Relative quasimaps and quasimap quantum Lefschetz

Data Seminario: 
Ora Inizio: 
Ora Fine: 
Luca Battistella
Imperial college, London

Quasimaps provide a compactification of the space of maps from smooth curves to a GIT quotient; Ciocan-Fontanine and Kim used them to prove beautiful wall-crossing formulae, comparing the resulting invariants with ordinary Gromov-Witten ones. In joint work with Navid Nabijou, we introduce the notion of genus zero relative quasimaps to a toric target X with respect to a smooth (but not necessarily toric) hyperplane section Y, extending work of Gathmann to this setting. Increasing the tangency requirement gives us smaller and smaller moduli spaces; their virtual classes are related by a simple recursion formula. Gathmann's inductive algorithm allows us to compute restricted invariants of Y from those of X. Under some positivity assumption, this can be made into a compact formula for generating series of quasimap invariants, which recovers a result of Ciocan-Fontanine and Kim. We are working towards an extension of this approach to reduced genus one invariants.