Modular data and regularity of Monge-Amp\`ere exhaustions and of Kobayashi distance

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.