Schobers, Mutations and Stability

ended

"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. Our main approach will be the construction of a perverse schober on a partial compactification of the SKMS. The existence of such a schober does not only confirm, but also clarifies the predicted action as it is now becomes the result of ``wall crossing'', i.e. moving outside the SKMS itself. To reach our objective we will approach the SKMS from different angles, most notably through its relation with the moduli space of stability conditions. "