Characteristic classes of proalgebraic varieties and motivic measures

Yokura, Shoji

doi:10.2140/agt.2012.12.601

Cited by 2 publications

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“…The paper [2] strongly stimulated and spurred our work, which started with [81][82][83][84][85][86] of the third author. Some of these papers are partly motivated by the final remark of MacPherson's survey article:…”

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confidence: 99%

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Hirzebruch Classes and Motivic Chern Classes for Singular Spaces

2010

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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...

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confidence: 99%

“…By our work, one can now introduce similar characteristic classes by using a "relative motivic measure"μ X with values in M (var/X) [51,Sec. 4], and the same for "motivic integrals" (compare also with [86]). …”

Section: Example 33mentioning

confidence: 99%