Communications in Analysis and Geometry

Volume 23 (2015)

Number 5

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

Pages: 923 – 949



Bert van Geemen (Dipartimento di Matematica, Università di Milano, Italy)

Kenji Koike (Faculty of Education, University of Yamanashi, Japan)


We study the geometry of a Picard modular fourfold which parametrizes abelian fourfolds of Weil type for the field of cube roots of unity. We find a projective model of this fourfold as a singular, degree ten, hypersurface $\mathcal{X}$ in projective 5-space. The Weyl group $W(E_6)$ acts on $\mathcal{X}$ and we provide an explicit description of this action. Moreover, we describe various special subvarieties as well as the boundary of $\mathcal{X}$.

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