Symmetric powers do not stabilize

Abstract:

We discuss the stabilization of symmetric products $\operatorname {Sym}^n(X)$ 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 conjectural Hodge-theoretic obstructions to the stabilization of symmetric products. We provide evidence for these obstructions by showing that the Newton polygon of the motivic zeta function associated to a curve equals the Hodge polygon of the curve.