Algebraic Geometry Seminar: Yunfeng Jiang

Let X and X' be two smooth Deligne-Mumford stacks. We call dash arrow X-->X' a Crepant Transformation if there exists a third smooth Deligne-Mumford stack Y and two morphisms \phi:Y-> X,  \phi': Y-> X' such that the pullbacks of canonical divisors are equivalent, i.e. $\phi^*K_{X}\cong \phi'^*K_{X'}$.  The crepant transformation conjecture says that the Gromov-Witten theory of X and X' is equivalent if X-->X' is a crepant transformation. This conjecture was well studied in two cases:  the first one is the case when X and X' are both smooth varieties; the other is the case that there is a real morphism X-> |X'| to the coarse moduli space of X', resolving the singularities of X'.  In this talk I will present some recent progress for this conjecture, especially in the case when both  X and X' are smooth Deligne-Mumford stacks.