O'Grady constructed a 6-dimensional irreducible holomorphic symplectic variety by taking a crepant resolution of some moduli space of stable sheaves on an abelian surface. In this paper, we naturally extend O'Grady's construction to fields of positive characteristic $p\neq 2$, called OG6 varieties. Assuming $p\geq 3$, we show that a supersingular OG6 variety is unirational, its rational cohomology group is generated by algebraic classes, and its rational Chow motive is of Tate type. These results confirm in this case the generalized Artin--Shioda conjecture, the supersingular Tate conjecture and the supersingular Bloch conjecture proposed in our previous work, in analogy with the theory of supersingular K3 surfaces.

中文翻译：

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维度六的超奇异奥格雷迪变种

O'Grady 通过对阿贝尔表面上稳定滑轮的一些模空间进行蠕变分辨率构造了一个 6 维不可约全纯辛簇。在本文中，我们自然地将 O'Grady 的构造扩展到具有正特征 $p\neq 2$ 的领域，称为 OG6 品种。假设 $p\geq 3$，我们证明超奇异 OG6 变体是无理的，它的有理上同调群是由代数类生成的，它的有理 Chow 动机是 Tate 类型的。在这种情况下，这些结果证实了我们之前工作中提出的广义 Artin-Shioda 猜想、超奇异 Tate 猜想和超奇异 Bloch 猜想，类似于超奇异 K3 表面的理论。