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We study the geometric structure of Lorentzian spin manifolds, which admitimaginary Killing spinors. The discussion is based on the cone construction anda normal form classification of skew-adjoint operators in signature $(2,n-2)$.Derived geometries include Brinkmann spaces, Lorentzian Einstein-Sasaki spacesand certain warped product structures. Exceptional cases with decomposableholonomy of the cone are possible.

Imaginary Killing spinors in Lorentzian geometry