Daniel Bergh: Categorical measures for equivariant varieties

I will give a report on a work in progress jointly with S. Gorshinsky, M. Larsen and V. Lunts.
The categorical measure was introduced by Bondal, Larsen and Lunts in 2004. Roughly speaking, it is a ring homomorphism from the Grothendieck group of varieties to the Grothendieck group of triangulated categories which maps a smooth projective variety to the class of its bounded derived category. We study this measure in an equivariant setting. In particular, we prove a conjecture by Galkin and Shinder about a categorical version of Kapranov's zeta function. We also give counterexamples to a recent conjecture by Polishchuk and Van den Bergh (arXiv:1503.04160).