Closed forms transverse to singular foliations (Q863442)

The transversal invariant measure \(\mu\) for an oriented \(p\)-dimensional foliation of \(M\) with singular set \(Z\) is called coherent relative to \(Z\) if for all \(\beta\in D^{p-1}(M)\) with \(d\beta\equiv 0\) in some neighbourhood of \(Z\) the current \(T_{\mu}\) associated to \(\mu\) satisfies \(T_{\mu}(d\beta) =0\). The author proves the following theorem: Let \(\mathcal{F}\) be a \(p\)-dimensional oriented foliation of the closed oriented manifold \(M\) with tame singular set \(Z\). Let \(i:Z\to M\) be the inclusion. Then a cohomology class \(\xi\in \ker i_p\subset H^p (M;\mathbb{R})\) admits a representative transversal to \(\mathcal{F}\) relative to \(Z\) if and only if for every measure \(\mu\) coherent relative to \(Z\) the associated homomorphism \(T_{\mu}:\ker i_p\to\mathbb{R}\) satisfies the condition \(T_{\mu}(\xi)>0\). The author derives from the above theorem results about the existence of singular symplectic forms and about the characterization of intrinsic harmonicity for forms.