Kevin are you aware of the Ran/Kawamata argument for this theorem? (1992 pair of articles in Journal of Algebraic Geometry) Kawamata's version is not only algebraic but shockingly cute. The only problem is the profs who told me about it couldn't really explain why having unobstructed formal deformations (the algebraic side) means also that the analytic obstruction vanishes/the Kuranishi space is smooth.
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