Let $\pi:X \to D^2$ be a family of diffeomorphic (but not isomorphic) complex manifolds. Each fiber is allowed to have boundary but is compact (maybe not Stein) and $D^2 \subset \mathbb{C}$ is a complex disk. Let $X_0:= \pi^{-1}(0)$ and let $f:X_0 \to \mathbb{C}$ be a holomorphic map.
I am looking for obstructions to extend $f$ to a smooth family of holomorphic maps $f_t:X_t \to \mathbb{C}$ with $t \in D^2$. That is, to find a complex-valued smooth function $F:X \to \mathbb{C}$ such that $F|_{X_t}$ is holomorphic for all $t \in D^2$.
I have seen some works on the literature about extending $f$ to a holomorphic map on all $X$ (or equivalently, that ask for holomorphicity on $t$) which is a lot stronger and in general not possible at all.
A good answer could also be a reference that deals with this kind of framework even if not exactly the same.
Another question is if the following is the right approach:
Take $T_R B$ to be the sheaf of germs of real analytic (or smooth) vector fields on $B$ and take the short exact sequence $$0 \to T_R X_0 \to T_RX_{|_{X_0}} \to \pi^*(T_R D^2)_{|_{X_0}} \to 0$$ then take the corresponding cohomology exact sequence and look for the connecting homomorphism $$\rho:H^0(X, \pi^*(T_R D^2)_{|_{X_0}}) \to H^1(X, T_RX).$$ Now if for $f \in H^0(X_0, \mathcal{O}_{X_0})$, one can find a vector $v \in H^0(X, \pi^*(T_R D^2)_{|_{X_0}})$ that doesn't kill $f$ then the extension that I am looking for exists. Is this true?
I am an amateur in deformation theory so sorry in advance if this is just all nonsense.
