Moduli spaces of stable quotients and wall-crossing phenomena

The moduli space of holomorphic maps from Riemann surfaces to the Grassmannian is known to have two kinds of compactifications: Kontsevich's stable map compactification and Marian-Oprea-Pandharipande's stable quotient compactification. Over a non-singular curve, the latter moduli space is Grothendieck's Quot scheme. In this paper, we give the notion of `$\epsilon$-stable quotients' for a positive real number $\epsilon$, and show that stable maps and stable quotients are related by the wall-crossing phenomena. We will also discuss Gromov-Witten type invariants associated to $\epsilon$-stable quotients, and investigate them under the wall-crossing.