Real structures and the $\mathrm{Pin}^-(2)$-monopole equations

TL;DR

This paper studies $ ext{Pin}^-(2)$-monopole invariants in symplectic and K"ahler 4-manifolds with real structures, establishing nonvanishing results and a Kobayashi-Hitchin correspondence.

Contribution

It introduces new invariants for real symplectic and K"ahler surfaces, proving a nonvanishing theorem and a Kobayashi-Hitchin type correspondence.

Findings

Proved nonvanishing theorem for real symplectic 4-manifolds.

Established Kobayashi-Hitchin correspondence for real K"ahler surfaces.

Extended Seiberg-Witten theory to real structures.

Abstract

We investigate the $\mathrm{Pin}^-(2)$-monopole invariants of symplectic $4$-manifolds and K\"{a}hler surfaces with real structures. We prove the nonvanishing theorem for real symplectic $4$-manifolds which is an analogue of Taubes' nonvanishing theorem of the Seiberg-Witten invariants for symplectic $4$-manifolds. Furthermore, the Kobayashi-Hitchin type correspondence for real K\"{a}hler surfaces is given.