A Siegel Modular 3-fold that is a Picard Modular 3-fold

Abstract

Let D be a bounded symmetric domain. ⊂ Aut( D) be a discrete, properly discontinuous group. If is cocompact and acts freely, it has been known for several decades (Kodaira: [K], Hirzebruch: [Hi]) that Г D is then an algebraic variety, and in fact of general type. The Hirzebruch proportionality theorem then tells us the (ratios of) Chern numbers of X = Г D, which allows us to recover D from the Chern numbers of X if we know only that X is of the form Г D for some D. The group is then of course just the fundamental group π1(X). So it can’t happen, for example, that X = Г D = Г\D′ for 2 non-isomorphic bounded symmetric domains D and D′.