Algebraicity of Weil classes on abelian sixfolds of Weil type for $\mathbb{Q}(\sqrt{-3})$ with discriminant $-2$
Let $K=\mathbb{Q}(\sqrt{-3})$. We prove that the Weil classes of every polarized abelian sixfold of Weil type for $K$ with Hermitian discriminant $-2$ are algebraic. This is a non-split discriminant, to which the theorems of Schoen and of Markman on split Weil classes do not apply. The proof uses Prym--Tyurin varieties of covers of $\mathbb{P}^1$ with monodromy the complex reflection group $G_{32}$, and combines known mechanisms in a new configuration. First, on every such variety the Weil classes of the six-dimensional isotypic factor are algebraic. They are obtained, by a Lefschetz-class…
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