University of Toronto's Symplectic Geometry Seminar

Abstract: Transversality property is an intersection theoretic property. But Tian related the non-transversality property to the singularity of the stable map when the target space is a rational projective surface. I will talk about ideas to calculate the pull-back cycles? intersection multiplicities when the Gromov-Witten invariant is enumerative. Surprisingly, intersection multiplicities are the same as the index of the differentiation of the product of evaluation maps. I will also introduce Tian?s establishment of the chamberwise real Gromov-Witten invariants and explain the real enumerative implications of transversality properties. If time allows, then I will explain how to calculate the full tangent space splitting at any point on the moduli space in an intuitive way. This calculation is a new generalization of the Konsevish?s normal bundle splitting result. This talk is about the first result in the ongoing joint project with Gang Tian.