I want to find an explicit solution for Seiberg-Witten equations over $\mathbb{C}P^{2}$ with Fubini-Study metric.
The Seiberg-Witten equations are defined by:
\begin{align} F_{A}^{+}&=q(\psi)=\psi \otimes \psi^{\ast}-\dfrac{|\psi|}{2}\\ D_{A}(\psi) &=0 \end{align}
where $D_A$ is the Dirac operator which over $\mathbb{C}P^{2}$ is $\sqrt{2}(\partial+\overline{\partial}).$
I know that this solution is trivial because the scalar curvature of $\mathbb{C}P^{2}$ is positive.