A PERTURBATION AND GENERIC SMOOTHNESS OF THE VAFA-WITTEN MODULI SPACES ON CLOSED SYMPLECTIC FOUR-MANIFOLDS

We prove a Freed-Uhlenbeck style generic smoothness theorem for the moduli space of solutions to the Vafa-Witten equations on a closed symplectic four-manifold by using a method developed by Feehan for the study of the PU(2)-monopole equations on smooth closed four-manifolds. We introduce a set of perturbation terms to the Vafa-Witten equations, and prove that the moduli space of solutions to the perturbed Vafa-Witten equations on a closed symplectic four-manifold for the structure group SU(2) or SO(3) is a smooth manifold of dimension zero for a generic choice of the perturbation parameters.