Localizing virtual fundamental cycles for semi-perfect obstruction theories

Recently H.-L. Chang and J. Li generalized the theory of virtual fundamental class to the setting of semi-perfect obstruction theory. A semi-perfect obstruction theory requires only the local existence of a perfect obstruction theory with compatibility conditions. In this paper, we generalize the torus localization, the cosection localization and their combination, to the setting of semi-perfect obstruction theory. As an application, we show that the Jiang-Thomas theory of virtual signed Euler characteristic works without the technical quasi-smoothness assumption from derived algebraic geometry.