Suppose $X,Y$ are two algebraic varieties over $\mathbb C$, and $f:X\to Y$ is a homomorphism. Then we can consider the relative de Rham complex $\mathcal H^i(X/Y):=\mathbf R^if_*(\Omega^{\bullet}_{X/\mathbb C})$. Then we get its Gauss Manin connection:$\epsilon :\mathcal H^i(X/Y)\to \Omega^1_{X/\mathbb C}\otimes \mathcal H^i(X/Y)$.
I just know the definition of it and have no basic knowledge about complex geometry. Could you please answer the following questions?
For any $y\in Y$, consider its fiber $X_y$ and the restriction of relative de Rham to $y$: $\mathcal H^i(X/Y)_y$. Do we have $\mathcal H^i(X/Y)_y\simeq H^i_{dR}(X_y/\kappa(y) )$?
In this paper, page 17, formula (3.7),(3.8) it claims: for any $y\in Y(\mathbb C)$, and pick a small neighborhood $U$ of $y$ in $Y(\mathbb C)$, then for any $y'\in U$, we have $H^i_{dR}(X_y/\kappa(y) )=H^i_{dR}(X_{y'}/\kappa(y') )$ by using Gauss Manin connection. I think it needs question 1, and maybe one can prove restrictions of relative de Rham are locally invariant. But I don't know how. Could you show the explicit reasoning?
By the way, if you have good references for answering these, it's also welcome to share .
Thanks ahead.