In order to shed light on Orlov’s conjecture that derived equivalent smooth, projective varieties have isomorphic Chow motives, we examine the zeta functions of derived equivalent varieties over finite fields; in this setting Orlov’s conjecture predicts equality of zeta functions. It is demonstrated that derived equivalent smooth, projective varieties over finite fields that are abelian or satisfy a certain condition on their cohomology. This condition is satisfied, for example, by a surface or Calabi–Yau 3–fold.
One of our approaches to comparing the zeta functions of derived equivalent varieties over finite fields comes from using the Lefschetz Fixed Point Theorem to turn the question into one of comparing the -adic ́etale cohomology of varieties. Cohomology groups are not in general preserved under the action of Fourier–Mukai equivalences on cohomology, but cohomological structures we call even and odd Mukai–Hodge structures, which are a realization of the Mukai motive, are preserved. Investigation into when isomorphism of these cohomological structures implies equality of zeta functions gives us our cohomological condition for equality of zeta functions.
We also develop a relative version of the map Fourier–Mukai transforms induce on cohomology and define a relative notion of even and odd Mukai–Hodge structures, and show these structures are preserved in a situation arising from the derived equivalence of smooth, projective varieties with semiample (anti-)canonical bundles. Using this result, it is demonstrated that when derived equivalent smooth, projective varieties have semiample (anti-)canonical bundles, the fibers over any fixed geometric point in their shared (anti-)canonical variety must also have isomorphic even and odd Mukai–Hodge structures. Hence, for any such varieties over finite fields, if their geometric fibers satisfy any of the conditions identified for isomorphism of Mukai–Hodge structures to imply equality of zeta functions, then the varieties themselves also have equal zeta functions.