Symmetric Powers Do Not Stabilize

TL;DR

This paper investigates the non-stabilization of symmetric products of certain algebraic varieties in the Grothendieck ring, revealing obstructions linked to Hodge theory and motivic zeta functions.

Contribution

It demonstrates that symmetric powers of smooth projective surfaces with non-zero canonical bundle do not stabilize, providing new insights into their motivic and Hodge-theoretic properties.

Findings

Symmetric products of surfaces with non-zero ^0(X, \u03a9_X) do not stabilize.

Hodge-theoretic obstructions prevent stabilization of symmetric products.

Relationship established between Newton polygons of motivic zeta functions and Hodge polygons.

Abstract

We discuss the stabilization of symmetric products Sym^n(X) of a smooth projective variety X in the Grothendieck ring of varieties. For smooth projective surfaces X with non-zero h^0(X, \omega_X), these products do not stabilize; we conditionally show that they do not stabilize in another related sense, in response to a question of R. Vakil and M. Wood. There are analogies between such stabilization, the Dold-Thom theorem, and the analytic class number formula. Finally, we discuss Hodge-theoretic obstructions to the stabilization of symmetric products, and provide evidence for these obstructions in terms of a relationship between the Newton polygon of a certain "motivic zeta function" associated to a curve, and its Hodge polygon.