---
abstract: '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 semi-positive, and prove them for semi-positive toric varieties, in particular for toric local Calabi-Yau targets. The proof also applies to local Calabi-Yau’s associated to some non-abelian quotients.'
address:
- 'School of Mathematics, University of Minnesota, 206 Church St. SE, Minneapolis MN, 55455, andSchool of Mathematics, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul, 130-722, Korea'
- 'School of Mathematics, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul, 130-722, Korea'
author:
- 'Ionuţ Ciocan-Fontanine'
- Bumsig Kim
title: 'Higher genus quasimap wall-crossing for semi-positive targets'
---

Introduction
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Overview
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When a complex affine algebraic variety $W$ is acted upon by a reductive group ${{\bf G}}$, a choice of a character $\theta$ of ${{\bf G}}$ determines a linearization of the action, and hence a GIT quotient ${W/\!\!/{{\bf G}}}=W/\!\!/_\theta {{\bf G}}$.

Under reasonable conditions on the triple $(W,{{\bf G}},\theta)$, certain stability conditions, depending on a parameter $\ke\in \QQ_{>0}\cup \{0+,\infty\}$ produce (relatively) proper Deligne-Mumford moduli stacks of $\ke$-[*stable quasimaps*]{} from pointed curves of genus 