Let $X$ be a smooth complex analytic space and let $D$ be the unit disk in $\mathbb{C}$. Let $\omega:Y \to D$ be a deformation of complex structures of $X$ in the sense that (1) $\omega^{-1}(0) \simeq X$ as complex spaces, (2) $\omega$ is a proper map of complex spaces of maximal rank at every point.
My questions is the following: if the complex structure $J$ on $X$ gives the same information as the sheaf of holomorphic functions on $X$. Is it true that a family of complex structures $J_{t}$ gives rise to an homotopy of all the complex functions of $X$ that end in holomorphic functions of $X_t$ equipped with $J_t$? In other words, if $g:X \to \mathbb{C}$ is holomorphic and I fix a path $\gamma:[0,1] \to D, \gamma(0)=0$, does there exist an homotopy $g_t$ along continuous functions such that $g_t:X_t \to \mathbb{C}$ is holomorphic for $J_t$?
If not, when is this true?
