A singular Riemann-Roch for Hirzebruch characteristics

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.

show abstract

“…Let π * (ch * (H p k )) ∪T * y (T B) (50) corresponding to the polarized variation of Hodge structures [R k f * R E , F ] as follows:…”

Section: Definition 2 Let α ∈ H * (D/ ) the Higher Genus χ [α]mentioning

confidence: 99%

Hodge genera of algebraic varieties, II

et al. 2009

View full text Add to dashboard Cite

show abstract

“…Moreover, G 0 (X) is the Grothendieck group of coherent sheaves of O X -modules. Here td (1+y) is a generalization given by Yokura [84] of the Todd class transformation td * used in the singular Riemann-Roch theorem of Baum-Fulton-MacPherson [9,33] (for Borel-Moore homology) or Fulton [31] (for Chow groups).…”

Section: Introductionmentioning

confidence: 99%

Hirzebruch Classes and Motivic Chern Classes for Singular Spaces

2010

Self Cite

View full text Add to dashboard Cite

In this paper we study some new theories of characteristic homology classes of singular complex algebraic (or compactifiable analytic) spaces.We introduce a motivic Chern class transformation mCy: K 0 (var/X) → G 0 (X) ⊗ Z[y], which generalizes the total λ-class λy(T * X) 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. A first construction of mCy is based on resolution of singularities and a suitable "blow-up" relation, following the work of Du Bois, Guillén, Navarro Aznar, Looijenga and Bittner. A second more functorial construction of mCy is based on some results from the theory of algebraic mixed Hodge modules due to M. Saito.We define a natural transformation Ty * : K 0 (var/X) → H * (X) ⊗ Q[y] commuting with proper pushdown, which generalizes the corresponding Hirzebruch characteristic. Ty * is a homology class version of the motivic measure corresponding to a 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 1 2 J.-P. Brasselet, J. Schürmann & S. Yokura transformation in the singular Riemann-Roch theorem of Baum-Fulton-MacPherson (for y = 0) and the L-class transformation of Cappell-Shaneson (for y = 1).We also explain the relation among the "stringy version" of our characteristic classes, the elliptic class of Borisov-Libgober and the stringy Chern classes of Aluffi and De Fernex-Lupercio-Nevins-Uribe.All our results can be extended to varieties over a base field k of characteristic 0.We introduce characteristic homology class transformations mC y and T y * on K 0 (var/X) related by the following commutative diagram:by taking fiberwise the (topological) Euler characteristic with compact support. Note that the homomorphisms e and mC 0 are surjective. Remark 0.1. Note that in the algebraic context the Chern-Schwartz-MacPherson transformation c * of [44] is defined only on spaces X embeddable into a smooth space (e.g. for quasi-projective varieties). But using the technique of "Chow envelopes" as in [31, Sec. 18.3], this transformation can uniquely be extended to all (reduced) separated schemes of finite type over spec(k). 4 J.-P. Brasselet, J. Schürmann & S. YokuraFor X a compact complex algebraic variety, we can also construct the following commutative diagram of natural transformations:4) with L * the homology L-class transformation of Cappell-Shaneson [21] (as reformulated by Yokura [81]). Here Ω(X) is the Abelian group of cobordism classes of self-dual constructible complexes. To make Ω(·) covariant functorial (correcting [81]), we use the definition of a "cobordism" given by Youssin [89] in the more general context of triangulated categories with duality. Note that the homology L-class transformation of Cappell-Shaneson [21] is defined o...

show abstract

“…This theory of relative Grothendieck groups has the same calculus as constructible functions, in particular, one also has a corresponding simple bivariant theory sK 0 . Note that the Hirzebruch class transformation T y * unifies, in a sense, the Chern, Todd and L-class transformations of singular spaces as asked in MacPherson's survey article [25] (also see [38]). …”

Section: The Existence and Uniqueness Of Grothendieck Transformationsmentioning

confidence: 98%

On Grothendieck transformations in Fulton–MacPherson’s bivariant theory

Brasselet

Schürmann

Yokura

2007

Journal of Pure and Applied Algebra

Self Cite

View full text Add to dashboard Cite

W. Fulton and R. MacPherson have introduced a notion unifying both covariant and contravariant theories, which they called a Bivariant Theory. A transformation between two bivariant theories is called a Grothendieck transformation. The Grothendieck transformation induces natural transformations for covariant theories and contravariant theories. In this paper we show some general uniqueness and existence theorems on Grothendieck transformations associated to given natural transformations of covariant theories. Our guiding or typical model is MacPherson's Chern class transformation c

show abstract

A singular Riemann-Roch for Hirzebruch characteristics

Cited by 17 publications

References 15 publications

Hodge genera of algebraic varieties, II

Hodge genera of algebraic varieties, II

Hirzebruch Classes and Motivic Chern Classes for Singular Spaces

On Grothendieck transformations in Fulton–MacPherson’s bivariant theory

Contact Info

Product

Resources

About