The Grothendieck ring of varieties and piecewise isomorphisms

Abstract

Let K0(Var k) be the Grothendieck ring of algebraic varieties over a field k .L et X, Y be two algebraic varieties over k which are piecewise isomorphic (i.e. X and Y admit finite partitions X1 ,..., Xn, Y1 ,..., Yn into locally closed subvarieties such that Xi is iso- morphic to Yi for all i ≤ n), then (X )=( Y ) in K0(Var k). Larsen and Lunts ask whether the converse is true. For characteristic zero and algebraically closedfield k, we answer positively this question when dim X ≤ 1o rX is a smooth connected projective surface or if X contains only finitely many rational curves.