Real structures and the Pin−(2)-monopole equations

We investigate the [Formula: see text]-monopole invariants of symplectic [Formula: see text]-manifolds and Kähler surfaces with real structures. We prove a nonvanishing theorem for real symplectic [Formula: see text]-manifolds which is an analogue of Taubes’ nonvanishing theorem of the Seiberg–Witten invariants for symplectic [Formula: see text]-manifolds. Furthermore, the Kobayashi–Hitchin type correspondence for real Kähler surfaces is given.