Hirzebruch χy\chi_y-genera of complex algebraic fiber bundles -- the multiplicativity of the signature modulo 44 --

Let $E$ be a fiber $F$ bundle over a base $B$ such that $E, F$ and $B$ are smooth compact complex algebraic varieties. In this paper we give explicit formulae for the difference of the Hirzebruch $\chi_y$-genus $\chi_y(E) - \chi_y(F)\chi_y(B)$. As a byproduct of the formulae we obtain that the signature of such a fiber bundle is multiplicative mod $4$, i.e. the signature difference $\sigma(E) -\sigma(F)\sigma(B)$ is always divisible by $4$. In the case of $\operatorname{dim}_{\mathbb C}E \leqq 4$ the $\chi_y$-genus difference $\chi_y(E) - \chi_y(F)\chi_y(B)$ can be concretely described only in terms of the signature difference $\sigma(E) -\sigma(F) \sigma(B)$ and/or the Todd genus difference $\tau(E) - \tau(F)\tau(B)$. Using this we can obtain that in order for $\chi_y$ to be multiplicative $\chi_y(E) = \chi_y(F)\cdot \chi_y(B)$ for any such fiber bundle $y$ has to be $-1$, namely only the Euler-Poincar\'e characteristic is multiplicative for any such fiber bundle.Comment: 12 pages, any comments and suggestions are welcome; typos are corrected and some revisions are made; more typos are correcte

Chern classes of proalgebraic varieties and motivic measures

Michael Gromov has recently initiated what he calls ``symbolic algebraic geometry", in which objects are proalgebraic varieties: a proalgebraic variety is by definition the projective limit of a projective system of algebraic varieties. In this paper we construct Chern--Schwartz--MacPherson classes of proalgebraic varieties, by introducing the notion of ``proconstructible functions " and "$\chi$-stable proconstructible functions" and using the Fulton-MacPherson's Bivariant Theory. As a "motivic" version of a $\chi$-stable proconstructible function, \Ga-stable constructible functions are introduced. This construction naturally generalizes the so-called motivic measure and motivic integration. For the Nash arc space \Cal L(X) of an algebraic variety $X$, the proconstructible set is equivalent to the so-called cylinder set or constructible set in the arc space.Comment: 37 page

A remark on Yoneda's Lemma

Yoneda'e Lemma is about the canonical isomorphism of all the natural transformations from a given representable covariant (contravariant, reps.) functor (from a locally small category to the category of sets) to a covariant (contravariant, reps.) functor. In this note we point out that given any representable functor and any functor we have the canonical natural transformation from the given representable functor to the "subset" functor of the given functor, "collecting all the natural transformations".Comment: 5 page

Motivic Milnor classes

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 $K^{\mathcal Prop}_{\ell.c.i}(\mathcal V/X \to S)$ 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.Comment: 18 pages, some revision was made with more reference

Oriented bivariant theories, I

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.Comment: 25 pages, to appear in International J. Mathematic

Enriched categories of correspondences and characteristic classes of singular varieties

For the category $\mathscr V$ of complex algebraic varieties, the Grothendieck group of the commutative monoid of the isomorphism classes of correspondences $X \xleftarrow f M \xrightarrow g Y$ with proper morphism $f$ and smooth morphism $g$ (such a correspondence is called \emph{a proper-smooth correspondence}) gives rise to an enriched category $\mathscr Corr(\mathscr V)^+_{pro-sm}$ of proper-smooth correspondences. In this paper we extend the well-known theories of characteristic classes of singular varieties such as Baum-Fulton-MacPherson's Riemann-Roch (abbr. BFM-RR) and MacPherson's Chern class transformation and so on to this enriched category $\mathscr Corr(\mathscr V)^+_{pro-sm}$. In order to deal with local complete intersection (abbr. $\ell.c.i.$) morphism instead of smooth morphism, in a similar manner we consider an enriched category $\mathscr Zigzag(\mathscr V)^+_{pro-\ell.c.i.}$ of \emph{proper-$\ell.c.i.$} zigzags and extend BFM-RR to this enriched category $\mathscr Zigzag(\mathscr V)^+_{pro-\ell.c.i.}$. We also consider an enriched category $\mathscr M_{*,*}(\mathscr V)^+_{\otimes}$ of proper-smooth correspondences $(X \xleftarrow f M \xrightarrow g Y; E)$ equipped with complex vector bundle $E$ on $M$ (such a correspondence is called \emph{a cobordism bicycle of vector bundle}) and we extend BFM-RR to this enriched category $\mathscr M_{*,*}(\mathscr V)^+_{\otimes}$ as well.Comment: 35 pages, comments are welcome; to appear in Fundamenta Mathematica

Poset-stratified space structures of homotopy sets

A poset-stratified space is a pair $(S, S \xrightarrow \pi P)$ of a topological space $S$ and a continuous map $\pi: S \to P$ with a poset $P$ considered as a topological space with its associated Alexandroff topology. In this paper we show that one can impose such a poset-stratified space structure on the homotopy set $[X, Y]$ of homotopy classes of continuous maps by considering a canonical but non-trivial order (preorder) on it, namely we can capture the homotopy set $[X, Y]$ as an object of the category of poset-stratified spaces. The order we consider is related to the notion of \emph{dependence of maps} (by Karol Borsuk). Furthermore via homology and cohomology the homotopy set $[X,Y]$ can have other poset-stratified space structures. In the cohomology case, we get some results which are equivalent to the notion of \emph{dependence of cohomology classes} (by Ren\'e Thom) and we can show that the set of isomorphism classes of complex vector bundles can be captured as a poset-stratified space via the poset of the subrings consisting of all the characteristic classes. We also show that some invariants such as Gottlieb groups and Lusternik--Schnirelmann category of a map give poset-stratified space structures to the homotopy set $[X,Y]$Comment: 23 pages, comments are welcome, to appear in Homology, Homotopy and Application

Naive motivic Donaldson-Thomas type Hirzebruch classes and some problems

Donaldson-Thomas invariant is expressed as the weighted Euler characteristic of the so-called Behrend (constructible) function. In \cite{Behrend} Behrend introduced a DT-type invariant for a morphism. Motivated by this invariant, we extend the motivic Hirzebruch class to naive Donaldson-Thomas type analogues. We also discuss a categorification of the DT-type invariant for a morphism from a bivariant-theoretic viewpoint, and we finally pose some related questions for further investigations

A survey of characteristic classes of singular spaces

A theory of characteristic classes of vector bundles and smooth manifolds plays an important role in the theory of smooth manifolds. An investigation of reasonable notions of characteristic classes of singular spaces started since a systematic study of singular spaces such as singular algebraic varieties. We make a quick survey of characteristic classes of singular varieties, mainly focusing on the functorial aspects of some important ones such as the singular versions of the Chern class, the Todd class and the Thom--Hirzebruch's L-class. Then we explain our recent "motivic" characteristic classes, which in a sense unify these three different theories of characteristic classes. We also discuss bivariant versions of them and characteristic classes of proalgebraic varieties, which are related to the motivic measures/integrations. Finally we explain some recent work on "stringy" versions of these theories, together with some references for "equivariant" counterparts.Comment: 58 page

Decomposition spaces and poset-stratified spaces

In 1920s R. L. Moore introduced \emph{upper semicontinuous} and \emph{lower semicontinuous} decompositions in studying decomposition spaces. Upper semicontinuous decompositions were studied very well by himself and later by R.H. Bing in 1950s. In this paper we consider lower semicontinuous decompositions $\mathcal D$ of a topological space $X$ such that the decomposition spaces $X/\mathcal D$ are Alexandroff spaces. If the associated proset (preordered set) of the decomposition space $X/\mathcal D$ is a poset, then the decomposition map $\pi:X \to X/\mathcal D$ is \emph{a continuous map from the topological space $X$ to the poset $X/\mathcal D$ with the associated Alexandroff topology}, which is nowadays called \emph{a poset-stratified space}. As an application, we capture the face poset of a real hyperplane arrangement $\mathcal A$ of $\mathbb R^n$ as the associated poset of the decomposition space $\mathbb R^n/\mathcal D(\mathcal A)$ of the decomposition $\mathcal D(\mathcal A)$ determined by the arrangement $\mathcal A$. We also show that for any locally small category $\mathcal C$ the set $hom_{\mathcal C}(X,Y)$ of morphisms from $X$ to $Y$ can be considered as a poset-stratified space, and that for any objects $S, T$ (where $S$ plays as a source object and $T$ as a target object) there are a covariant functor $\frak {st}^S_*: \mathcal C \to \mathcal Strat$ and a contravariant functor $\frak {st}^*_T$ $\frak {st}^*_T: \mathcal C \to \mathcal Strat$ from $\mathcal C$ to the category $\mathcal Strat$ of poset-stratified spaces. We also make a remark about Yoneda's Lemmas as to poset-stratified space structures of $hom_{\mathcal C}(X,Y)$.Comment: commnets are welcom