theorem. Similar formulae for Hodge-theoretic invariants of algebraic varieties were announced by the first and third authors in [4, 11]. 1.

Abstract. In this paper, for a possibly singular complex variety X, generating functions of total orbifold Chern homology classes of the symmetric products S n X are given. Those are very natural “class versions” of known generating function formulae of (generalized) orbifold Euler characteristics of S n X. The classes work covariantly for proper morphisms. We state the result more generally. Let G be a finite group and Gn the wreath product G ∼ Sn. For a G-variety X and a group A, we give a“Dey-Wohlfahrt type formula ” for equivariant Chern-Schwartz-MacPherson classes associated to Gn-representations of A (Theorem 1.1 and 1.2). In particular, if X is a point, this recovers a known exponential formula for counting numbers |Hom(A,Gn)|. 1.

This is a survey on a universal bivariant theory MC S (X → Y), which is a prototype of a bivariant analogue of Levine–Morel’s algebraic cobordism, and its application to constructing a bivariant theory FΩ(X → Y) of cobordism groups. Before giving such a survey, we recall the genus such as signature, which is the main important invariant defined on the cobordism group, i.e, a ring homomorphism from the cobordism group to a commutative ring with a unit. We capture the Euler–Poincaré characteristic and genera as a drastic generalization of the very natural counting finites sets.

Abstract. The aim of this paper is to study the behavior of intersection homology genera and their associated characteristic classes under morphisms of projective algebraic varieties. We obtain formulae that relate (parametrized families of) global invariants of a projective variety X to such invariants of singularities of proper algebraic maps defined on X. Such formulae severely constrain, both topologically and analytically, the singularities of complex maps, even between smooth varieties. Similar results were announced by the first and third author in [10, 22].

Abstract. We study the behavior of Hodge-theoretic genera under morphisms of complex algebraic varieties. We prove that the additive χy-genus which arises in the motivic context satisfies the so-called “stratified multiplicative property”, which shows how to compute the invariant of the source of a proper surjective morphism from its values on various varieties that arise from the singularities of the map. By considering morphisms to a curve, we obtain a Hodge-theoretic analogue of the Riemann-Hurwitz formula. We also consider the contribution of monodromy to the χy-genus of a smooth projective family, and prove an Atiyah-Meyer formula for twisted χy-genera. This formula measures the deviation from multiplicativity of the χy-genus, and expresses the correction terms as higher-genera associated to cohomology classes of the quotient of the total period domain by the action of the monodromy group. By making use of Saito’s theory of mixed Hodge modules, we also obtain characteristic class formulae of Atiyah-Meyer type. In the last section we use intersection homology for the study of twisted Hodge-theoretic genera in the singular setting: we extend some results of [10] regarding the stratified multiplicative property of the Iχy-genus, and conjecture a Atiyah-Meyer type formula for twisted Iχy-genera. Contents

ABSTRACT. In 1981 W. Fulton and R. MacPherson introduced the notion of bivariant theory (BT), which is a sophisticated unification of covariant theories and contravariant theories. This is for the study of singular spaces. In 2001 M. Levine and F. Morel introduced the notion of algebraic cobordism, which is a universal oriented Borel–Moore functor with products (OBMF) of geometric type, in an attempt to understand better V. Voevodsky’s (higher) algebraic cobordism. In this paper we introduce a notion of oriented bivariant theory (OBT), a special case of which is nothing but the oriented Borel–Moore functor with products. The present paper is a first one of the series to try to understand Levine–Morel’s algebraic cobordism from a bivariant-theoretical viewpoint, and its first step is to introduce OBT as a unification of BT and OBMF. 1.

ABSTRACT. Motivic characteristic classes of possibly singular algebraic varieties are homology class versions of motivic characteristics, not classes in the so-called motivic (co) homology. This paper is a survey on them with more emphasis on capturing infinitude finitely and on the motivic nature, in other words, the scissor relation or additivity. 1.

ABSTRACT. The Milnor class is a generalization of the Milnor number, defined as the difference (up to sign) of Chern–Schwartz–MacPherson’s class and Fulton–Johnson’s canonical Chern class of a local complete intersection variety in a smooth variety. In this paper we introduce a “motivic ” Grothendieck group K Prop ℓ.c.i (V/X h − → S) and natural transformations from this Grothendieck group to the homology theory. We capture the Milnor class, more generally Hirzebruch–Milnor class, as a special value of a distinguished element under these natural transformations. We also show a Verdier-type Riemann–Roch formula for our motivic Hirzebruch-Milnor class. We use Fulton–MacPherson’s bivariant theory and the motivic Hirzebruch class. 1.

ABSTRACT. The Milnor class is a generalization of the Milnor number, defined as the difference (up to sign) of Chern–Schwartz–MacPherson’s class and Fulton–Johnson’s canonical Chern class of a local complete intersection variety in a smooth variety. In this paper we introduce a “motivic ” Grothendieck group K Prop ℓ.c.i (V/X h − → S) and natural transformations from this Grothendieck group to the homology theory. We capture the Milnor class, more generally Hirzebruch–Milnor class, as a special value of a distinguished element under these natural transformations. We also show a Verdier-type Riemann–Roch formula for our motivic Hirzebruch-Milnor class. We use Fulton–MacPherson’s bivariant theory and the motivic Hirzebruch class. 1.

Abstract. In this note we survey Hodge-theoretic formulae of Atiyah-Meyer type for genera and characteristic classes of complex algebraic varieties, and derive some new and interesting applications. We also present various extensions to the singular setting of the Chern-Hirzebruch-Serre signature formula. 1.