Abstract
Recently, Chang and 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 of Graber–Pandharipande [T. Graber and R. Pandharipande, Localization of virtual cycles, Invent. Math. 135(2) (1999) 487–518], the cosection localization [Y.-H. Kiem and J. Li, Localizing virtual cycles by cosections, J. Amer. Math. Soc. 26(4) (2013) 1025–1050] and their combination [H.-L. Chang, Y.-H. Kiem and J. Li, Torus localization and wall crossing for cosection localized virtual cycles, Adv. Math. 308 (2017) 964–986], to the setting of semi-perfect obstruction theory. As an application, we show that the Jiang-Thomas theory [Y. Jiang and R. Thomas, Virtual signed Euler characteristics, preprint (2014), arXiv:1408.2541] of virtual signed Euler characteristic works without the technical quasi-smoothness assumption from derived algebraic geometry.
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