Abstract We realize a graded variant K₀ (Var₊^) of the Grothendieck ring of varieties as a quadratic extension of the subring K₀ (Var₊^sp) spanned by classes of smooth and proper varieties. As such, there exists a natural involution D on K₀ (Var₊^). We show that D commutes with the symmetric power operations Sym^m up to zero divisors. Moreover, we study varieties that are smooth up to cut-and-paste relations, which we call D-singular varieties, and we give applications to compactifications of varieties and the irrationality of Kapranov zeta functions.
Andrew Burke (Fri,) studied this question.