Title: Log Gromov-Witten theory for symplectic category

2017.05.11 |

Date | Fri 26 May |

Time | 14:15 — 15:15 |

Location | 1532-122 Aud-G2 |

**Abstract**

Extending Gromov-Witten (GW) theory to “smoothable” normal crossings (NC) varieties and to smooth varieties relative to NC divisors has been a serious challenge since the inception of this theory in 1990s. For smooth divisors, *the relative compactificatio*n (Li, Ionel-Parker, and Li- Ruan) includes maps with image inside an *expanded degeneration* of the target. The relative compactification is “virtually” smooth and can be used to define a GW theory on smoothable simple NC varieties with two components. The GW degeneration formula of J. Li relates GW invariants of the singular space to GW invariants of a smoothing. The idea of expanded degenerations does not extend to NC varieties with more than two components. On the algebraic side, the log GW theory of Gross-Siebert and Abramovich-Chen, developed in the past 10 years, replaces the idea of expanded degenerations with an extra log structure on the target and yields a satisfactory GW theory which admits a similar degeneration formula. On the other hand, GW invariants are expected to be symplectic invariants. In this talk, after a quick review of the earlier approaches, I will introduce an indirect analogue of the log moduli spaces relative to arbitrary simple NC symplectic divisors. This construction does not require any extra structure on or modification of the target, and can be used to construct a GW theory on NC symplectic varieties. Unlike the relative case, log moduli spaces are often virtually singular. However, the singularities are toroidal and I will describe an explicit toric model for the normal cone to each stratum. I will provide some examples for clarification. This talk is an introduction to a research in progress.