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

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