Involution on the graded Grothendieck ring of varieties and 𝔻-singularities

We realize a graded variant K₀(Varₖ) of the Grothendieck ring of varieties as a quadratic extension of the subring K₀(Varₖˢᵖ) spanned by classes of smooth and proper varieties. As such, there exists a natural involution D on K₀(Varₖ). We show that D commutes with the symmetric power operations Symᵐ 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.