The existence of bivariant Chern classes was conjectured by W.Fulton and R.MacPherson and proved by J.P.Brasselet for cellular morphisms of analytic varieties. In this paper we show that restricted to morphisms whose target varieties are possible singular but (rational) homology manifolds, the bivariant Chern classes (with rational coefficients) are uniquely determined. In the final sections we construct a unique bivariant Chern class satisfying a suitable normalization condition. In fact, it will be a special case of a general construction of unique Grothendieck transformations, which in a sense gives a positive answer to a corresponding question posed by Fulton and MacPherson. This construction generalizes and unifies previous attemps, in particular the main result of the paper math.AG/0209299.
We define bivariant algebraic K-theory and bivariant derived Chow on the homotopy category of derived schemes over a smooth base. The orientation on the latter corresponds to virtual Gysin homomorphisms. We then provide a morphism between these two bivariant theories and compare the two orientations. This comparison then yields a homological and cohomological Grothendieck-Riemann-Roch formula for virtual classes.
Fulton and MacPherson introduced the notion of bivariant theories and Grothendieck transformations related to Riemann-Roch-theorems. But there are many situations, where such a bivariant theory or a corresponding Grothendieck transformation is only partially known: characteristic classes of singular spaces (e.g. Stiefel-Whitney or Chern classes), cohomology operations (e.g. singular Adams Riemann-Roch and Steenrod operations for Chow groups) or equivariant theories (e.g. Lefschetz Riemann-Roch). We introduce in this paper a simpler notion of partial (weak) bivariant theories and partial Grothe…
We produce a Grothendieck transformation from bivariant operational $K$-theory to Chow, with a Riemann-Roch formula that generalizes classical Grothendieck-Verdier-Riemann-Roch. We also produce Grothendieck transformations and Riemann-Roch formulas that generalize the classical Adams-Riemann-Roch and equivariant localization theorems. As applications, we exhibit a projective toric variety $X$ whose equivariant $K$-theory of vector bundles does not surject onto its ordinary $K$-theory, and describe the operational $K$-theory of spherical varieties in terms of fixed-point data. In an appendix, V…
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…
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 cl…
A procedure for constructing bivariant theories by means of Grothendieck duality is developed. This produces, in particular, a bivariant theory of Hochschild (co)homology on the category of schemes that are flat, separated and essentially of finite type over a fixed noetherian scheme S. The theory takes values in the category of symmetric graded modules over the graded-commutative ring \oplus_i H^i(S,O_S). In degree i, the cohomology and homology H^0(S,O_S)-modules thereby associated to such an x: X -> S, with Hochschild complex H_x, are Ext^i(H_x, H_x) and Ext^{-i}(H_x, x^!O_S). This lays the…
The purpose of this work is to study the notion of bivariant theory introduced by Fulton and MacPherson in the context of motivic stable homotopy theory, and more generally in the broader framework of Grothendieck six functors formalism. We introduce several kinds of bivariant theory associated with a suitable ring spectrum and we construct a system of orientations (called fundamental classes) for global complete intersection morphisms between arbitrary schemes. This fundamental classes satisfies all the expected properties from classical intersection theory and lead to Gysin morphisms, Rieman…
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 …
In this paper we prove a categorification of the Grothendieck-Riemann-Roch theorem. Our result implies in particular a Grothendieck-Riemann-Roch theorem for Toën and Vezzosi's secondary Chern character. As a main application, we establish a comparison between the Toën-Vezzosi Chern character and the classical Chern character, and show that the categorified Chern character recovers the classical de Rham realization.
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…
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 "$χ$-stable proconstructible functions" and using the Fulton-MacPherson's Bivariant Theory. As a "motivic" version of a $χ$-stable proconstructible function, $\Ga$-stable constructible functions are introduced…
We study the singular Bott-Chern classes introduced by Bismut, Gillet and Soule. Singular Bott-Chern classes are the main ingredient to define direct images for closed immersions in arithmetic K-theory. In this paper we give an axiomatic definition of a theory of singular Bott-Chern classes, study their properties, and classify all possible theories of this kind. We identify the theory defined by Bismut, Gillet and Soule as the only one that satisfies the additional condition of being homogeneous. We include a proof of the arithmetic Grothendieck-Riemann-Roch theorem for closed immersions that…
We show that the characteristic polynomial of a hyperplane arrangement can be recovered from the class in the Grothendieck group of varieties of the complement of the arrangement. This gives a quick proof of a theorem of Orlik and Solomon relating the characteristic polynomial with the ranks of the cohomology of the complement of the arrangement. We also show that the characteristic polynomial can be computed from the total Chern class of the complement of the arrangement; this has also been observed by Huh. In the case of free arrangements, we prove that this Chern class agrees with the Chern…
We construct a functor, from the category of schemes to the category of graded rings, that is an initial object for having a theory of Chern classes with an additive first Chern class. For any scheme $X$, the graded ring that our functor associates to $X$ is related to the associated graded ring of the $γ$-filtration on the Grothendieck ring of finite rank locally free sheaves on $X$ via a Grothendieck-Riemann-Roch type theorem.
Classical Chern-Weil theory proves existence of a ring homomorphism - the Chern character - from the Grothendieck ring of locally free finite rank A-modules to the algebraic de Rham cohomology of A. In this paper we prove existence of a ring homomorphism from the Grothendieck ring of locally free finite rank A-modules with an L-connection to the Lie-Rinehart cohomology of L, where L is a Lie-Rinehart algebra. As a corollary we get the classical Chern character.
We construct Chern classes of vector bundles in the universal precobordism theory of Annala--Yokura over an arbitrary Noetherian base ring of finite Krull dimension. As an immediate corollary of this, we show that the Grothendieck ring of vector bundles can be recovered from the universal precobordism ring, and that we can construct candidates for Chow rings satisfying an analogue of the classical Grothendieck--Riemann--Roch theorem. We also strengthen the weak projective bundle formula of Annala--Yokura to work for arbitrary projective bundles.
We study the operational bivariant theory associated to the covariant theory of Grothendieck groups of coherent sheaves, and prove that it has many geometric properties analogous to those of operational Chow theory. This operational K-theory agrees with Grothendieck groups of vector bundles on smooth varieties, admits a natural map from the Grothendieck group of perfect complexes on general varieties, satisfies descent for Chow envelopes, and is A^1-homotopy invariant. Furthermore, we show that the operational K-theory of a complete linear variety is dual to the Grothendieck group of coherent …
The moduli space of principally polarized abelian varieties $A_g$ of genus g is defined over the integers and admits a minimal compactification $A_g^*$, also defined over the integers. The Hodge bundle over $A_g$ has its Chern classes in the Chow ring of $A_g$ with rational coefficients. We show that over the prime field $F_p$, these Chern classes naturally lift to $A_g^*$ and do so in the best possible way: despite the highly singular nature of $A_g^*$ they are represented by algebraic cycles on $A_g^*\otimes F_p$ which define elements in its bivariant Chow ring. This is in contrast to the si…
We obtain a precise relation between the Chern-Schwartz-MacPherson class of a subvariety of projective space and the Euler characteristics of its general linear sections. In the case of a hypersurface, this leads to simple proofs of formulas of Dimca-Papadima and Huh for the degrees of the polar map of a homogeneous polynomial, extending these formula to any algebraically closed field of characteristic~0, and proving a conjecture of Dolgachev on 'homaloidal' polynomials in the same context. We generalize these formulas to subschemes of higher codimension in projective space. We also describe a…
In this paper, we introduce six axioms for relative Bott-Chern secondary characteristic classes and prove the uniqueness and existence theorem for them. Such a work provides us a natural way to understand and hence to prove the arithmetic Grothendieck-Riemann-Roch theorem.
We study the additivity of various geometric invariants involved in Reimann-Roch type formulas and defined via the trace map. To do so in a general context we prove that given any Grothendieck category A, the derived category D(A) has a compatible triangulation in the sense of [May, J.P. :The Additivity of Traces in Triangulated Categories, Advances in Mathematics 163, (2001), 34-73], but not resorting to model categories. The result is proved just using the structural properties inherent to D(A). In the second part of the paper we apply compatibility to prove additivity of traces firstly and …
In this article, we construct Chern classes in rational Deligne cohomology for coherent sheaves on a smooth complex compact manifold. We prove that these classes verify the functoriality property under pullbacks, the Whitney formula and the Grothendieck-Riemann-Roch theorem for projective morphisms between smooth complex compact manifolds.
In this note we prove a conjecture of Kashiwara, which states that the Euler class of a coherent analytic sheaf F on a complex manifold X is the product of the Chern character of F with the Todd class of X. As a corollary, we obtain a functorial proof of the Grothendieck-Riemann-Roch theorem in Hodge cohomology for complex manifolds.
We study the behavior of the Chern classes of graph hypersurfaces under the operation of deletion-contraction of an edge of the corresponding graph. We obtain an explicit formula when the edge satisfies two technical conditions, and prove that both these conditions hold when the edge is multiple in the graph. This leads to recursions for the Chern classes of graph hypersurfaces for graphs obtained by adding parallel edges to a given (regular) edge. Analogous results for the case of Grothendieck classes of graph hypersurfaces were obtained in previous work. Both Grothendieck classes and Chern c…
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 …
We define an `enriched' notion of Chow groups for algebraic varieties, agreeing with the conventional notion for complete varieties, but enjoying a functorial push-forward for arbitrary maps. This tool allows us to glue intersection-theoretic information across elements of a stratification of a variety; we illustrate this operation by giving a direct construction of Chern-Schwartz-MacPherson classes of singular varieties, providing a new proof of an old (and long since settled) conjecture of Deligne and Grothendieck.
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 cla…
In this paper we give an introduction to our recent work on characteristic classes of complex hypersurfaces based on some talks given at conferences in Strasbourg, Oberwolfach and Kagoshima. We explain the relation between nearby cycles for constructible functions or sheaves as well as for (relative) Grothendieck groups of algebraic varieties and mixed Hodge modules, and the specialization of characteristic classes of singular spaces like the Chern-, Todd-, Hirzebruch- and motivic Chern-classes. As an application we get a description of the differences between the corresponding virtual and fun…
In this paper we study some new theories of characteristic homology classes for singular complex algebraic varieties. First we introduce a natural transformation T_{y}: K_{0}(var/X) -> H_{*}(X,Q)[y] commuting with proper pushdown, which generalizes the corresponding Hirzebruch characteristic. 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. T_{y} is a homology class version of the motivic measure corresponding to a suitable specialization of the well known Hodge poly…
Let [X/G] be a smooth Deligne-Mumford quotient stack. In a previous paper the authors constructed a class of exotic products called inertial products on K(I[X/G]), the Grothendieck group of vector bundles on the inertia stack I[X/G]. In this paper we develop a theory of Chern classes and compatible power operations for inertial products. When G is diagonalizable these give rise to an augmented $λ$-ring structure on inertial K-theory. One well-known inertial product is the virtual product. Our results show that for toric Deligne-Mumford stacks there is a $λ$-ring structure on inertial K-theory.…
In this paper we express the class of the structure sheaves of the closures of Deligne--Lusztig varieties as explicit double Grothendieck polynomials in the first Chern classes of appropriate line bundles on the ambient flag variety. This is achieved by viewing such closures as degeneracy loci of morphisms of vector bundles.
We prove a `motivic' analogue of the Weyl character formula, computing the Euler characteristic of a line bundle on a generalized flag manifold $G/B$ multiplied either by a motivic Chern class of a Schubert cell, or a Segre analogue of it. The result, given in terms of Demazure-Lusztig (D-L) operators, recovers formulas found by Brubaker, Bump and Licata for the Iwahori-Whittaker functions of the principal series representation of a $p$-adic group. In particular, we obtain a new proof of the classical Casselman-Shalika formula for the spherical Whittaker function. The proofs are based on local…
CORRECTION. One of the main results in this paper contains a fatal error. We cannot conclude the existence of nontrivial vector bundles on X from the nontriviality of its K-group. The K-group that is computed here is the Grothendieck group of perfect complexes and not vector bundles. Since the varieties are not quasi-projective, existence of nontrivial perfect complexes says nothing about the existence of nontrivial vector bundles. We thank Sam Payne for drawing our attention to the error and Christian Haesemeyer for explanations about the K-theory. Abstract: Following Sam Payne's work, we stu…
In this paper we discuss some questions about geometry over the field with one element, motivated by the properties of algebraic varieties that arise in perturbative quantum field theory. We follow the approach to F1-geometry based on torified schemes. We first discuss some simple necessary conditions in terms of Euler characteristic and classes in the Grothendieck ring, then we give a blowup formula for torified varieties and we show that the wonderful compactifications of the graph configuration spaces, that arise in the computation of Feynman integrals in position space, admit an F1-structu…
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 Che…
