Why in homological mirror symmetry, we restrict us to a projective variety (Calabi-Yau manifold)? Because in physics we don't need this condition.
What's the general picture for general Calabi-Yau manifold?
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2$\begingroup$ I think that this is much more likely to get answers if posted on mathoverflow. $\endgroup$ – Matt E Oct 11 '10 at 4:29
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Yau gives some quick explanation of this in his scholaropedia page for CY manifolds (which is really a great read if you're interested in such things). In particular, speaking about spacetime manifolds $\mathbb{R}^{3,1}\times X$ (with standard Minkowski metric on the first part), he says "for the most basic product models $N=1$ supersymmetry, the space $X$ must be a Calabi-Yau manifold of complex dimension $3$".
He references a 1985 paper of Candelas, Horowitz, Strominger and Witten.
