Torus localization and wall crossing for cosection localized virtual cycles

A Vanishing Associated With Irregular MSP Fields

“…We quote the relative version of cosection localized pullback in [CKL,Prop. 2.11], stated in [CKL,Remark 2.12]. The proof of [CKL,Prop.…”

“…2.11], stated in [CKL,Remark 2.12]. The proof of [CKL,Prop. 2.11] carries word by word to our case, such as W Γ /Y Γ,ν,e satisfies the "virtually smooth" condition in [CKL,(2.1)] because of (6.1).…”

“…The proof of [CKL,Prop. 2.11] carries word by word to our case, such as W Γ /Y Γ,ν,e satisfies the "virtually smooth" condition in [CKL,(2.1)] because of (6.1). The cosections setup are also consistent.…”

Localizing virtual fundamental cycles for semi-perfect obstruction theories

“…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 [17], the cosection localization [21] and their combination [6], to the setting of semi-perfect obstruction theory. As an application, we show that the Jiang-Thomas theory [20] of virtual signed Euler characteristic works without the technical quasi-smoothness assumption from derived algebraic geometry.A few effective techniques to handle virtual fundamental classes were discovered during the past two decades, such as the torus localization of Graber-Pandharipande [17], the degeneration method of J. Li [30] and the cosection localization [21].…”

“…Often combining these techniques turns out to be quite effective. In [6], it was proved that the torus localization works for the cosection localized virtual fundamental classes and this combined localization turned out to be quite useful for the Landau-Ginzburg/Calabi-Yau correspondence [10].…”