On the Grothendieck ring of varieties in positive characteristic
Kirti Joshi
TL;DR
The paper investigates whether the Grothendieck ring of varieties in positive characteristic is a domain by working with the smooth-complete variant ${\rm K}^0_{\rm bl}(\mathcal{CV}^{sm}_k)$. It constructs two motivic measures: the Albanese motivic measure ${\rm Alb}:{\rm K}^0_{\rm bl}(\mathcal{CV}^{sm}_k)\to {\mathbb Z}[AV_k]$ and proves, in characteristic $p>0$ with $p>13$ or $p=11$, that the class of a supersingular elliptic curve is a zero divisor via Deligne’s isomorphisms; this shows ${\rm K}^0_{\rm bl}(\mathcal{CV}^{sm}_k)$ contains zero divisors. It also establishes that the étale fundamental group defines a motivic measure ${\Pi}:{\rm K}^0_{\rm bl}(\mathcal{CV}^{sm}_k)\to {\mathbb Z}[\mathscr{Prof}]$, extending to ${\rm K}^0(\mathcal{V}_k)$ in characteristic zero, and verifies the required compatibility with Bittner relations under blowups and products. Together, these results connect arithmetic-geometric invariants with motivic measures and demonstrate non-domain structure in positive characteristic, while providing tools that transfer to the classical Grothendieck ring in characteristic zero.
Abstract
This paper proves two theorems (1) Let $k$ be an algebraically closed field of characteristic $p>0$. I prove (Theorem 2.1.1) that if, $p > 13$ or $p = 11$, then the isomorphism class of any supersingular elliptic curve is a zero divisor in the ring of smooth, complete $k$-varieties and Bittner relations. In particular, this ring contains zero divisors. The proof proceeds via establishing (in Theorem 2.2.1) that the Albanese variety functor is a motivic measure. (2) I prove (Theorem 3.1) that the etale fundamental group of a smooth, proper variety over any alg. clsoed field k (in any characteristic) also provides a motivic measure on this ring. In particular, the etale fundamental group is a motivic measure on the Grothendieck ring of varieties over complex numbers.