COMPACTLY SUPPORTED A¹-EULER CHARACTERISTICS OF SYMMETRIC POWERS OF CELLULAR VARIETIES

The compactly supported A¹ -Euler characteristic, introduced by Hoyois and later refined by Levine and others, is an analogue in motivic homotopy theory of the classical Euler characteristic of complex topological manifolds. It is an invariant on the Grothendieck ring of varieties K₀(Varₖ) taking values in the Grothendieck-Witt ring GW(k) of the base field k . The former ring has a natural power structure induced by symmetric powers of varieties. In a recent preprint, the first author and Pál construct a power structure on GW(k) and show that the compactly supported A¹ -Euler characteristic respects these two power structures for $0$ -dimensional varieties, or equivalently étale k -algebras. In this paper, we define the class Symₖ of symmetrisable varieties to be those varieties for which the compactly supported A¹ -Euler characteristic respects the power structures and study the algebraic properties of the subring K₀(Symₖ) of symmetrisable varieties. We show that it includes all cellular varieties, and even linear varieties as introduced by Totaro. Moreover, we show that it includes non-linear varieties such as elliptic curves. As an application of our main result, we compute the compactly supported A¹ -Euler characteristics of symmetric powers of Grassmannians and certain del Pezzo surfaces.