I will talk about finiteness properties of hyperbolic varieties (some old ones, some new ones, and some we expect to be true, but can't prove yet). Our starting point is the theorem of de Franchis: given a variety Y and a hyperbolic Riemann surface C, the set of non-constant maps from Y to C is finite. The main question I'd like us to ask is simply "to what extent does this finiteness statement hold for higher-dimensional hyperbolic targets"? It obviously fails for surfaces (take C x C), but it surprisingly holds for the (orbifold) moduli space of compact hyperbolic Riemann surfaces of genus g (g>1). I will propose an (in my honest opinion) reasonable conjecture for *all* hyperbolic varieties. The main result will be a proof of this conjecture for all moduli spaces of polarized varieties. Joint work with Steven Lu, Ruiran Sun, and Kang Zuo.
