On the existence of pseudo-Riemannian metrics on the moduli space of symplectic structures (Q2486073)

The authors consider the moduli space of symplectic structures up to isotopy on a given compact symplectic manifold \((M, \omega )\). It is locally parametrized by an open subset of \(H^2(M)\). In a previous paper [\textit{J. Fricke} and \textit{L. Habermann}, Manuscr. Math. 109, 405--417 (2002; Zbl 1027.53110)] they discussed a symmetric covariant tensor field \(g\) on the above moduli space. They prove that \(g_\omega\) is non-degenerate if and only if every cohomology class in \(H^{2n-2}(M)\) has an \(\omega\)-harmonic representative, where \(2n =\dim M\). A result of \textit{O. Mathieu} [Comment. Math. Helv. 70, 1--9 (1995; Zbl 0831.58004)] asserts that any element of \(H^2(M)\) has an \(\omega\)-harmonic representative. As a corollary, they conclude that \(g\) is non-degenerate if \(\dim M = 4\). Finally, they apply the above non-degeneracy criterion to some examples of nilmanifolds of dimension six.