A negative Kähler-Einstein threefold with non-integrable infinitesimal Einstein deformations

Abstract

We construct a smooth canonically polarized threefold, not biholomorphic to a product of positive-dimensional varieties, whose normalized Kähler-Einstein metric admits a non-integrable infinitesimal Einstein deformation. The same tangent direction is non-integrable as an infinitesimal complex deformation. In fact, the space of infinitesimal Einstein deformations in our example has real dimension 8, its integrable directions form a real 6-dimensional subspace, and every direction outside that subspace is obstructed. This answers both parts of a suitably generalized version of a question posed by Dai, Wang, and Wei in real dimension 6.