Transversality of the perturbed reduced Vafa-Witten moduli spaces on 4-manifolds

Previously we finish the establishment of the transversality of the general part of the Vafa-Witten moduli spaces, in this paper, we deal with the rest, i.e., the reduced part. We consider Vafa-Witten equation on closed, oriented and smooth Riemann 4-manifolds with $C\equiv 0$, and construct perturbation to establish the transversality of the perturbed equation. We show that for a generic choice of the perturbation terms, the moduli space of solutions to the perturbed reduced Vafa-Witten equation for the structure group $SU\left(2\right)$ or $SO\left(3\right)$ on a closed 4-manifold is a smooth manifold of dimension zero. Finally we prove that for two generic orientation-preserving parameters, the corresponding moduli spaces are cobordant, and the method can also be applied to the general part.