Hodge numbers are not derived invariants in positive characteristic
arXiv:2106.09949 · doi:10.1007/s00208-022-02474-w
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.
arXiv comment32 pages, Macaulay2 and Magma code as ancillary files, with an appendix by Alexander Petrov. final version which appeared a while ago in Math Ann
References in corpus (10)
- Derived categories of coherent sheaves
- Cubic fourfolds fibered in sextic del Pezzo surfaces
- The Brauer group is not a derived invariant
- The Spectral Construction for a (1,8)-Polarized Family of Abelian Varieties
- Derived Equivalent Calabi-Yau 3-folds from Cubic 4-folds
- The Hodge ring of varieties in positive characteristic
- Crystalline aspects of geography of low dimensional varieties I: Numerology
- Computing representation matrices for the action of Frobenius to cohomology groups
- Rigid-analytic varieties with projective reduction violating Hodge symmetry
- Calabi-Yau threefolds over finite fields and torsion in cohomologies