Intersection homology, hypergeometric functions, and moduli spaces as ball quotients

We use the Hodge theory of intersection homology valued in a rank 1 local system on a punctured complex projective line $P1$1{bsol}S to define generalized hypergeometric functions, in the spirit of the pioneering work of Deligne and Mostow. These functions sometimes provide an isomorphism between a moduli space $M$ and a complex ball quotient Γ{bsol}$Bn$, where Γ is a discrete subgroup of PU(1, n). Consequently $M$ admits an orbifold complex hyperbolic metric.; We describe two moduli space ↔ ball quotient examples, which we call the Gaussian and the Eisenstein ancestral examples. They are, respectively, the geometric invariant theory (GIT) moduli spaces of binary forms of degree 8 and of degree 12. Their corresponding Deligne-Mostow ball quotient structures are defined over the ring $Z$[ι] of Gaussian integers and the ring $Z$[ω] of Eisenstein integers. We show that the ancestral examples contain a host of other moduli space ↔ ball quotient examples, both known and novel, as distinguished subspaces. In particular, we introduce a technique, based on the study of intersection homology and self-maps of $P1$, to construct many such subspaces.; In the course of classifying the simplest such subspaces, we show that most of the examples in the original list of Mostow are “descendants” of the ancestral examples. More precisely, when taken together these examples constitute the orbifold points, naturally stratified, of the ancestral examples. Eight of the descendants do not show up on Mostow's list, so we partially correct it, in agreement with an independent computer calculation due to Thurston. We then use one of these examples to show that the moduli space of cubic surfaces is a ball quotient of Deligne-Mostow type. This recovers in a much more direct fashion recent results of Allcock, Carlson, and Toledo.; In the course of classifying some slightly more complicated distinguished subspaces of the ancestral examples, we prove that certain GIT moduli spaces of paired binary forms inherit the structure of local ball quotients. In particular this immediately reproduces the ball quotient structure on the moduli space of rational elliptic surfaces due to Heckman and Looijenga.