Integration of Voevodsky motives

Abstract: In this paper, we construct four different theories of integration, two that are for Voevodsky motives, one for mixed $\ell$-adic sheaves, and a fourth theory of integration for rational mixed Hodge structures. We then show that they circumvent some of the complications of classical motivic integration, leading to new arithmetic and geometric results concerning K-equivalent $k$-varieties. For example, in addition to recovering known results regarding K-equivalent smooth projective complex varieties, we show that K-equivalent smooth projective $\mathbb{F}_q$-varieties have isomorphic rational $\ell$-adic Galois representations (up to semisimplification), and so also the same zeta functions (the equality of zeta functions is true even without projectivity). This is an arithmetic result inaccessible to classical motivic integration. This paper also gives more evidence for a conjecture of Chin-Lung Wang suggesting the equivalence of integral motives of K-equivalent smooth projective varieties. Furthermore, we connect our theory of integration of rational Voevodsky motives to the existence of motivic $t$-structures for geometric Voevodsky motives; we show that the existence of a motivic t-structure implies that K-equivalent smooth projective varieties have equivalent rational (Chow) motives. We also connect this to a conjecture of Orlov concerning bounded derived categories of coherent sheaves. This makes progress on showing that all cohomology theories should agree for K-equivalent smooth projective varieties (at least rationally and for suitable base fields).