A three dimensional ball quotient

provide interesting examples of algebraic varieties. Up to finitely many cases they are expected to be of general type. For example, there are finitely many modular curves which are rational or elliptic curves. General type results have been proved for several series of modular varieties such as the Hilbert modular surfaces [HZ], Siegel modular varieties [T] and certain orthogonal modular threefolds [GHS]. There have been studied also particular examples which are not of general type. For example in a recent paper [FS1] we constructed many Siegel threefolds which admit a Calabi--Yau model. Some of them seem to be new. All Euler numbers which we obtained are non-negative. \smallskip This was a motivation for us to look for other kinds of examples of modular varieties with a Calabi--Yau model. A promising class is given by the ball quotients which belong to the unitary group $\U(1,3)$. Its arithmetic subgroups are called Picard modular groups. In this paper we determine the structure of a very particular example of a Picard modular variety of general type. On its non-singular models there exist many holomorphic differential forms. In a forthcoming paper [FS2] we shall show that one can construct Calabi-Yau manifolds by considering quotients of this variety and resolving singularities. \smallskip The unitary group $\U(1,n)$ can be a considered as a subgroup of the real orthogonal group $\O(2,2n)$ . By restricting via the natural embedding $\U(1,n)\subset \O(2,2n)$, the theory of Borcherds products can be also applied to the unitary group . There are several examples in the literature where Borcherds products have been used for the investigation of the structure of orthogonal or unitary varieties. For example they are used in [GHS] and for the study of classical geometric configurations [AF], [Ko1], [Ko2]. We determine here the structure of a three dimensional ball quotient.