Abstract: |
One central question in Gromov-Witten (GW) theory is to relate the GW invariants of a hypersurface to the GW invariants of the ambient space. In genus zero, this is usually done by the so-called quantum Lefschetz principle, which uses the twisted GW invariants of the ambient space. This approach is analogous to the classical theorem that the number of lines inside a cubic surface can be obtained by computing the Euler number of a certain vector bundle on the space of lines inside \mathbb P^3 (which is the Grassmannian Gr(2,4)). Thus, the quantum Lefschetz principle provides an effective way to calculate the GW invariants of the hypersurface when the twisted GW invariants of the ambient space are known (e.g. toric stacks). However, this approach requires a technical assumption called convexity for the line bundle over the ambient space defining the hypersurface. Hence, when convexity fails, the GW invariants of a hypersurface are much less known for a long time. In this talk, I will present a way (mirror theorem) to obtain the genus zero GW invariants of positive hypersurfaces in toric stacks for which the convexity may fail. One key ingredient in the proof is to resolve the genus zero quasimap wall-crossing conjecture proposed by Ciocan-Fontaine and Kim for these targets. |