Hodge numbers are not derived invariants in positive characteristic. (English) Zbl 1532.14036
Summary: 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
M. Gross and S. Popescu [Compos. Math. 127, No. 2, 169–228 (2001; Zbl 1063.14051)],
A. Bak [“The spectral construction for a \((1,8)\)-polarized family of abelian varieties”, Preprint, arXiv:0903.5488],
and
C. Schnell [in: Derived categories in algebraic geometry. Proceedings of a conference held at the University of Tokyo, Japan in January 2011. Zürich: European Mathematical Society (EMS). 279–285 (2012; Zbl 1284.14024)].
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
R. Abuaf’s result [Int. Math. Res. Not. 2017, No. 22, 6943–6960 (2017; Zbl 1405.14044)]
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.
MSC:
| 14F08 | Derived categories of sheaves, dg categories, and related constructions in algebraic geometry |
| 14C30 | Transcendental methods, Hodge theory (algebro-geometric aspects) |
| 18G80 | Derived categories, triangulated categories |
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