Student AG Seminar: Moduli spaces of (…) are negatively curved

The coarse moduli space of curves of genus $g$,  $g$ greater than 1, over the complex numbers is a hyperbolic analytic space and hence negatively curved. Curvature is hard to translate to algebraic terms, but hyperbolicity has aspects that lend themselves to algebraic analogs. For instance, Brody hyperbolicity asks whether there are non-trivial holomorphic maps from $\mathbb{C}^*$ to a given space. This more or less corresponds to whether there are non-trivial regular morphisms from $\mathbb{A}^1 \setminus \{0\}$ or abelian varieties. This later notion is purely algebraic and has received considerable attention in the past several decades.