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Classes in Zakharevich K-groups constructed from Quillen K-theory

PD Dr. Oliver Bräunling

Freitag, 22. Oktober 2021 10:30Uhr

(joint work with M. Groechenig)

The Grothendieck ring of varieties K_0(Var) is defined a lot like K_0 in Quillen K-theory. However, vector bundles get replaced by varieties and instead of quotienting out exact sequences, we quotient out Z -> X <- U relations, where Z is a closed subvariety and U its open complement. This ring plays a crucial role in motivic integration, as in the proof that K-equivalent [that's yet another meaning of K...] varieties have the same Hodge numbers.

Even though things look very analogous, K_0(Var) is not the K_0 group of some abelian category (or Waldhausen etc). Usual K-theory foundations do not apply. Zakharevich and Campbell established that there is an analogous theoretical formalism nonetheless, so there are also higher K-groups corresponding to K(Var). However, until recently, it was not known whether any such K_n(Var) for n>0 contains any non-zero element beyond torsion classes. Some months ago, we managed to give the first construction of such, indeed showing that for all odd n>=3 the group K_n(Var) is infinite-dimensional. To do this, we develop two new tools. Joint with M. Groechenig and A. Nanavaty, motivic realizations give rise to maps out of K(Var), and (joint just with M. Groechenig) there is a kind of exponential map from Quillen K-theory to K(Var), allowing us to import Quillen K-theory classes to give rise to classes living on abelian varieties in K(Var).