I have a problem with Kuranishi's theorem in deformation theory. I'll try to formulate it in general terms, and then describe the particular situation.
Let $\pi : M \to S$ be a smooth fiber bundle - i.e. $M$ and $S$ are smooth manifolds, and $\pi$ is a surjective submersion. There is an associated surjective morphism of vector bundles $ \pi_* : T_M \to \pi^* T_S$. I want to find a lifting of $\pi^* T_S$ into $T_M$.
Suppose I can find a smooth map $f : S \to M$ which satisfies $\pi \circ f = id_S$. This induces an injective map $T_S \to f^*T_M$. Can I lift this to a map $\pi^* T_S \to T_M$? How about if some extra data is given, like a metric on $S$, or a family of metrics $g_s$ on the fibers $T_M |_{M_s}$ (where $s$ is a parameter in $S$)?
Basically I'm trying to use Kuranishi's theorem to get something like Siu's canonical lifts. In this situation $M$ is the product of a fixed smooth manifold and the space of its complex structures, and $S$ is a complex manifold (open ball, even). The map $\pi$ is the passing to the quotient by the action of the group of diffeomorphisms. If we fix a hermitian metric $h$ on $M_0$, then Kuranishi gives a map $f : S \to M$ which satisfies the above hypothesis. I'm told that Kuranishi should induce a lifting of $\pi^*T_S$ into $T_M$, but I can't seem to figure out how.