The theory of virtual fundamental class defines important
invariants such as the Gromov–Witten and the Donaldson–
Thomas invariants. It has been generalized to the cosection
localized virtual cycle which has applications in Seiberg–
Witten, Fan–Jarvis–Ruan–Witten and other invariants. In
this paper, we prove the formulas of virtual pullback, torus
localization and wall crossing for cosection localized virtual
cycles.