Cosection Localization for D-Manifolds and $(-2)$-Shifted Symplectic Derived Schemes, Revisited

This is a continuation of prior work of the author on cosection localization for d-manifolds. We construct reduced virtual fundamental classes for derived manifolds with surjective cosections and cosection localized virtual fundamental classes for $(-2)$-shifted symplectic derived schemes in larger generality. Moreover, using recent results of Oh-Thomas, we show that the algebraic and differential geometric constructions of reduced and cosection localized virtual fundamental classes of $(-2)$-shifted symplectic derived schemes yield the same result in homology. We obtain applications towards the construction and integrality of reduced invariants in Donaldson-Thomas theory of Calabi-Yau fourfolds.