Voevodsky Motives, Stable Homotopy Theory, and Integration

Abstract

In this thesis, we study applications and connections of Voevodsky's theory of motives to stable homotopy theory, birational geometry, and arithmetic. On the one hand, we show that we can use the stable $\infty$-category of Voevodsky motives to develop a theory of integration similar to classical motivic integration. We show that this theory allows us to circumvent some of the complications of classical motivic integration that obstruct the path to concrete arithmetic and geometric results. One main application of this part is that up to a common direct sum, K-equivalent smooth projective $k$-varieties have the same integral Chow motives (once we invert the exponential characteristic of $k$), partially answering a conjecture of Chin-Lung Wang. In addition to generalizing a theorem of Kontsevich on the equality of Hodge numbers of K-equivalent smooth projective complex varieties, we show that such varieties have isomorphic \textit{integral} singular cohomology groups and that K-equivalent complex varieties have isomorphic rational mixed Hodge structures. On the arithmetic side, we show that K-equivalent smooth $\mathbb{F}_q$-varieties have isomorphic $\ell$-adic Galois representations (up to semi-simplification). Furthermore, we connect this theory of integration of Voevodsky motives to the existence of motivic $t$-structures for geometric Voevodsky motives; we show that if the expected motivic $t$-structure on rational geometric Voevodsky motives exists, then K-equivalent smooth projective varieties, in particular birational Calabi-Yau smooth projective varieties over a field admitting resolution of singularities, have equivalent rational (Chow) motives. This implies that all cohomology theories (considering only the group structures) agree for K-equivalent varieties. We also connect this to a conjecture of Orlov concerning bounded derived categories of coherent sheaves. The other part of this thesis is concerned with if classical stable homotopy theory in algebraic topology sits inside stable motivic homotopy theory over an algebraically closed field. We show that up to inverting the exponential characteristic, the latter subsumes the former theory. In order to do this, we construct a stable \'etale realization functor. As a consequence of this construction, we also prove a homotopy theoretic generalization of the \'etale version of the Suslin-Voevodsky comparison theorem comparing \'etale and motivic cohomology, both with suitable torsion coefficients.