On the Grothendieck ring of varieties in positive characteristic

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.