Suppose $C/\mathbb{Q}$ is an algebraic curve (not necessarily complete) defined over $\mathbb{Q}$, and $p$ is a $\mathbb{Q}$ valued point of $C$. Suppose there is a algebraic fibration
\begin{equation}
\pi: Z \rightarrow C
\end{equation}
such that $Y=\pi^{-1}(p)$ is the only singular fiber, i.e.
\begin{equation}
\pi: X \rightarrow S
\end{equation}
where $X=Z \setminus Y$ and $S=C \setminus p$, is a smooth morphism between smooth varieties. Then by theorem 1 of the paper "On the differentiation of de Rham cohomology classes with respect parameters" by Katz and Oda, there is a canonical integrable connetion $\nabla_{GM}$ –Gauss-Manin connection–on the relative de Rham cohomology sheaf $R^q\,\pi_*(\Omega_{X/S}^*)$.
If we take extension of field and then analytification, we got maps
\begin{equation}
\pi: Z_{\mathbb{C}}^{an} \rightarrow C_{\mathbb{C}}^{an}\\
\pi:X_{\mathbb{C}}^{an} \rightarrow S_{\mathbb{C}}^{an}
\end{equation}
The Gauss-Manin connection now is the the analytification of $\nabla_{GM}$, and let us denote it by $\nabla_{GM}^{an}$. For the purpose of this question, let me assume that $\nabla_{GM}^{an}$ has regular singularities at $p$.
Then from the works of Griffiths and Schmid on variations of Hodge structures, there exists Deligne's canonical extension of the locally free sheaf $R^q\pi_*(\Omega_{X/S}^{*an})$ on $S_{\mathbb{C}}^{an}$ to a locally free sheaf on $C_{\mathbb{C}}^{an}$. The process of extending it over $p$ could be described explicitly as follows.
Choose a local coordinate in a neighbourhood of $p$, say $t$ and $p$ is the point with $t=0$, i.e. origin. Choose muli-valued local sections $e=(e_1,e_2,\cdots,e_n)^{t}$ of $R^q\pi_*(\Omega_{X/S}^{*an})$ in a neighbourhood of $0$ which form a multi-valued local frame. Suppose the monodromy matrix around $t=0$ with respect to thess muli-valued sections is $T$, define $N= \log T$, the vector
\begin{equation}
\exp (-\frac{\log t}{2 \pi i} \,N)\,e
\end{equation}
is single valued and we use it as a local frame to do the extension. This is Deligne's canonical extension in analytical case.
My questions are
1, With the assumption of $\nabla_{GM}^{an}$ is regular, is $\nabla_{GM}$ regular. Actually I am not clear with the definition of regularity of $\nabla_{GM}$ when it is defined over $\mathbb{Q}$, could someone explain it?
2, Could $R^q\,\pi_*(\Omega_{X/S}^*)$ be canonically extended to a locally free sheaf on $C$?
3, If such an extension exists, is there an explicit way like the analytical case to describe this extension using monodromy matrix?
4, If such an extension exists, after $\otimes_{\mathbb{Q}} \mathbb{C}$ and analytification, is this extension compatible with the Deligne's canonical extension in the analytical case?
Any comments and references will be appreciated.
Best Answer
A short answer is that Deligne's definition of canonical extension (https://publications.ias.edu/sites/default/files/71_Localbehavior.pdf) is not analytic, and works fine over $\mathbb Q$. Rather than the exponential you define, he defines a new connection via an algebraic formula, which is constant, and whose horizontal sections map the sections you define. Then he defines the canonical extension to be generated by the horizontal sections. This can be done algebraically by finding horizontal sections in a formal power series ring, say.
If more explanation is needed, my way of thinking of this is to consider everything in terms of the action of the operator $t \nabla$ on the sections of the vector bundle $R^q \pi_*$ over the field of formal Laurent series, $\mathbb Q((t))$ and $\mathbb C((t))$. The condition that $\nabla$ have regular singularities is, I believe, equivalent to the condition that $t \nabla$ can be put in Jordan normal form - i.e. an arbitrary section can be written as an infinite sum of sections that lie in a generalized eigenspace of $t \nabla$.
So the descent of this to $\mathbb Q((t))$ is the condition that an arbitrary section can be written as an infinite sum of sections in a finite-dimensional $t\nabla$-stable subspace.
In the case when the connection comes from algebraic geometry, one knows that all the eigenvalues of the monodromy, which are the same as the eigenvalues of $e^{ 2\pi i t\nabla}$, are roots of unity, so the eigenvalues of $t \nabla$ are rational numbers.
Hence one can simply define the canonical extension to be the sum of the subspaces generated by elements in generalized eigenspaces whose eigenvalue is a nonnegative rational number. (We want nonnegative because multiplying by $t$ adds $1$ to the eigenvalue.)
In fact, I think you want $N$ to be nilpotent, in which case all eigenvalues of $t\nabla$ are integers, and you can do the same thing with nonnegative integers. In fact you can take it to be the submodule generated by the subspace of sections in generalized eigenspaces with eigenvalue zero. (If a generator has eigenvalue a positive integer, we can lower it to zero by dividing by the corresponding power of $t$, so we may take generators to be those sections with eigenvalue $0$.) Using this description, you should be able to see it matches Deligne's characterization that the matrix of 1-forms defining the connection has logarithmic poles with nilpotent residues.