  Since its introduction in 1995 by Li-Tian and Behrend-Fantechi, the theory of
virtual fundamental class has played a key role in algebraic geometry, defining
important invariants such as the Gromov-Witten invariant and the
Donaldson-Thomas invariant. Quite a few methods for handling the virtual
fundamental classes were discovered such as torus localization, degeneration,
virtual pullback and cosection localization. Often combining these methods
turns out to be quite effective. In this paper, we prove virtual pullback,
torus localization and wall crossing formulas for cosection localized virtual
cycles.
