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We study the geometry of a Picard modular fourfold which parametrizes abelianfourfolds of Weil type for the field of cube roots of unity. We find aprojective model of this fourfold as a singular, degree ten, hypersurface X inprojective 5-space. The Weyl group W(E_6) acts on X and we provide an explicitdescription of this action. Moreover, we describe various special subvarietiesas well as the boundary of X.

A Picard modular fourfold and the Weyl group W(E_6)