As far as I understand, mirror symmetry is an involution on the set of Calabi-Yau manifolds which acts at Hodge numbers by $h^{p,q} \leftrightarrow h^{q,p}$.
Kontsevich in 1994 conjectured an equivalence between Fukaya category (encoding information on the special Lagrangian submanifolds) and derived category of coherent sheaves. Later it was extended to Fano varieties and it's expected that mirror to a Fano variety is a LG model (a noncompact algebraic variety equipped with a regular function $W:X\rightarrow \mathbb{C}$ and a symplectic form).
Results of Auroux et al. establish one half of HMS: derived category of coherent sheaves on $\mathbb{CP}^{2}$ is equivalent to the derived category of vanishing cycles on $X= \{xyz=1 \} \subset \mathbb{C^{*}}^{3}$.
Are there any results for the second half of HMS (if we equip $\mathbb{CP}^{2}$ with Fubini-Study form)?
Thank you.
According to Zaslow's answer here -https://mathoverflow.net/questions/30000/do-you-understand-syz-conjecture , this paper by Chan and Leung should have your answer = http://www.sciencedirect.com/science/article/pii/S0001870809002886.