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arXiv · 2609.38801

Supersingular Tate conjecture for irreducible symplectic varieties of known types: I

Lie Fu·Xuanlin Huang·Zhiyuan Li
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Abstract

We prove that for a supersingular irreducible symplectic variety admitting suitable lifting to characteristic zero has Tate Chow motive if it is of deformation type $K3^{[n]}$, OG6 (with Artin invariant $\neq$ 4), or OG10 (with Artin invariant $\neq$ 12), and has supersingular abelian Chow motive if it is of $\mathrm{Kum}^n$-type (with Artin invariant $\neq$ 3). In particular, any product of those irreducible symplectic varieties satisfies the supersingular Tate conjecture for the whole $\ell$-adic or crystalline cohomology ring.

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Lie Fu, Xuanlin Huang, Zhiyuan Li. 2026-09-30. Supersingular Tate conjecture for irreducible symplectic varieties of known types: I. https://arxiv.org/abs/2609.38801

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