The canonical spectral sequences for Poisson manifolds
Let \((M,G)\) be a \(C^\infty\) Poisson manifold, where \(G\) is the Poisson tensor. Let \(\Lambda^k(M)\) denote the space of the differential \(k\)-forms, \(i(G)\) the interior product and \(d\) the exterior differentiation, let \(\delta=i(G)\circ d-d\circ i(G):\Lambda^{k+1}\rightarrow\Lambda ^k(M)\) define the canonical complex \[ \ldots\rightarrow\Lambda^{k+1}(M)\rightarrow\Lambda^k(M) \rightarrow\Lambda^{k-1}(M)\rightarrow\ldots \] The homology of this complex is denoted by \(H_*^{\text{can}}(M)\) and called the { canonical homology} of \(M\). One considers also the { periodic double complex} \({\mathcal E}^{\text{per}}_{*,*}(M)\) defined by \({\mathcal E}^{\text{per}}_{p.q}(M)=\Lambda^{q-p}(M)\), for all \(p,q\in {\mathbb Z}\), which has \(d\) for horizontal differential and \(\delta\) for vertical differential, both of degree \(-1\). Associated with the latter, one has two spectral sequences \(\{E^r(M)\}\) and \(\{' E^r(M)\}\), both converging to the total homology \(H_*^D(M)\), i.e., to the homology of the total complex \(({\mathcal E}_k(M),D)\), where \({\mathcal E}_k(M)=\bigoplus_{p+q=r} \Lambda^{q-p}(M)\) and \(D=d+\delta\). It is known after Brylinski that when \(M\) is a compact \(2n\)-dimensional symplectic manifold, the canonical homology group \(H_k^{\text{can}}(M)\cong H_{\text{de Rham}}^{2n-k}(M)\), and that the first spectral sequence \(\{E^r(M)\}\) degenerates at \(E^1(M)\) (i.e., \(E^1(M)\cong E^\infty(M))\). In this paper, the authors show that the preceding theorem does not hold for arbitrary Poisson manifolds. In fact, they exhibit an example of a 5-dimensional compact Poisson manifold \(M^5\) (which is cosymplectic), for which \(H_1^{\text{can}}(M^5)\not\cong H^4(M^5)\), and \(E^1(M^5) \not\cong E^2(M^5)\).
scientific article; zbMATH DE number 1216489
scientific article