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(1) What is a "fake Calabi-Yau threefold"? Can I describe as a complex or symplectic manifold with trivial canonical bundle, but no compatible Kahler structure? Some mathematicians actually seem to think there is a large generalization of mirror symmetry applying to these kinds of spaces...(I note that none of the infinite class being discussed are thought to be T^3 fibrations in any obvious sense, which gives cause for at least mild suspicion about uses in mirror symmetry).

(2) There is a natural reason for mathematicians to think some mirror symmetry may relate complex and symplectic manifolds which don't come from superconformal sigma models (reference request). I am very excited with the basic idea that such pairs can be obtained from "mirror" smoothings of singular limits of Calabi-Yau's which are mirror. (Can one make sense of this physically too, but not in a way that is accessible to known sigma model techniques?).

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Perhaps you can say where you saw the term "fake Calabi-Yau." I can think of several possibilities, but I can't guess without context. –  Mark Gross Jun 28 '13 at 14:09

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