Definition A compact complex manifold M is called rigid if $H^1(M,\Theta_M) = 0$
$\Theta_M$ is the sheaf of holomorphic sections of the complex tangent bundle.
Let's consider a proper map $f : M \rightarrow B$ whose differential is everywhere surjective. Let's suppose $B$ is connected and choose a point $b \in B$, set $M_b = f^{-1}(b)$. My question is:
If $M$ is rigid then $H^1(M_b, \Theta_{M_b}) = 0$ for every choice of $b \in B$?
I guess my claim is true but I don't find a way to prove it. I thought I could use some exact sequence of sheaves and then the induced long exact sequence but I didn't succeed in applying this idea.
Any help would be very appreciated. Thank you in advance!