  Regularity properties of intrinsic objects for a large class of Stein
Manifolds, namely of Monge-Amp\`ere exhaustions and Kobayashi distance, is
interpreted in terms of modular data. The results lead to a construction of an
infinite dimensional family of convex domains with squared Kobayashi distance
of prescribed regularity properties. A new sharp refinement of Stoll's
characterization of $\mathbb C^n$ is also given.
