Higher-dimensional ball quotients and the invariant quintic (Q1576596)

In this article, the author continues his work on a certain quintic hypersurface in projective 5-space. Let \(W(E_{6})\) be the Weyl group of type \(E_{6}\). It acts in a natural way on \({\mathbb{P}}^{5}\). Let \({\mathcal I}_5\) be the zero set of the invariant homogeneous polynomial of degree 5 on six variables (it is unique up to a non-zero constant). A smooth projective \(\widetilde {\mathcal I}_{5}\) is constructed by blowing up singularities of \({\mathcal I}_{5}\). The author shows that \(\widetilde {\mathcal I}_{5}\) is the compactification of a ball quotient. More precisely, \(\widetilde {\mathcal I}_{5}\) contains a divisor with normal crossings \(\widetilde\Delta\) such that \(\widetilde {\mathcal I}_{5}\setminus\widetilde\Delta\) is isomorphic to the quotient of a four-ball by a discrete subgroup of \(SU(4,1)\). This result was conjectured by the author [\textit{B. Hunt}, ``The geometry of some special arithmetic quotients'', Lect. Notes Math. 1637 (1996; Zbl 0904.14025)]. To prove this conjecture, a first result which gives a criterion for checking whether a variety of dimension at least three is a ball quotient is proven.