Involution on the Graded Grothendieck Ring of Varieties and \mathbbD-Singularities

2025/08/25 by Burke, Andrew
#14B05 #14G10 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.17587

Abstract

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

Citations

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Related

• Quotient singularities in the Grothendieck ring of varieties
• Singularities of symmetric powers and irrationality of motivic zeta functions
• Zeta functions and 'Kontsevich invariants' on singular varieties
• The standard realizations for the K-theory of varieties
• Resolution of Singularities of Germs in Characteristic Positive Associated with Valuation Rings of Iterated Divisor Type