Positive-dimensional Seiberg-Witten moduli spaces

Question

I am looking for examples of (symplectic or not) 4-dimensional manifolds $X$ that have positive dimensional Seiberg-Witten moduli spaces (and $b^{2+}>1$). Of course, the result/conjecture is that the resulting SW invariants should be zero (see page 261 and Corollary 13.22 of Salamon's book).

Answer 1

Exercise: The non-symplectic $K3\#n(S^1\times S^3)$ for $n>0$ with the spin-c structure induced from the canonical one on $K3$; it will have $n$-dimensional SW moduli.

If you want symplectic examples: Assume for simplicity on a symplectic 4-manifold $(X,\omega)$ that $[\omega]\in H^2(X;\mathbb Z)$. A theorem of Donaldson says that for $k\in\mathbb N$ sufficiently large ($k\ge k_0$ where $k_0$ depends not only on the geometry of $X$ but also on $\omega$), $k[\omega]$ is Poincaré dual to an embedded symplectic surface $\Sigma\subset X$. We can build an $\omega$-compatible $J$ (hence many) for which $\Sigma$ is a $J$-holomorphic curve corresponding to the spin-c structure $s_k=s_\omega+k[\omega]$, where $s_\omega$ is the canonical spin-c structure. Then for $J$ generic the $\omega$-perturbed SW moduli space with respect to $s_k$ must be nonempty (via Taubes’ SW = Gr theorem), of even index $k(k[\omega]^2-K_\omega\cdot[\omega])>0$ where $K_\omega$ is the canonical class.

Other examples are given in Jabuka's paper "Symplectic surfaces and generic J-holomorphic structures on 4-manifolds".