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The module with integrable connection $(\mathcal{H}^i, \nabla^i_{GM})$ corresponds to the local system $Rf^{an}_* \mathbb{C}_{Y^{an}}$ on $S^{an}$, so the cohomology of the de Rham complex of $(\mathcal{H}^i, \nabla^i_{GM})$ will compute cohomology $H^j(S^{an}, R^if^{an}_* \mathbb{C}_{Y^{an}})$. By definition, the stalks of $R^if^{an}_* \mathbb{C}_{Y^{an}}$ are (singular) cohomology groups of the fibers of $f^{an}$. There is a Leray spectral sequence $$H^j(S^{an}, R^if^{an}_* \mathbb{C}_{Y^{an}}) \Rightarrow H^{i+j}(Y^{an}, \mathbb{C}_{Y^{an}})$$ which these isomorphisms identify with $$H^j(S, (\mathcal{H}^i, \nabla^i_{GM})) \Rightarrow H^{i+j}_{dR}(Y/\mathbb{C}).$$

Fix $s\in S$, then $\pi_1(S, s)$ homotopically acts on $Y^{an}_s = (f^{an})^{-1}(s)$, hence you get an action on $H^i(Y^{an}_s, \mathbb{C})$, called the monodromy action. For $j=0$, you get should get $$H^0(S^{an}, \mathcal{H}^{i, an})^{\nabla^{i, an}_{GM}} = H^0(S^{an}, (\mathcal{H}^{i, an}, \nabla^{i, an}_{GM})) = H^0(S^{an}, R^i f^{an}_* \mathbb{C}_{Y^{an}}) = H^i(S^{an}, Y_s)^{\pi_1(SH^i(Y^{an}_s, \mathbb{C})^{\pi_1(S, s)}.$$

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The Gauss-Manin module with integrable connection on $\mathcal{H}^i$ (\mathcal{H}^i, \nabla^i_{GM})$corresponds to the local system $Rf_* Rf^{an}_* \mathbb{C}_Y$, mathbb{C}_{Y^{an}}$ on $S^{an}$, so the cohomology of the de Rham complex of this connection $(\mathcal{H}^i, \nabla^i_{GM})$ will compute cohomology $H^j(S^{an}, R^if^{an}_* \mathbb{C}_{Y^{an}})$. By definition, the stalks of $R^if^{an}_* \mathbb{C}_{Y^{an}}$ are (singular) cohomology groups of the fibers of $f^{an}$. There is a Leray spectral sequence $H^j(S^{an},$H^j(S^{an}, R^if^{an}_* \mathbb{C}_{Y^{an}}) \Rightarrow H^{i+j}(Y^{an}, \mathbb{C}_{Y^{an}})$mathbb{C}_{Y^{an}})$$ which these isomorphisms identify with H^j(S, H^j(S, (\mathcal{H}^i, \nabla^i_{GM})) \Rightarrow H^{i+j}_{dR}(Y/\mathbb{C}). H^{i+j}_{dR}(Y/\mathbb{C}).$$ For$j=0$, you get should get $$H^0(S^{an}, \mathcal{H}^{i, an})^{\nabla^{i, an}_{GM}} = H^0(S^{an}, (\mathcal{H}^{i, an}, \nabla^{i, an}_{GM})) = H^0(S^{an}, R^i f^{an}_* \mathbb{C}_{Y^{an}}) = H^i(S^{an}, Y_s)^{\pi_1(S, s)}.$$ 1 The Gauss-Manin connection on$\mathcal{H}^i$corresponds to the local system $Rf_* \mathbb{C}_Y$, so the cohomology of the de Rham complex of this connection will compute cohomology $H^j(S^{an}, R^if^{an}_* \mathbb{C}_{Y^{an}})$. By definition, the stalks of $R^if^{an}_* \mathbb{C}_{Y^{an}}$ are (singular) cohomology groups of the fibers of$f^{an}$. There is a Leray spectral sequence $H^j(S^{an}, R^if^{an}_* \mathbb{C}_{Y^{an}}) \Rightarrow H^{i+j}(Y^{an}, \mathbb{C}_{Y^{an}})$ which these isomorphisms identify with $H^j(S, (\mathcal{H}^i, \nabla^i_{GM})) \Rightarrow H^{i+j}_{dR}(Y/\mathbb{C})\$.