Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves

We study homological mirror symmetry for Del Pezzo surfaces and their mirror Landau-Ginzburg models. In particular, we show that the derived category of coherent sheaves on a Del Pezzo surface X-k obtained by blowing up CP2 at k points is equivalent to the derived category of vanishing cycles of a certain elliptic fibration W-k :M-k -> C with k+3 singular fibers, equipped with a suitable symplectic form. Moreover, we also show that this mirror correspondence between derived categories can be extended to noncommutative deformations of X-k , and give an explicit correspondence between the deformation parameters for X-k and the cohomology class [B+i omega]epsilon H-2 (M-k , C).