I thought in asking this question on Math StackExchange, but by my experience I don' t think anyone will notice me. Recently, I started studying deformation of complex manifolds in the sense of Kodaira and Spencer, but some things relating infinitesimal approximations are not clear for me.


Let $\varpi: \mathscr{M} \twoheadrightarrow D$ be a analytic family of compact complex manifolds such that $D \in \mathbb{C}^m$ is disk centered at the origin, $M = M\_0$ is covered by coordinate charts $\{ U\_i \}$ and the radius of $D$ is small enough to $\{ U\_i \times D \}$ cover $\mathscr{M}$. Now, let $(\zeta\_i^1 (z\_i , t), …,\zeta\_i^n(z\_i^n, t), t^1, …,t^m)$ denote the coordinates of $U\_i \times D$, then, in the literature, without any formal justificative, it's stated that $$\frac{\partial \zeta\_i^j}{\overline{\partial z\_i}^k} = \sum\_{l} \varphi\_{lk}^i (t) \frac{\partial \zeta\_i^j}{\partial z\_i^l}$$ 
holds. The informal justificative usually done is by saying that if $t$ is close enough to $0$, then $pr^{(0, 1)}|\_{T\_t^{(0, 1)}} : T\_t^{(0, 1)} \rightarrow T^{(0, 1)} $ defines an isomorphism. So it is possible to construct a map $$\varphi(t) = pr^{(1, 0)}\circ (pr^{(0, 1)}|\_{T\_t^{(0, 1)}})^{-1}: T^{(0, 1)} \longrightarrow T^{(1, 0)} $$ and it satisfies $(1 + \varphi(t)) (v) \in T\_t^{(0, 1)}$ for all $v \in T^{(0, 1)}$. 


Clearly, $pr^{(0, 1)}|\_{T\_t^{(0, 1)}} : T\_t^{(0, 1)} \rightarrow T^{(0, 1)} $ being an isomorphism is the same as the complex strutcture being the same for all $t$, **so how is possible to make this construction of $\varphi$ more formal? Why the equality above holds?**


Maybe the assumption "t close enough to 0" should be formalized in terms of nilpotent elements, something like a kind of prorepresentable functor, such that $pr^{(0, 1)}|\_{T\_t^{(0, 1)}} : T\_t^{(0, 1)} \rightarrow T^{(0, 1)} $ being an isomorphism makes sense.


Thanks in advance.


**EDIT** As noted by Peter Dalakov, the assertion that the splitting of $T\_{\mathbb{C}}$ does not change, is totally wrong. So the remaining question is: why $$\frac{\partial \zeta\_i^j}{\overline{\partial z\_i}^k} = \sum\_{l} \varphi\_{lk}^i (t) \frac{\partial \zeta\_i^j}{\partial z\_i^l}$$ holds?



Kodaira-Spencer theory of deformation done right
I am not sure which book you are using . But from Kodaira, Morrow's book $\it complex$ $\it manifolds$ page 149-151, the construction of vector-value $(0,1)-$form $\varphi$ will tell you why the equality holds, also you will see why $t$ is required to be close enough to 0. 


