Characteristic classes of mixed Hodge modules and applications

Abstract

Abstract. These notes are an extended version of the authors' lectures at the 2013 CMI Workshop "Mixed Hodge Modules and Their Applications". We give an overview, with an emphasis on applications, of recent developments on the interaction between characteristic class theories for singular spaces and Saito's theory of mixed Hodge modules in the complex algebraic context.

Topological and Hodge L-classes of singular covering spaces and varieties with trivial canonical class

“…In view of the algebraic results of [7] and [11], the conclusions of the present paper are also relevant in the context of a question raised by Jörg Schürmann in [54]: Is the Siegel-Sullivan orientation ∆(X) of a pure-dimensional compact complex algebraic variety X the image of the intersection homology (mixed) Hodge module on X under the motivic Hodge Chern class transformation MHC 1 : K 0 (MHM(X)) → K coh 0 (X) of Brasselet-Schürmann-Yokura [14], followed by the K-theoretical Riemann-Roch transformation of Baum, Fulton and MacPherson?…”

Positivity for Hodge modules and geometric applications

“…Besides [Sai93] invoked in the text, I will highlight the papers [Sai91b], which proves a conjecture of Steenbrink on the spectrum of certain isolated singularities, [BS05], which makes a connection between multiplier ideals and the V -filtration, and [DMS11], which extends the latter to arbitrary subvarieties; see also the references therein. Hodge modules have also proved useful in the study of characteristic classes of singular varieties; see the survey [MS13].…”

The uniqueness theorem for Gysin coherent characteristic classes of singular spaces

“…In [6], the first author derived Verdier–Riemann–Roch‐type formulae for the Gysin restriction of both the topological characteristic classes $$L_{\ast }$$ of Goresky and MacPherson [21] and the Hodge‐theoretic intersection Hirzebruch characteristic classes $$IT_{1 \ast }$$ of Brasselet, Schürmann, and Yokura [9]. For an introduction to characteristic classes of singular spaces via mixed Hodge theory in the complex algebraic context, see Schürmann's expository paper [32]. The formulae in [6] have the potential to yield new evidence for the equality of the characteristic classes $$L_{\ast }$$ and $$IT_{1 \ast }$$ for pure‐dimensional compact complex algebraic varieties, as conjectured by Brasselet, Schürmann, and Yokura in [9, Remark 5.4].…”