A smooth non-product canonically polarized threefold is constructed whose Kähler-Einstein metric has an 8-dimensional space of infinitesimal Einstein deformations, with exactly a 6-dimensional integrable subspace.
Koiso,Einstein metrics and complex structures, Invent
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A negative K\"ahler-Einstein threefold with non-integrable infinitesimal Einstein deformations
A smooth non-product canonically polarized threefold is constructed whose Kähler-Einstein metric has an 8-dimensional space of infinitesimal Einstein deformations, with exactly a 6-dimensional integrable subspace.