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There are the following two notions of "Gauss-Manin connection":

  1. The complex-analytic one: let $f:X\to S$ be a smooth family of complex manifolds. Then we obtain a local system $R^nf_{\ast}\mathbb{C}$ of complex vector spaces on $S$, defining a holomorphic vector bundle $\mathcal{V}=R^nf_{\ast}\mathbb{C}\otimes\mathcal{O}_S$ on $S$ with an integrable connection $\nabla :\mathcal{V}\to\mathcal{V}\otimes\Omega_S^1$. Now the vector bundle $\mathcal{V}$ can be identified with the relative de Rham cohomology $\mathcal{H}_{dR}^n(X/S)$ of the family, so we get a connection on the latter.

  2. The algebraic one: let $f:X\to S$ be a smooth morphism of smooth schemes over a field $k$. Now Katz and Oda in "On the differentiation of De Rham cohomology classes with respect to parameters" (J. Math. Kyoto Univ. 8 (1968), pp. 199-213) construct an integrable connection on $\mathcal{H}_{dR}^n(X/S)$ as some boundary map in a certain spectral sequence.

It is implicit in the literature that these two constructions are compatible, i.e. for a smooth family of smooth varieties over the complex numbers, the connection described in 1. is just the analytification of the one in 2. This sounds pretty reasonable as well. But thinking a bit about it, I was unable to come up with an argument, so could perhaps someone give me a hint where to find this or how to do it?

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Well since you only asked for a hint: Try constructing an analytic or perhaps $C^\infty$ version of Katz-Oda, which would easier to compare with 1. A somewhat round about alternative is deduce it from the statement that the Riemann-Hilbert equivalence commutes with direct image. This should be in Borel's book on D-modules. –  Donu Arapura Feb 12 '11 at 22:14

1 Answer 1

up vote 7 down vote accepted

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 $\mathscr{O}$-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 $(\mathscr O_X \otimes_{\mathbb{C}} V,\nabla)$.

Your second definition is a special case of the direct image $\mathcal{H}^n(f_+\mathscr O_X)$ 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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Thanks. However, this is not quite my point. One still has to show that the down-to-earth local systems construction gives the same as the spectral sequence construction, and this is where I am stuck. –  Robert Kucharczyk Feb 12 '11 at 23:17
    
I added a bit about compatibility of Riemann-Hilbert and direct images with references to Borel's Algebraic D-modules. Hope that answers your question. –  YBL Feb 13 '11 at 0:34
    
Thanks, this was the missing bit. –  Robert Kucharczyk Feb 13 '11 at 0:38

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