Equivariant characteristic classes of external and symmetric products of varieties

Abstract

Abstract. We obtain refined generating series formulae for equivariant characteristic classes of external and symmetric products of singular complex quasi-projective varieties. More concretely, we study equivariant versions of Todd, Chern and Hirzebruch classes for singular spaces, with values in delocalized Borel-Moore homology of external and symmetric products. As a byproduct, we recover our previous characteristic class formulae for symmetric products and obtain new equivariant generalizations of these results, in particular also in the context of twisting by representations of the symmetric group.

Lagrangian planes in hyperkähler varieties of $K3^{[n]}$-type

“…Proof. The first goes back to a PhD thesis of Moonen, see [30,Lemma 5.11] and the references therein. However, it can also be derived directly quite easily, see the discussion in [31].…”

“…In the general case we have the following 'multiplicativity' [30,Example 3.1] . Let g ∈ S n be of cycle type (k 1 , .…”

Geometry and Topology of External and Symmetric Products of Varieties

“…This conjecture, which by now admits several different proofs, predicted a 3-dimensional analogue of Göttsche's generating series formula (Göttsche, 1990) for the Euler characteristics of Hilbert schemes of points on smooth projective surfaces. Some of the above-mentioned results have been refined in two papers (Maxim & Schürmann, 2018, 2020 mentioned earlier via the equivariant study of external products of varieties and coefficients. This approach is inspired by the BKR correspondence of Bridgeland et al (2001).…”

“…In Maxim and Schürmann (2018), we obtained generating series formulae for equivariant characteristic classes, introduced in Cappell et al ( 2012), of external and symmetric products of singular complex quasi-projective varieties. More concretely, we considered equivariant versions of homology Todd, Chern, and Hirzebruch classes with values in the delocalized Borel-Moore homology of external and symmetric products.…”

“…This is an expository note intended to give a brief description of some of the main results of our paper Maxim and Schürmann (2020) (see also Maxim & Schürmann, 2018) concerning invariants of external products of suitable coefficients on possibly singular complex quasi-projective varieties. Along the way, we provide a historical overview of the study of geometry and topology of various spaces of objects associated with a given variety.…”