This thesis is devoted to the study of A-branes on symplectic tori and to the Mirror Symmetry conjecture. Using a method called Seidel's mirror map, we are able to reconstruct the homogeneous coordinate ring of a complex abelian variety using Lagrangian intersection theory on the mirror symplectic torus. Moreover, we extend Seidel's method by reconstructing singular Kummer surfaces embedded in a projective space of dimension three and complex tori embedded in noncommutative deformations of projective space. In particular we describe how this procedure defines a new deformation of the eight dimensional projective space. Finally, we show that Seidel's mirror map can reconstruct coherent sheaves mirror to a given (non-necessarily Lagrangian) A-brane, in agreement with the conjectural definition of the category of A-branes and with the Homological Mirror symmetry conjecture.
My Research
