Let $U$ be a smooth algebraic variety defined over $k \hookrightarrow \mathbb{C}$. Let $\mathcal{E}$ be a locally free sheaf on $U$ equipped with an integrable connection

$\nabla: \mathcal{E} \to \mathcal{E} \otimes_{\mathcal{O}_U} \Omega^1_U$.

Let $X$ be a smooth variety including $U$ as the complement of a divisor $D$ with simple normal crossings.

**Definition**. $(\mathcal{E}, \nabla)$ has regular singularities along $D$ if there exists a coherent $\mathcal{O}_X$-module $\mathcal{E}_X$ and a logarithmic integrable connection $\nabla_X: \mathcal{E}_X \to \mathcal{E}_X \otimes \Omega^1_X(\log D)$ extending $(\mathcal{E}, \nabla)$, i.e. there exists an isomorphism $(\mathcal{E}_X, \nabla_X)_{|U} \cong (\mathcal{E}, \nabla)$.

For such connections, one can form the de Rham complex $$ DR(\mathcal{E}_X)=[\mathcal{E}_X \stackrel{\nabla_X}{\longrightarrow} \mathcal{E}_X \otimes \Omega^1_X(\log D) \to \cdots] $$ and take the hypercohomology of $X$ with values in this complex. Let me call it

$H_{dR}^i(U, (\mathcal{E}, \nabla)):=\mathbb{H}^i(X, DR(\mathcal{E}_X))$

By a fundamental theorem of Deligne, $H^i_{dR}(U, (\mathcal{E}, \nabla)) \otimes \mathbb{C}$ is isomorphic to $H^i(U^{an}, \ker(\nabla^{an}))$, the cohomology of $U^{an}$ with values in the local system of complex vector spaces $ker(\nabla^{an})$.

That's for the general theory. My question aims to understand what do these groups actually compute when $\mathcal{E}$ is the Gauss-Manin connection, which has regular singularities by a theorem of Griffiths. To fix the ideas, let $f: Y \to S$ be a smooth proper family of algebraic varieties and look at the (locally free) sheaves of relative cohomology $\mathcal{H}^i(Y/S)$. One has the Gauss-Manin connection

$\nabla^i_{GM}: \mathcal{H}^i(Y/S) \to \mathcal{H}^i(Y/S) \otimes \Omega^1_S$.

**Question**: What do the groups $H^j_{dR}(S, (\mathcal{H}^i(Y/S), \nabla^i_{GM}))$ compute? For instance, what is the relation between these groups and the de Rham cohomology of one particular fiber $Y_s$?

Somehow, I will expect a relation between $H^0_{dR}(S, (\mathcal{H}^i(Y/S), \nabla^i_{GM}))$, that is the global sections horizontal with respect to the connection and $H^i_{dR}(Y_s)$, but I don't know how to precise the relation.

I will be happy to understand the case of curves and abelian varieties. Do things get simpler? For instance, if $f: Y \to S$ is a family of curves over a curve S=$\bar{S}$ minus a finite set of points, does the first cohomology group of the connection on $\mathcal{H}^1(Y/S)$ vanishes?

Thanks for your help!