When I was reading "Advances in Moduli Theory" by Shimizu Yuji, I´ve found a weird way of writing the Kodaira-Spencer map $\rho$. For a given analytic family of complex compact manifolds $\pi :\mathcal{V} \twoheadrightarrow \mathcal{W} $, the author uses the notation $$\rho : \Theta_\mathcal{W} \longrightarrow R^1 \pi_* \Theta_{\mathcal{V/W}}$$ for the Kodaira-Spencer map given in each fiber by $\rho_w : \Theta_w \longrightarrow H^1(V_w,\Theta_{\mathcal{V_w}})$, such that $\Theta$ is the sheaf of holomorphic tangent vector fields. However what´s $\Theta_\mathcal{V/W}$ in the context of manifolds????!!!!
In scheme theory, $\Theta_{X/Y} = Hom_{\mathcal{O}_X} (\Omega^1_{X/Y}, \mathcal{O}_X)$, however for manifolds what´s the correct definition of relative differentials?
Is it possible to define $\Omega^1_{X/Y}$ for manifolds as people do for schemes??
Trying the analogous approach for some $f: X \rightarrow Y$ would lead in considering the map $\Delta_{X/Y}: X \longrightarrow X \times_Y X$ and, then pulling back the cotangent space $T^{*}(\Delta_{X/Y} (X))$. Apparently the action of pulling back and picking the dual commutes (I think so, but maybe it's wrong), so it would be enough to test if $\Delta_{X/Y}^* (T(\Delta_{X/Y})) \cong TX /Ker(f_*)$, however for a proper surjective submersion $f$ (as in the original case $\pi$) it looks like that $\Delta_{X/Y}^* (T(\Delta_{X/Y})) \cong TX$ (when drawing some "sketches", considering that $f$ is a locally trivial fibration).
Thanks in advance.
I'm not sure why you want to consider the fiber product but it is easy to define $\Omega_{X/Y}^1$: it is just the dual of $\Theta_{X/Y}$. The complex $\Omega_{X/Y}^\bullet = \Lambda^\bullet \Omega_{X/Y}^1$ has a natural differential, the vertical $\partial$ operator. This is defined on p. 116 of "Advances in Moduli Theory" as acting on holomorphic functions as the $\partial$ operator on the total space $X$ followed by restriction $\Omega_X^1 \to \Omega_{X/Y}^1$. I think this must be the relative differential you're asking about.
There is also a more intrinsic way to get this. The space $\Theta_{X/Y}$ is a holomorphic Lie algebroid via the Lie bracket of vector fields and the action on holomorphic functions by differentiation. In general if $A$ is any Lie algebroid then the dual exterior algebra $\Lambda^\bullet A^*$ becomes a dga with the differential given by $$ (d_A \mu)(v_0,\ldots,v_k) = \sum_{i=0}^k (-1)^i v_i \cdot \mu(v_0,\ldots,\hat{v_i},\ldots,v_k) \\+ \sum_{i<j} (-1)^{i+j}\mu([v_i,v_j], v_0, \ldots, \hat{v_i},\ldots,\hat{v_j},\ldots,v_k), $$ where $\mu \in \Lambda^k A^*, v_0,\ldots,v_k \in A$.