Algebraic Topology Seminar
Inna Zakharevich
University of Chicago
Analyzing the Grothendieck ring of varieties using K-theory
Abstract: The Grothendieck ring of varieties, $K_0[\mathcal{V}_k]$ was
introduced by Grothendieck in a 1964 letter to Serre. It is defined
to be the free abelian group generated by varieties (over a fixed
field $k$), under the relation that if $Y$ is a closed subvariety of
$X$ then $[X] = [Y] + [X \backslash Y]$. Multiplication is defined
via the Cartesian product. Thus, in particular, if $X$ and $Y$ are
varieties which are piecewise isomorphic then $[X] = [Y]$ in the ring.
There are two important structural questions about this ring:
1. If two varieties $X$ and $Y$ satisfy $[X] = [Y]$, are they
piecewise isomorphic?
2. Is the class of the affine line a zero divisor?
The answers to these questions are "no" and "yes", respectively; this
was shown in a recent paper of Borisov, where he constructed an
example to resolve question (2), which also coincidentally answered
question (1).
In this talk we will analyze the structure of this ring using
techniques of algebraic K-theory. In particular, we will construct a
spectrum $K(\mathcal{V}_k)$ whose $\pi_0$ is the Grothendieck ring of
varieties, and whose higher homotopy groups contain further geometric
information about piecewise isomorphisms of varieties. We will then
analyze the geometry of this spectrum to show that the kernel of
multiplication by the affine line is generated by varieties $X$ and
$Y$ such that $[X] = [Y]$ but $X$ and $Y$ are not piecewise
isomorphic, thus showing that Borisov's coincidence is not a
coincidence at all.
Monday April 27, 2015 at 3:00 PM in SEO 1227