Deformations of log-symplectic structures (Q2874656)

A Poisson structure or a Poisson \(2\)-tensor \(\pi\) on a \(2n\)-dimensional manifold \(M\) is called log-symplectic when it is nondegenerate or symplectic on the complement of a codimension-\(1\) submanifold \(Z\) along which \(\wedge ^{n}\pi\) is transversal to the zero section of the dual volume bundle \(\wedge^{2n}TM\). Developed in the context of \(b\)-manifold geometry, this paper deals with the classification of Poisson structures on a compact manifold perturbed from \(\pi\) by small diffeomorphisms, showing that such Poisson structures are parametrized by the second \(b\)-cohomology group \(H_{Z} ^{2}\left( M\right) \).NEWLINENEWLINEIn more explicit terms, when expressing \(\pi^{-1}\) as \(\alpha+d\log\left( \lambda\right) \wedge p^{\ast}\left( \theta\right) \) with a closed \(2\)-form \(\alpha\) on \(M\), \(\theta\in\Omega^{1}\left( Z\right) \) a closed \(1\)-form on \(Z\), \(p\;\)the projection to \(Z\) from a normed tubular neighborhood \(E\), and \(\lambda\) a function equal to the norm on the unit neighborhood of \(Z\), perturbations of \(\pi\) by small diffeomorphisms are found to be of the form \(\pi_{\gamma}^{\varpi}\) with \(\left( \pi_{\gamma}^{\varpi}\right) ^{-1} =\pi^{-1}+\varpi+d\log\left( \lambda\right) \wedge p^{\ast}\left( \gamma\right) \) for \(\left( \varpi,\gamma\right) \in\Omega^{2}\left( M\right) \times\Omega^{1}\left( Z\right) \), where the original cosymplectic structure \(\left( \alpha|_{Z},\theta\right) \) on \(Z\) with the volume form \(\theta \wedge\left( \alpha|_{Z}\right) ^{n-1}\) of \(Z\) is perturbed to the cosymplectic structure \(\left( \left( \alpha+\varpi\right) |_{Z} ,\theta+\gamma\right) \).