Research Seminar: Complex Geometry

1.11.2017 Helge Ruddat (Mainz)

Local and relative Gromov-Witten invariants

Abstract: Given a smooth projective X with a smooth nef divisor D, the Gromov-Witten invariants of the total space of O(−D) are called local invariants. On the other hand, one may consider Gromov-Witten invariants of X relative to the divisor D. In a joint work with Michel van Garrel and Tom Graber, we show these two invariants are related by a simple formula. I will explain how the proof uses the degeneration formula to achieve an identification of virtual fundamental classes. If time permits, I will also explain the generalization where D is simple normal crossings and the local invariants are related to the sum of line bundles given by the components of D. Tropical geometry and log Gromov-Witten theory are the key new tools in achieving these results. Our result has applications to the computation of symplectic cohomology.