Convergence for a family of poisson structures

Question

I would like to get some references. I hope that somebody helps.

Let $(M,\Pi)$ be a smooth Poisson manifold. Let $\delta:\mathcal{V}^{.}(M)\to \mathcal{V}^{.}(M)$ be a differential operator on the space of multivector fields on $M$, defined by $\delta(A)=[\Pi, A]$. Then we have a differential complex \begin{align*} \cdots \to \mathcal{V}^{p-1}(M)\xrightarrow{\delta} \mathcal{V}^p(M)\xrightarrow{\delta}\mathcal{V}^{p+1}(M)\to \cdots \end{align*}

Then third Poisson cohomology $H_{\Pi}^3(M)$ can be interpreted as an obstruction. Now we assume that $H_{\Pi}^3(M)=0$ and we have a $\Lambda \in \mathcal{V}^2(M)$ such that $[\Pi,\Lambda]=0$. Then we can get a formal family $\Lambda(t)=\Pi+t\Lambda+t^2\Lambda_2+\cdots$ of poisson structures such that $[\Lambda(t),\Lambda(t)]=0$ inductively.

I am interested in convergent issue. Can we get a smooth family for sufficiently small $t$ if we choose carefully $\Lambda_2, \Lambda_3,...$. I hope that someone give me some references on Analysis to deal with this convergent issue if it exists.

Answer 1

In general, this seems not to be possible. Consider the case of a vanishing Poisson structure $\Pi = 0$. Then every bivector field $\Lambda$ is closed for the differential $\delta$, which is just the zero map $\delta = 0$. So the wanted $\Lambda(t)$ should satisfy the Jacobi identity $[\Lambda(t), \Lambda(t)] = 0$ and it should be smooth (or even analytic) in $t$ with first derivative at $t = 0$ given by $\Lambda$. Differentiating this at $t = 0$ twice gives the Jacobi identity $[\Lambda, \Lambda] = 0$ for the original $\Lambda$. So this is a real obstruction... So this only shows that the continuation of an infinitesimal deformation to a honest (either formal or even analytic) deformation is generally very difficult.