Hodge numbers are not derived invariants in positive characteristic

TL;DR

This paper demonstrates that Hodge numbers are not invariants under derived equivalence in positive characteristic, providing explicit examples and extending known results from complex geometry.

Contribution

It shows that Hodge numbers can differ for derived equivalent Calabi-Yau threefolds in positive characteristic, challenging assumptions from complex geometry.

Findings

Hodge numbers differ in characteristic 3 for derived equivalent threefolds

Provides examples where Hodge numbers are not derived invariants in positive characteristic

Extends the understanding of derived invariants beyond complex numbers

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) = D^b(M), building on work of Gross, Popescu, Bak, and Schnell. Over the complex numbers, X is simply connected while pi_1(M) = (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 H^*(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.