Complex geometry motivated by mirror symmetry

Abstract

This is a proposal to develop research in complex geometry motivated by the Mirror Symmetry Conjecture. One of the various dualities predicted by this conjecture states that to every complex variety X there corresponds a symplectic variety X( (its mirror), together with the structure of a symplectic Lefschetz fibration W : X( ’ C, such that the derived category of coherent sheaves over X is equiv- alent to the Fukaya category generated by the Lagrangian vanishing cicles of the fibration W. The research themes of this proposal correspond to the investigation of various open questions brought to light by this conjecture. They are:* Level-rank duality and geometric engineering. * Mirror symmetry for adjoint orbits and Lefschetz fibrations. * Local analitic invariants and stratifications of moduli stacks. * Applications of computational algebraic geometry to mathematical physics. * Moduli stacks and their non-commutative deformations. (AU)