Spaces of Hermitian triples and the Seiberg-Witten equations (Q2759283)

Let \(X\) be a smooth, closed, orientable \(4\)-manifold. A Hermitian triple for \(X\) is a triple \((g,J,\omega)\) formed by a Riemannian metric \(g\), an almost complex structure \(J\) and a non-degenerate (not necessarily closed) \(2\)-form \(\omega\) with \(\omega(u,v)= g(u,Jv)\), for all tangent vectors \(u,v\). NEWLINENEWLINENEWLINEThe canonical map \(\tau\) sends a Hermitian triple \((g,J,\omega)\) to the cohomology class defined by the harmonic projection of \(\omega\) with respect to \(g\). Seiberg-Witten solutions allow to construct Hermitian triples with non-zero value of \(\tau\) and within a prescribed homotopy class of almost-complex structures (that is, fixing the canonical class \(K_{J}\)). NEWLINENEWLINENEWLINEIf \(K_{0}\) is simultaneously a Seiberg-Witten basic class and a canonical class (which is the case if \(X\) admits a symplectic structure), then the image under \(\tau\) of the almost complex structures with \(K_{J}=K_{0}\) is a dense subset of the positive cone \(H^{+}\) inside the degree-\(2\) cohomology. NEWLINENEWLINENEWLINETwo natural \(\text{Diff}^{+}(X)\)-invariant equations are discussed at the end of the paper: \(\tau=[0]\) and \(\tau=K_{J}\).