After reading through part of Victor Ginzburg's notes on Calabi-Yau algebras, I have a question about a principle in mirror symmetry. Let $(X,X')$ be a mirror pair of Calabi-Yau varieties then mirror symmetry predicts a bijection between

$$M_\mathbb{C}(X) \leftrightarrow M_K(X'),$$
where $M_{\mathbb{C}}(X)$ are smooth CY-deformations of $X$ and $M_{K}(X')$ is the 'stringy' Kahler resolutions of $X'$. The homological mirror symmetry conjecture of Maxim Kontsevich predicts an equivalence of categories $D^b(coh(X_c))\simeq D(Fuk(X'_{c'}))$ where $c\leftrightarrow c'$ under the bijection on the moduli spaces. The problem is that there are singular Calabi-Yau varieties without any smooth deformations or any smooth crepant resolutions. So the homological mirror symmetry conjecture doesn't seem to have a lot of substance in these cases. An insight of Michael Van den Bergh is that every Calabi-Yau variety should have a *noncommutative* deformation or *noncommutative* crepant resolution. Then under the bijection on moduli spaces it is then possible for a noncommutative deformation to map to commutative Kahler resolution and vice verse. To extend homological mirror conjecture to include these noncommutative spaces it seems plausible to define the category of coherent sheaves on a noncommutative space using finitely generated projective modules. So here is my question:

**The definition of the Fukaya category on a symplectic manifold uses techniques that only seem to be available in the geometric context, so is there a plausible definition of the `Fukaya category' of a noncommutative space in order to make the noncommutative homological mirror symmetry conjecture hold?**