Calabi-Yau threefolds over finite fields and torsion in cohomologies

Lam, Yeuk Hay Joshua

doi:10.48550/arxiv.2009.09951

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2021

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“…Our X shows that h 2,0 = 0 is possible. The quotient X/µ 3 shows that h 1,0 = 0 is possible as well, exactly like Lam's example of a quintic threefold modulo µ 5 in characteristic 5 [39,Thm. 1.2].…”

Section: Introductionsupporting

confidence: 56%

Hodge numbers are not derived invariants in positive characteristic

Addington¹,

Bragg²

2021

Preprint

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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 π1(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 higherdimensional examples to show that h 0,3 is not a derived invariant in any positive characteristic.

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Section: Introductionsupporting

confidence: 56%