ERC Project: Schemes: Schobers, Mutations and Stability (R-11012)

Mirror symmetry is a manifestation of string theory that predicts a certain symmetry between complex geometry and symplectic geometry. Mirror symmetry is justified on physical grounds but makes nonetheless strong and testable predictions about purely mathematical concepts. A celebrated example is the prediction by physicists of the number of rational curves of a given degree in a generic quintic threefold which went far beyond classical enumerative geometry. The main actor in this proposal is the "Stringy Kähler Moduli Space" which is the moduli space of complex structures of the mirror partner of a Calabi-Yau manifold. The SKMS is not rigorously defined as mirror symmetry itself is not rigorous, but in many cases there are precise heuristics available to characterize it. Mirror symmetry predicts the existence of an action of the fundamental group of the SKMS on the derived category of coherent sheaves of a Calabi-Yau manifold. This prediction has only been verified in a limited number of cases. We will attempt to confirm the prediction for algebraic varieties occurring in geometric invariant theory and the minimal model program.

Keywords:Algebraic and complex geometry, Lie Algebras, Lie groups

Disciplines:Algebraic geometry, Associative rings and algebras, Category theory, homological algebra, Commutative rings and algebras, Field theory and polynomials