Singularities of symmetric powers and irrationality of motivic zeta functions

2025/08/20 by Shein, Vladimir · 1 citation
#14E05 (Secondary) #14G10 (Primary) 14B05 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.15065

Abstract

Let K₀(V_K) be the Grothendieck ring of varieties over a field K of characteristic zero, and let \mathbbL = [\mathbbA¹_K] denote the Lefschetz class. We prove that if a K-variety has \mathbbL-rational singularities, then all its symmetric powers also have \mathbbL-rational singularities. We then use this result to show that, for a smooth complex projective variety X of dimension greater than one, the rationality of its Kapranov motivic zeta function Z(X, t) (viewed as a formal power series over K₀(V_ℂ)) implies that the Kodaira dimension of X is negative and that X does not admit global nonzero differential forms of even degree. This extends the irrationality part of the Larsen-Lunts rationality criterion from the surface case to arbitrary dimension. We also discuss some applications of these results.

Citations

• Quotient singularities in the Grothendieck ring of varieties
• The motivic nearby fiber and degeneration of stable rationality
• Symmetric powers of Severi-Brauer varieties
• Zeta Functions of Curves with no Rational Points
• Symmetric Powers Do Not Stabilize
• Rationality criteria for motivic zeta-functions
• The universal Euler characteristic for varieties of characteristic zero
• Motivic measures and stable birational geometry
• On the motive of the Hilbert scheme of points on a surface
• Kodaira dimension of symmetric powers
• The elliptic curve in the S-duality theory and Eisenstein series for Kac-Moody groups
• Torification and Factorization of Birational Maps

Cited by

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

Related

• Rationality criteria for motivic zeta-functions
• Zeta functions and 'Kontsevich invariants' on singular varieties
• Quotient singularities in the Grothendieck ring of varieties
• Involution on the Graded Grothendieck Ring of Varieties and \mathbbD-Singularities
• Rationalite des series de Poincare et des fonctions Zeta motiviques