Algebraicity of Weil classes on non-split abelian sixfolds of Weil type over $\mathbb{Q}(\sqrt{-3})$ with Hermitian determinant $-2$
Let $K=\mathbb{Q}(\sqrt{-3})$. We give a proof that every Weil class is algebraic on every complex abelian sixfold of Weil type for $K$ whose Hermitian form has signature $(3,3)$ and determinant $-2$ modulo norms. These forms are not split, so neither Schoen's theorem on sixfolds nor the theorem on sixfolds claimed in a preprint of Markman, both of which concern split forms, applies. We model the sixfolds by the part of the Jacobian of a cover $C\to\mathbb{P}^1$, branched at thirteen points, on which its group, the complex reflection group $G_{32}$, acts through its reflection…
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