Characteristic classes of symmetric products of complex quasi-projective varieties
type: Item
instance of: scholarly article
title: Characteristic classes of symmetric products of complex quasi-projective varieties
zbMATH Open document ID: 1400.14024
author: Jörg Schürmann
author: Julius L. Shaneson
author: Shoji Yokura
author: Sylvain Cappell
author: Laurentiu Maxim
published in: Journal für die Reine und Angewandte Mathematik
publication date: 2017-07-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1008.4299
review text: Let \(X\) be a complex quasi-projective variety and let \(X^{(n)}\) denote its symmetric product. Let \(T_{y*}:K_0(var/X)\rightarrow H_{ev}^{BM}(X)\oplus\mathbb{Q}[y]\) be the un-normalized motivic Hirzebruch class transformation, where \(H_{ev}^{BM}(X)\) denotes the Borel-Moore homology in even degrees and \(K_0(var/X)\) the relative Grothendieck group of complex algebraic varieties over \(X\) (see [\textit{J.-P. Brasselet} et al., J. Topol. Anal. 2, No. 1, 1--55 (2010; Zbl 1190.14009)]). For any positive integer \(r\), let \(d^r:X\rightarrow X^{(r)}\) denote the composition of the diagonal embedding \(X\hookrightarrow X^r\) with the projection \(X^r\rightarrow X^{(r)}\), and let \(\Psi_r\) denote the \(r\)-th homological Adams operation, which on \(H_{2k}^{BM}(X^{(r)},\mathbb{Q})\) is defined by multiplication by \(1/r^k\), together with \(y\mapsto y^r\). The main result of this paper is the folloing generating series formula for the Hirzebruch classes of the symmetric powers \(\mathcal{M}^{(n)}\in D^b\text{MHM}(X^{(n)})\) of a fixed complex of mixed Hodge modules on the variety \(X\): \(\sum_{n\geq 0}T_{(-y)*}(\mathcal{M}^{(n)})\cdot t^n=\text{exp}\left(\sum_{r\geq 1}\Psi_r(d_*^rT_{(-y)*}(\mathcal{M}))\cdot\frac{t^r}{r}\right)\). If one lets \(\mathcal{M}\) be the constant Hodge sheaf \(\mathbb{Q}_X^H\), the above formula becomes the expression \(\sum_{n\geq 0}T_{(-y)*}(X^{(n)})\cdot t^n=\text{exp}\left(\sum_{r\geq 1}\Psi_r(d_*^rT_{(-y)*}(X))\cdot\frac{t^r}{r}\right)\) for the motivic Hirzebruch classes.
reviewed by: Fumio Hazama
Mathematics Subject Classification ID: 14C17
Mathematics Subject Classification ID: 14C25
zbMATH DE Number: 6744767
zbMATH Keywords: symmetric product
zbMATH Keywords: characteristic classes
zbMATH Keywords: Grothendieck group
MaRDI profile type: Publication
OpenAlex ID: W2963365901
arXiv ID: 1008.4299
cites work: POWER OPERATIONS IN K-THEORY
cites work: Riemann-Roch for singular varieties
cites work: Lefschetz-Riemann-Roch for singular varieties
cites work: Q4669276
cites work: HIRZEBRUCH CLASSES AND MOTIVIC CHERN CLASSES FOR SINGULAR SPACES
cites work: Characteristic classes of Hilbert schemes of points via symmetric products
cites work: Equivariant Characteristic Classes of Singular Complex Algebraic Varieties
cites work: Hodge genera of algebraic varieties I
cites work: Stratifiable Maps and Topological Invariants
cites work: Complexe de de Rham filtré d'une variété singulière
cites work: Q3701536
cites work: Q5513537
cites work: Hodge theory meets the minimal model program: a survey of log canonical and Du Bois singularities
cites work: Q4531285
cites work: Q5733144
cites work: Chern classes for singular algebraic varieties
cites work: Q3050549
cites work: Symmetric products of mixed Hodge modules
cites work: Q2998305
cites work: Twisted genera of symmetric products
cites work: The Hirzebruch-Riemann-Roch theorem.
cites work: Generating functions of orbifold Chern classes I: symmetric products
cites work: Modules de Hodge polarisables. (Polarisable Hodge modules)
cites work: Induced ${\cal D}$-modules and differential complexes
cites work: Mixed Hodge modules
cites work: Mixed Hodge complexes on algebraic varieties
cites work: Characteristic classes of mixed Hodge modules
cites work: Q3595861
cites work: Q5187814
cites work: Motivic characteristic classes
cites work: Equivariant Pontrjagin classes and applications to orbit spaces. Applications of the G-signature theorem to transformation groups, symmetric products and number theory
DOI: 10.1515/CRELLE-2014-0114
Recommended article: Q2998305
Recommended article: Symmetric products of mixed Hodge modules
Recommended article: Characteristic classes of Hilbert schemes of points via symmetric products
Recommended article: Motivic characteristic classes
Recommended article: Twisted genera of symmetric products
Recommended article: Equivariant Hodge-Deligne polynomials of symmetric products of algebraic groups
Recommended article: Real Non-Abelian Mixed Hodge Structures for Quasi-Projective Varieties: Formality and Splitting
Recommended article: On the theory of mixed Hodge modules
Recommended article: Characteristic classes of mixed Hodge modules
Recommended article: Equivariant Characteristic Classes of Singular Complex Algebraic Varieties
description: scientific article