Hodge numbers are not derived invariants in positive characteristic

Abstract

We study a pair of Calabi–Yau threefolds X and M, fibered in non-principally polarized Abelian surfaces and their duals, and an equivalence \(D^b(X) \cong D^b(M)\), building on work of Gross, Popescu, Bak, and Schnell. Over the complex numbers, X is simply connected while \(\pi _1(M) = (\mathbf {Z}/3)^2\). In characteristic 3, we find that X and M have different Hodge numbers, which would be impossible in characteristic 0. In an appendix, we give a streamlined proof of Abuaf’s result that the ring \(\mathrm{H}^{*}({\mathscr {O}})\) is a derived invariant of complex threefolds and fourfolds. A second appendix by Alexander Petrov gives a family of higher-dimensional examples to show that \(h^{0,3}\) is not a derived invariant in any positive characteristic.