Tangent space of the space of compatible complex structures Let $(M,\omega)$ be a symplectic manifold, and let $\mathcal{J}(M,\omega)$ denote the space of complex structures on $M$ which are compatible with $\omega$. I have been told the following fact:

We have an isomorphism $T_J\mathcal{J}(M,\omega)\cong\Omega^{0,1}(M)$, where $\Omega^{0,1}(M)$ is intended with respect to the complex structure $J\in\mathcal{J}(M,\omega)$.

I am trying to prove this fact, but without success until now. Does anyone have ideas, or better still a nice reference?
I will post if I find anything.

Note: The case that interests me the most is when $M$ is a compact, oriented surface (of genus $g_M\ge2$).
 A: Write $\mathcal{J}$ for the set of linear complex structure compatible by the standard symplectic form $\omega_0$ on $\mathbb{R}^{2}$. Given any symplectic vector space $(V, \omega)$ of dimension $2$, the existence of a symplectic basis implies that it is symplectomorphic to $(\mathbb{R}^{2}, \omega_0)$ and as such, $\mathcal{J}(V, \omega)$ - the set of linear complex structure on $V$ compatible with $\omega$ -  is diffeomorphic to $\mathcal{J}$.
Let $(M^{2}, \omega)$ be any symplectic surface and write $\mathcal{J}(M, \omega)$ for the space of complex structure on $M$ compatible with $\omega$. Consider the fiber bundle $\pi : E(M, \omega) \to M$ defined as the subbundle of $\mathrm{End}(TM) \to M$ whose fiber over $m \in M$ is $\mathcal{J}(T_mM, \omega)$ ; The typical fiber of $E(M, \omega)$ is thus $\mathcal{J}$. It appears that $\mathcal{J}(M, \omega)$ is the space of sections of $\pi$. The tangent bundle $TE(M, \omega)$ admits a subbundle $p : V E(M, \omega) \to E(M, \omega)$, the 'fiber tangent bundle', defined as the kernel of the pushforward $\pi_{\ast} : TE(M, \omega) \to TM$.
Let $J$ be a compatible almost complex structure on $(M, \omega)$, i.e. a section of $\pi$, and consider the restriction of $VE(M, \omega)$ to the image of $J$ in $E(M, \omega)$ : it can be described as the pullback bundle $J^{\ast}p : J^{\ast}VE(M, \omega) \to M$. The point is that $T_J \mathcal{J}(M, \omega)$ is identified with the space of sections of $J^{\ast}p$. As such, we have to show that $J^{\ast}VE(M, \omega) \cong T^{0,1}M $ as bundles over $M$, where the decomposition $T^{\ast}_{\mathbb{C}}M \cong T^{1,0}M \oplus T^{0,1}M$ of the complex cotangent bundle $T^{\ast}_{\mathbb{C}}M := T^{\ast}M \otimes_{\mathbb{R}} \mathbb{C}$ is with respect to $J$. Indeed, note that by definition, $(J^{\ast}p)^{-1}(m) \cong V_{J(m)}E(M, \omega) \cong T_{J(m)}\mathcal{J}(T_mM, \omega)$.
In order to do so, it suffices to show (in a 'natural way') that $(J^{\ast}p)^{-1}(m) \cong T^{0,1}_mM$. This follows from a general argument by Sévennec ; the proof is worked out for instance in Chapter II of the collective notes Holomorphic curves in symplectic geometry edited by Audin-Lafontaine. Sévennec's proposition goes as follow :
Fix a complex structure $J_0 \in \mathcal{J}(\mathbb{R}^{2n}, \omega_0)$. The map $J \mapsto (J+J_0)^{-1} \circ (J-J_0)$ is a diffeomorphism from $\mathcal{J}(\mathbb{R}^{2n}, \omega_0)$ onto the open unit ball in the vector space of (real) symmetric $n \times n$-matrices $S$ such that $J_0S + SJ_0=0$, that is the set of (real) symmetric $J_0$-antilinear matrices. It follows that $T_{J_0}\mathcal{J}(\mathbb{R}^{2n}, \omega_0)$ is isomorphic to this vector space.
In the special case when $n=1$, the set of such $S$'s (associated to $(T_mM, \omega, J)$) is a real two-dimensional vector space and it is naturally identifiable with the complex one-dimensional vector space $T^{0,1}_mM$. This proves the result. It also suggests what should be the result when $(M, \omega)$ is an arbitrary symplectic manifold.
N.B. - While Sévennec's diffeomorphism maps $\mathcal{J}(M, \omega)$ onto an open set of an affine space, the definition of $\mathcal{J}(M, \omega)$ does not make it anything near an affine subset of $\mathrm{End}(TM)$.
