Singularities of symmetric powers and irrationality of motivic zeta functions

Key Points

• The research aims to explore the relationship between the singularities of varieties and their symmetric powers, as well as the implications for motivic zeta functions.
• Proved that if a K-variety has L-rational singularities, all its symmetric powers also possess L-rational singularities.
• Analyzed the rationality of the Kapranov motivic zeta function for smooth complex projective varieties of dimension greater than one.
• Investigated the link between the rationality of motivic zeta functions and the Kodaira dimension.
• Demonstrated that a smooth complex projective variety with a rational Kapranov motivic zeta function has a negative Kodaira dimension.
• Established that such varieties do not admit global nonzero differential forms of even degree.
• Extended the irrationality part of the Larsen–Lunts rationality criterion from surfaces to varieties of arbitrary dimension.