Abstract:

The mirror of a Fano variety M is not a manifold in the usual sense, but rather a "Landau-Ginzburg model", i.e. a non-compact manifold X equipped with a complex-valued function w (the "superpotential"). In this context, the homological mirror symmetry conjecture predicts that the derived category of coherent sheaves over M is equivalent to a derived category of Lagrangian vanishing cycles associated to the critical points of the superpotential w on its mirror. In the special case where the critical points of w are non-degenerate, the existence of a rigorous definition due to Seidel makes it possible to test the homological mirror symmetry conjecture on various examples by determining both categories explicitly.

In this talk, based on joint work with L. Katzarkov and D. Orlov, we will first review the necessary background, and then focus on a specific family of examples: weighted projective planes. We will show that the homological mirror conjecture holds in this case. Moreover, non-exact deformations of the symplectic structure on the Landau Ginzburg model correspond to non-commutative deformations of the weighted projective plane.