${\mathbb{A}}^1$-Euler Characteristic of Low Symmetric Powers and Split Toric Varieties

Abstract

For a smooth, projective scheme $X$ over a field $k$ or any variety $X$ if $k$ has characteristic zero, we compute the compactly supported ${\mathbb{A}}^1$-Euler characteristic of ${\mathrm{Sym}}^2(X)$ if $\mathrm{char}(k)\ne 2$ and of ${\mathrm{Sym}}^3(X)$ if $\mathrm{char}(k)\ne 2,3$. We do so by extending the definition of a $G$-equivariant quadratic Euler characteristic first studied by Pajwani-Pál to arbitrary characteristic and by studying its relation to the ${\mathbb{A}}^1$-Euler characteristic of quotients. As an application, we show that the compactly supported ${\mathbb{A}}^1$-Euler characteristic of ${\mathrm{Sym}}^n(X)$ agrees with the prediction from the power structure constructed by Pajwani-Pál for $n=2,3$. Furthermore, we compute the compactly supported ${\mathbb{A}}^1$-Euler characteristic of split toric varieties and show that the compactly supported ${\mathbb{A}}^1$-Euler characteristic of all of their symmetric powers agrees with the prediction from the power structure constructed by Pajwani-Pál.

68 pages; comments welcome!