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 \((-2)\) $(-2)$ -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 \((-2)\) $(-2)$ -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.