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I have read (cursorily) Berger's paper, and did not find anything which was helpful, but it's 94 pages long. Could you be so kind to point it more precisely. The paper is here: numdam.org/item/?id=ASENS_1957_3_74_2_85_0 Many thanks!
I also need to prove that the development map is globally defined. For example, for an orbifold CP^1 with one conical point, there is no globally defined development map, because it is simply connected.
Sure: for non-smooth varieties it is actually a definition, see Guaraldo, F., Macri, P., Tancredi, A., {\em Topics on real analytic spaces}, Advanced lectures in mathematics, Braunschweig: F. Vieweg, 1986.
Many thanks. They don't seem to use the smoothness, in fact, the argument is almost literally the same as I gave. Still, the reference to the full strength statement would be extremely helpful.
