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In previous work we have conjectured wall-crossing formulas for genus zero quasimap invariants of GIT quotients and proved them via localization in many cases. We extend these formulas to higher genus when the target is semipositive, and prove them for semipositive toric varieties, in particular for toric local Calabi–Yau targets. The proof also applies to local Calabi–Yau's associated to some nonabelian quotients.
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Ionuţ Ciocan-Fontanine, Bumsig Kim, Higher genus quasimap wall-crossing for semipositive targets. J. Eur. Math. Soc. 19 (2017), no. 7, pp. 2051–2102DOI 10.4171/JEMS/713