A classification of star log symplectic structures on a compact oriented surface (Q2287432)

On a manifold \(M\), a bi-vector field \(\pi\in C^{\infty}\left( M,\wedge ^{2}TM\right) \) with a vanishing Schouten-Nijenhuis bracket \(\left[ \pi ,\pi\right] =0\) is called a Poisson bi-vector, which defines a Poisson structure on \(M\), and the nondegenerate Poisson bi-vectors correspond dually to the symplectic 2-forms on \(M\). It is well known that on a compact connected surface \(S\), all bi-vector fields are Poisson and the nondegenerate ones are classified by the de Rham cohomology classes of the corresponding symplectic 2-forms. Roughly speaking, we may say that this paper classifies Poisson bi-vectors on \(S\) that are degenerate exactly on finitely many closed curves whose intersections are locally modeled as lines meeting at one point, extending \textit{O. Radko}'s classification [J. Symplectic Geom. 1, No. 3, 523--542 (2002; Zbl 1093.53087)] of Poisson bi-vectors linearly degenerate along a curve. Indeed, for a compact connected surface \(S\) with a given star divisor \(D\), i.e., a finite collection \(D=\left\{ Z_{1},\dots,Z_{n}\right\} \) of smooth curves in \(S\) that are pairwise transverse, the author first constructs a rank-2 vector bundle \(^{b}TS\rightarrow S\), the so-called \(b\)-tangent bundle, with smooth cross-sections realizing vector fields tangent along curves in \(D\) so that \(^{b}TS\rightarrow S\) becomes a Lie algebroid with a canonical anchor map \(^{b}TS\rightarrow TS\). A non-vanishing cross-section \(\omega\) of \(\wedge^{2}\left( ^{b}T^{\ast}S\right) \) with \(d\omega=0\) is called a log symplectic form, and the corresponding bi-vector \(\pi\in C^{\infty}\left( S,\wedge^{2}\left( ^{b}TS\right) \right) \) via duality defines a log symplectic Poisson structure on \(S\), which is degenerate exactly on curves in \(D\). Then it is shown that two log-symplectic forms on \(\left( S,D\right) \) are symplectomorphic if and only if they are cohomologous in the Lie algebroid cohomology \(^{b}H^{2}\left( S\right) \) of \(^{b}TS\rightarrow S\) and they give the same orientation on \(^{b}TS\). It is found that \(^{b}H^{2}\left( S\right) \) is a direct sum of \(H^{2}\left( S\right) \), \(H^{1}\left( Z_{i}\right) \), and \(H^{0}\left( Z_{i}\cap Z_{j}\right) \) with \(i<j\). Furthermore the Poisson cohomology of such a log symplectic Poisson structure \(\pi\) is computed. In particular, the degree-2 cohomology group \(H_{\pi}^{2}\left( S\right) \) is expressed as a direct sum of \(H^{2}\left( S\right) \), \(H^{1}\left( Z_{i}\right) \), \(\left[ H^{0}\left( Z_{i}\cap Z_{j}\right) \right] ^{2}\), \(\left[ H^{0}\left( Z_{i}\cap Z_{j}\cap Z_{k}\right) \right] ^{3}\), and \(\left[ H^{0}\left( Z_{i_{1}}\cap \cdots\cap Z_{i_{\ell}}\right) \right] ^{4}\) with \(\ell\geq4\), for each intersection of two, three, or more curves in \(D\) with strictly increasing indices. This result illustrates that the Poisson cohomology \(H_{\pi}^{2}\left( S\right) \) yields more information than the \(b\)-cohomology \(^{b}H^{2}\left( S\right) \).