arXiv subjects

Shoji Yokura

Publications and source records attributed to Shoji Yokura.

At least 37 records · Page 2Linked to original sources

arXiv2017

Oriented bivariant theory II- Algebraic cobordism of $S$-schemes -

This is a sequel to our previous paper of oriented bivariant theory [14]. In 2001 M. Levine and F. Morel constructed algebraic cobordism $Ω_*(X)$ for schemes $X$ over a field $k$ in an abstract way and later M. Levine and R. Pandhairpande reconstructed it more geometrically. In this paper in a similar manner we construct an algebraic cobordism $Ω^*(X \xrightarrow {π_X} S)$ for a scheme $X$ over a fixed scheme $S$ in such a way that if the target scheme $S$ is the point $pt = \operatorname{Spec} k$, then $Ω^{-i}(X \xrightarrow {π_X} pt)$ is isomorphic to Levine-Morel's algebraic cobordism $Ω_i(X)$

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arXiv2017

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".

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arXiv2017

Hirzebruch $χ_y$-genera modulo $8$ of fiber bundles for odd integers $y$

I. Hambleton, A. Korzeniewski and A. Ranicki proved that the signature of a fibre bundle of closed, connected, compatibly oriented PL manifolds is always multiplicative modulo 4. In this paper, we consider the Hirzebruch $χ_y$-genera for odd integers $y$ for a smooth fiber bundle such that the base, fibre, and total space are compact complex algebraic manifolds (in the complex analytic topology, not in the Zariski topology). We show that the Hirzebruch $χ_y$-genera of such a fibre bundle are always multiplicative modulo 4. We also investigate multiplicativity modulo 8 and show that if $y$ is congruent to 3 modulo 4, then the $χ_y$-genera are multiplicative modulo 8. We also show that when $y$ is congruent to 1 modulo 4, the Hirzebruch $χ_y$-genera of such a fiber bundle are multiplicative modulo 8 if and only if the signature is multiplicative modulo 8, and that the non-multiplicativity modulo 8, in this case, is identified with an Arf-Kervaire invariant.

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arXiv2016

Motivic and derived motivic Hirzebruch classes

In this paper we give a formula for the Hirzebruch $χ_y$-genus $χ_y(X)$ and similarly for the motivic Hirzebruch class $T_{y*}(X)$ for possibly singular varieties $X$, using the Vandermonde matrix. Motivated by the notion of secondary Euler characteristic and higher Euler characteristic, we consider a similar notion for the motivic Hirzebruch class, which we call a \emph{derived motivic Hirzebruch class}

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arXiv2016

Hirzebruch $χ_y$-genera of complex algebraic fiber bundles -- the multiplicativity of the signature modulo $4$ --

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 $χ_y$-genus $χ_y(E) - χ_y(F)χ_y(B)$. 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 $σ(E) -σ(F)σ(B)$ is always divisible by $4$. In the case of $\operatorname{dim}_{\mathbb C}E \leqq 4$ the $χ_y$-genus difference $χ_y(E) - χ_y(F)χ_y(B)$ can be concretely described only in terms of the signature difference $σ(E) -σ(F) σ(B)$ and/or the Todd genus difference $τ(E) - τ(F)τ(B)$. Using this we can obtain that in order for $χ_y$ to be multiplicative $χ_y(E) = χ_y(F)\cdot χ_y(B)$ for any such fiber bundle $y$ has to be $-1$, namely only the Euler-Poincaré characteristic is multiplicative for any such fiber bundle.

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arXiv2013

Motivic Bivariant Characteristic Classes

Let K_0(V/X) be the relative Grothendieck group of varieties over X in obj(V), with V the category of (quasi-projective) algebraic (resp. compact complex analytic) varieties over a base field k. Then we constructed the motivic Hirzebruch class transformation in the algebraic context for k of characteristic zero and in the compact complex analytic context. It unifies the well-known three characteristic class transformations of singular varieties: MacPherson's Chern class, Baum-Fulton-MacPherson's Todd class and the L-class of Goresky-MacPherson and Cappell-Shaneson. In this paper we construct a bivariant relative Grothendieck group K_0(V/-) and in the algebraic context (in any characteristic) two Grothendieck transformations mC_y resp. T_y defined on K_0(V/-). Evaluating at y=0, we get a motivic lift T_0 of Fulton-MacPherson's bivariant Riemann-Roch transformation. The associated covariant transformations agree for k of characteristic zero with our motivic Chern- and Hirzebruch class transformations defined on K_0(V/X). Finally, evaluating at y=-1, we get for k of characteristic zero, a motivic lift T_{-1} of Ernström-Yokura's bivariant Chern class transformation.

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arXiv2013

C-symplectic poset structure on a simply connected space

For a field $\K$ of characteristic zero, we introduce a cohomologically symplectic poset structure ${\mathcal P}_{\K}(X)$ on a simply connected space $X$ from the viewpoint of $\K$-homotopy theory. It is given by the poset of inclusions of subgroups preserving c-symplectic structures in the group ${\mathcal E}(X_{\K})$ of $\K$-homotopy classes of $\K$-homotopy self-equivalences of $X$, which is defined by the $\K$-Sullivan model of $X$. We observe that the height of the Hasse diagram of ${\mathcal P}_{\K}(X)$ added by 1, denoted by c-s-${\rm depth}_{\K}(X)$, is finite and often depends on the field $\K$. In this paper, we will give some examples of ${\mathcal P}_{\K}(X)$.

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arXiv2013

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.

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arXiv2012

Characteristic classes of symmetric products of complex quasi-projective varieties

We prove generating series formulae for suitable twisted characteristic classes of symmetric products of a singular complex quasi-projective variety. More concretely, we study homology Hirzebruch classes for motivic coefficients, as well as for complexes of mixed Hodge modules. As a special case, we obtain a generating series formula for the (intersection) homology Hirzebruch classes of symmetric products. In some cases, the latter yields a similar formula for twisted homology L-classes generalizing results of Hirzebruch-Zagier and Moonen. Our methods also apply to the study of Todd classes of (complexes of) coherent sheaves, as well as Chern classes of (complexes of) constructible sheaves, generalizing to arbitrary coefficients results of Moonen and resp. Ohmoto.

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arXiv2012

Characteristic classes of Hilbert schemes of points via symmetric products

We obtain a formula for the generating series of (the push-forward under the Hilbert-Chow morphism of) the Hirzebruch homology characteristic classes of the Hilbert schemes of points for a smooth quasi-projective variety of arbitrary pure dimension. This result is based on a geometric construction of a motivic exponentiation generalizing the notion of motivic power structure, as well as on a formula for the generating series of the Hirzebruch homology characteristic classes of symmetric products. We apply the same methods for the calculation of generating series formulae for the Hirzebruch classes of the push-forwards of "virtual motives" of Hilbert schemes of a threefold. As corollaries, we obtain counterparts for the MacPherson (and Aluffi) Chern classes of Hilbert schemes of a smooth quasi-projective variety (resp. for threefolds). For a projective Calabi-Yau threefold, the latter yields a Chern class version of the dimension zero MNOP conjecture.

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arXiv2011

Grothendieck groups and a categorification of additive invariants

A topologically-invariant and additive homology class is mostly not a natural transformation as it is. In this paper we discuss turning such a homology class into a natural transformation; i.e., a "categorification" of it. In a general categorical set-up we introduce a generalized relative Grothendieck group from a cospan of functors of categories and also consider a categorification of additive invariants on objects. As an example, we obtain a general theory of characteristic homology classes of singular varieties.

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arXiv2011

Motivic bivariant characteristic classes

The relative Grothendieck group $K_0(\m V/X)$ is the free abelian group generated by the isomorphism classes of complex algebraic varieties over $X$ modulo the "scissor relation". The motivic Hirzebruch class ${T_y}_*: K_0(\m V /X) \to H_*^{BM}(X) \otimes \bQ[y]$ is a unique natural transformation satisfying that for a nonsingular variety $X$ the value ${T_y}_*([X \xrightarrow {\op {id}_X} X])$ of the isomorphism class of the identity $X \xrightarrow {id_X} X$ is the Poincaré dual of the Hirzebruch cohomology class of the tangent bundle $TX$. It "unifies" the well-known three characteristic classes of singular varieties: MacPherson's Chern class, Baum-Fulton-MacPherson's Todd class (or Riemann-Roch) and Goresky-MacPherson's L-class or Cappell-Shaneson's L-class. In this paper we construct a bivariant relative Grothendieck group $\bK_0(\m V/X \to Y)$ so that it equals the original relative Grothendieck group $K_0(\m V/X)$ when $Y$ is a point. We also construct a unique Grothendieck transformation $T_y: \bK_0(\m V/X \to Y) \to \bH(X \to Y) \otimes \bQ[y]$ satisfying a certain normalization condition for a smooth morphism so that it equals the motivic Hirzebruch class ${T_y}_*: K_0(\m V /X) \to H_*^{BM}(X) \otimes \bQ[y]$ when $Y$ is a point. When $y =0$, $T_0: \bK_0(\m V/X \to Y) \to \bH(X \to Y) \otimes \bQ$ is a "motivic" lift of Fulton-MacPherson's bivariant Riemann-Roch $\ga_{td}^{\op {FM}}:\bK_{alg}(X \to Y) \to \bH(X \to Y) \otimes \bQ$.

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arXiv2009

Motivic characteristic classes

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.

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arXiv2009

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.

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arXiv2008

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.

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arXiv2006

Characteristic 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 introduce characteristic classes of proalgebraic varieties, using Grothendieck transformations of Fulton--MacPherson's Bivariant Theory, modeled on the construction of MacPherson's Chern class transformation of proalgebraic varieties. We show that a proalgebraic version of the Euler--Poincaré characteristic with values in the Grothendieck ring is a generalization of the so-called motivic measure.

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arXiv2005

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.

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arXiv2005

Hirzebruch classes and motivic Chern classes for singular spaces

In this paper we study some new theories of characteristic homology classes for singular complex algebraic (or compactifiable analytic) spaces. We introduce a motivic Chern class transformation mC_{*}: K_{0}(var/X)-> G_{0}(X)[y], which generalizes the total λ-class of the cotangent bundle to singular spaces. Here K_{0}(var/X) is the relative Grothendieck group of complex algebraic varieties over X as introduced and studied by Looijenga and Bittner in relation to motivic integration, and G_{0}(X) is the Grothendieck group of coherent sheaves of O_{X}-modules. We define a natural transformation T_{y*}: K_{0}(var/X)-> H_{*}(X,Q)[y] commuting with proper pushdown, which generalizes the corresponding Hirzebruch characteristic. T_{y*} is a homology class version of the motivic measure corresponding to suitable specialization of the well known Hodge polynomial. This transformation unifies the Chern class transformation of MacPherson and Schwartz (for y=-1), the Todd class transformation of the singular Riemann-Roch theorem of Baum-Fulton-MacPherson (for y=0) and the L-class transformation of Cappell-Shaneson (for y=1). In the simplest case of a normal Gorenstein variety with ``canonical singularities'' we also explain a relation among the ``stringy version'' of our characteristic classses, the elliptic class of Borisov-Libgober and the stringy Chern classes of Aluffi and De Fernex-Lupercio-Nevins-Uribe. Moreover, all our results can be extended to varieties over a base field k of characteristic 0.

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