Benjamin Severn

math.AG — Algebraic Geometry

Algebraicity of Weil classes on abelian sixfolds of Weil type for $\mathbb{Q}(\sqrt{-3})$ with discriminant $-2$

Contributed by Benjamin Severn

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 correspondence supplied by Milne's theorem, from Schoen's algebraic Weil classes on fifty-one cyclic intermediate quotients; the choice of quotients is governed by an integral identity among characters of a torus, verified by an exact computation. Second, the ten-dimensional family of these covers dominates the nine-dimensional moduli space: we exhibit an explicit cover at which the period map has rank nine, by a certified computation. Third, a Hilbert-scheme and Baire-category argument spreads algebraicity from an open set to the whole moduli space. The discriminant of the family is identified exactly. All computations are supplied as reproducible certificates. For the members whose special Mumford--Tate group is the full special unitary group, an invariant-theoretic lemma then gives the Hodge conjecture for every power.