Suppose that $X_t$ and $Y_t$ are two families of complex manifolds, parametrized by maps $\sigma : \mathcal X \to T$ and $\tau : \mathcal Y \to T$. Suppose too I have holomorphic maps $f_t : X_t \to Y_t$ defined for all $t \neq 0$. I want to know what it means for the maps $f_t$ to converge to a limit $g : X_0 \to Y_0$. In particular I'd like to know the right analogs of i) pointwise convergence and ii) uniform convergence on compact subsets. The latter should guarantee that $g$ is holomorphic.
What are the magic words I should be looking up?
The basic problem is that without an identification of $X_s$ with $X_t$, I don't see a sensible notion of "pointwise" convergence. Similarly there's not a good way to transform compacts from fiber to fiber. My first guess would be: for any section $\alpha : U \to \sigma^{-1}(U)$, where $U \ni 0$ is an open in $T$, we get $f_u(\alpha(u))$ converges to $g(\alpha(0))$ as $u \to 0$. For uniform convergence, I don't know.