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The two constructions are compatible.

Your first definition of the Gauss-Manin connexion is $DR^{-1} (R f_* \mathbb{C}_X )$. Here $DR : D^b_{hr}(\mathcal{D}_X) \to D^b_c( \mathbb{C}_X )$ and $DR(\mathcal{M}) = \omega_X \otimes^L_{D_X} \mathcal{M}$ is the analytic de Rham complex. This is an equivalence by the Riemann-Hilbert correspondance. It sends an $\mathcal{O}$-coherent \mathscr{O}$-coherent$\mathcal{D}$-module (i.e. a vector bundle with an integrable connexion) to a local system (i.e. a locally constant sheaf). The inverse functor sends a locally constant$V$to the vector bundle$\mathscr O_X \otimes_{\mathbb{C}} V$together with the only connexion so that$V$is the local system of horizontal sections in$(O_X (\mathscr O_X \otimes_{\mathbb{C}} V,\nabla)$. Your second definition is a special case of the direct image$\mathcal{H}^n(f_+O_X)$\mathcal{H}^n(f_+\mathscr O_X)$ in the sense of D-modules for $f$ smooth. The (algebraic or analytic) de Rham complex on $X$ is filtered by $$L^r\Omega_X^\bullet = f^*\Omega_Y^r \otimes \Omega_X^{\bullet-r}$$ This induces a spectral sequence $$R^pf_*(Gr_L^q \Omega_X^\bullet) \Rightarrow R^{p+q}f_*(\Omega_X^\bullet)$$ But $Gr_L^q \Omega_X^\bullet = \Omega^q_S \otimes \Omega_{X/S}^{\bullet-q}$ and the Gauss-Manin connexion can be interpreted as the differential $$R^nf_*\Omega_{X/S} \to \Omega^1_S \otimes R^{n}f_*\Omega_{X/S}$$ in the spectral sequence. Now the analyfication functor is compatible with inverse and direct images of $\mathcal{O}$-modules and it sends $\Omega^i_X$ to $\Omega^i_{X^{an}}$ so it sends one spectral sequence to the other. This shows that the analytic and algebraic Gauss-Manin connexions are compatible.

It remains to prove is that $DR$ is compatible with direct images $$DR f_+ \mathcal{M} \overset{\sim}{\to} Rf_* DR \mathcal{M}$$ (for $f$ a smooth morphism of complex analytic varieties and $\mathcal{M} =\mathscr O_X$ a regular connexion).

For $f:X\to S$ an open immersion this is a theorem of Deligne (cf. Borel IV.6.1) and it is actually equivalent to regularity (by a theorem of Mebkhout I think).

For $f:X\to S$ proper this is done in (Borel VIII.15): analyfication induces the natural transformation $DR f_+ \to Rf_* DR$ and it is an isomorphism because of the projection formula. In the case $\mathcal{M} =\mathscr O_X$ you can make this a little bit more concrete. It is enough to prove that the natural transformation is an isomorphism on the fibers. But by proper base change, this means you can suppose $S$ is a point so this is equivalent to $H^n_{dR}(X^{an}) \to H^{n}(X^{an};\mathbb{C})$ being an isomorphism which is the Poincaré Lemma.

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The two constructions are compatible.

Your first definition of the Gauss-Manin connexion is $DR^{-1} (R f_* \mathbb{C}_X )$.

Here $DR : D^b_{hr}(\mathcal{D}_X) \to D^b_c( \mathbb{C}_X )$ $DR(\mathcal{M}) = \omega_X \otimes^L_{D_X} \mathcal{M}$ is the analytic de Rham complex. This is an equivalence by the Riemann-Hilbert correspondance. It sends an $\mathcal{O}$-coherent $\mathcal{D}$-module (i.e. a vector bundle with an integrable connexion) to a local system (i.e. a locally constant sheaf). The inverse functor sends a locally constant $V$ to the vector bundle $O_X \otimes_{\mathbb{C}} V$ together with the only connexion so that $V$ is the local system of horizontal sections in $(O_X \otimes_{\mathbb{C}} V,\nabla)$.

Your second definition is a special case of the direct image $\mathcal{H}^n(f_+O_X)$ in the sense of D-modules for $f$ smooth. The (algebraic or analytic) de Rham complex on $X$ is filtered by $$L^r\Omega_X^\bullet = f^*\Omega_Y^r \otimes \Omega_X^{\bullet-r}$$ This induces a spectral sequence $$R^pf_*(Gr_L^q \Omega_X^\bullet) \Rightarrow R^{p+q}f_*(\Omega_X^\bullet)$$ But $Gr_L^q \Omega_X^\bullet = \Omega^q_S \otimes \Omega_{X/S}^{\bullet-q}$ and the Gauss-Manin connexion can be interpreted as the differential $$R^nf_*\Omega_{X/S} \to \Omega^1_S \otimes R^{n}f_*\Omega_{X/S}$$ in the spectral sequence. Now the analyfication functor is compatible with inverse and direct images of $\mathcal{O}$-modules and it sends $\Omega^i_X$ to $\Omega^i_{X^{an}}$ so it sends one spectral sequence to the other. This shows that the analytic and algebraic Gauss-Manin connexions are compatible.

It remains to prove is that $DR$ is compatible with direct images $$DR f_+ \mathcal{M} \overset{\sim}{\to} Rf_* DR \mathcal{M}$$ (for $f$ a smooth morphism of complex analytic varieties and $\mathcal{M} = O_X$ a regular connexion).

For $f:X\to S$ an open immersion this is a theorem of Deligne (cf. Borel IV.6.1) and it is actually equivalent to regularity (by a theorem of Mebkhout I think).

For $f:X\to S$ proper this is done in (Borel VIII.15): analyfication induces the natural transformation $DR f_+ \to Rf_* DR$ and it is an isomorphism because of the projection formula. In the case $\mathcal{M} = O_X$ you can make this a little bit more concrete. It is enough to prove that the natural transformation is an isomorphism on the fibers. But by proper base change, this means you can suppose $S$ is a point so this is equivalent to $H^n_{dR}(X^{an}) \to H^{n}(X^{an};\mathbb{C})$ being an isomorphism which is the Poincaré Lemma.

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The two constructions are compatible.

Your first definition of the Gauss-Manin connexion is $DR^{-1} (R f_* \mathbb{C}X mathbb{C}_X )$. Here $DR : D^b{hr}(\mathcal{D}_X) D^b_{hr}(\mathcal{D}_X) \to D^b_c( \mathbb{C}_X )$ $DR(\mathcal{M}) = \omega_X \otimes^L_{D_X} \mathcal{M}$ is the analytic de Rham complex. This is an equivalence by the Riemann-Hilbert correspondance. It sends an $\mathcal{O}$-coherent $\mathcal{D}$-module (i.e. a vector bundle with an integrable connexion) to a local system (i.e. a locally constant sheaf). The inverse functor sends a locally constant $V$ to the vector bundle $O_X \otimes_{\mathbb{C}} V$ together with the only connexion so that $V$ is the local system of horizontal sections in $(O_X \otimes_{\mathbb{C}} V,\nabla)$.

Your second definition is a special case of the direct image $\mathcal{H}^n(f_+O_X)$ in the sense of D-modules for $f$ smooth. The (algebraic or analytic) de Rham complex on $X$ is filtered by $$L^r\Omega_X^\bullet = f^*\Omega_Y^r \otimes \Omega_X^{\bullet-r}$$ This induces a spectral sequence $$R^pf_*(Gr_L^q \Omega_X^\bullet) \Rightarrow R^{p+q}f_*(\Omega_X^\bullet)$$ But $Gr_L^q \Omega_X^\bullet = \Omega^q_S \otimes \Omega_{X/S}^{\bullet-q}$ and the Gauss-Manin connexion can be interpreted as the differential $$R^nf_*\Omega_{X/S} \to \Omega^1_S \otimes R^{n}f_*\Omega_{X/S}$$ in the spectral sequence. Now the analyfication functor is compatible with inverse and direct images of $\mathcal{O}$-modules and it sends $\Omega^i_X$ to $\Omega^i_{X^{an}}$ so it sends one spectral sequence to the other. This shows that the analytic and algebraic Gauss-Manin connexions are compatible.

It remains to prove is that $DR$ is compatible with direct images $$DR f_+ \mathcal{M} \overset{\sim}{\to} Rf_* DR \mathcal{M}$$ (for $f$ a smooth morphism of complex analytic varieties and $\mathcal{M} = O_X$ a regular connexion).

For $f:X\to S$ an open immersion this is a theorem of Deligne (cf. Borel IV.6.1) and it is actually equivalent to regularity (by a theorem of Mebkhout I think).

For $f:X\to S$ proper this is done in (Borel VIII.15): analyfication induces the natural transformation $DR f_+ \to Rf_* DR$ and it is an isomorphism because of the projection formula. In the case $\mathcal{M} = O_X$ you can make this a little bit more concrete. It is enough to prove that the natural transformation is an isomorphism on the fibers. But by proper base change, this means you can suppose $S$ is a point so this is equivalent to $H^n_{dR}(X^{an}) \to H^{n}(X^{an};\mathbb{C})$ being an isomorphism which is the Poincaré Lemma.

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