I was hoping for an answer in general terms avoiding things like holonomy, Chern classes, Kahler manifolds, fibre bundles and terms of similar ilk. Simply, what are the compelling reasons for restricting the landscape to admittedly bizarre Calabi-Yau manifolds? I have Yau's semi-popular book but haven't read it yet, nor, obviously, String Theory Demystified :)
Answer
There is one simple reason: in such scenario the physics at the string scale has supersymmetry. Supersymmetry (more technically $N=1$ supersymmetry) has some nice phenomenological features that make it an attractive bridge between low energy physics and string theory. The existence of this symmetry translates directly to the requirement that the compactification manifold is Calabi-Yau.
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