Hello,


Let $X$ be a smooth variety in char. 0. Let us call a $D$-module on $X$ constant, if it is isomorphic to a finite direct sum of the $D$-modules $O$ (the sheaf of regular functions with the usual $D$-module structure).
Then a subquotient of a constant $D$-module should be constant. But how to show it?
The reason why it should be true is that Riemann-Hilbert correspondence (when over the complex numbers) translates as to representations of a the fundamental group, and there such a statement is trivial.


Thanks,
Sasha



A submodule of a constant D-module is constant
If your base field $k$ is algebraically closed (or if $X$ has a rational point), then this follows directly from the fact that the category of $\mathcal{O}\_X$-coherent flat connections is neutral Tannakian, i.e. equivalent to the category of finite dimensional $k$-representations of some affine $k$-group scheme $G$. For this fact the Riemann-Hilbert correspondence is not needed, it just gives you additional information about $G$.


Here is a more direct argument: If $E$ is a $D$-module which is coherent as an $\mathcal{O}\_X$-module, then $E$ is automatically locally free. In particular, if $E$ is a sub $D$-module of the "constant" $D$-module $\mathcal{O}\_X^n$, then the short exact sequence of $D$-modules


$$0\rightarrow E \rightarrow \mathcal{O}^n\_X \rightarrow \mathcal{O}^n\_X/E\rightarrow 0$$
is locally split as a sequence of $\mathcal{O}\_X$-modules.


We see that if $n=1$, then $E=0$ or $E=\mathcal{O}\_X$. We proceed by induction.


Since a $D$-module is trivial if and only if there exists a dense open subset of $X$ on which it is trivial, we may work in the local ring of a closed point $x\in X$ (lets assume $X$ to be connected…).


If $e\_1,\ldots, e\_n$ is a horizontal basis, then the horizontal sections of $\mathcal{O}\_X^n$ are precisely those sections which are in the $k$-span of $e\_1,\ldots, e\_n$. The group of $D$-automorphisms of $\mathcal{O}\_X^n$ then identifies with $GL\_n(k)$. Thus if $E$ contains a horizontal section, we are done by induction. The case $n=1$ implies that if $E\cap e\_i\mathcal{O}\_X\neq 0$ for some $i$, then $e\_i\in E$. If $f\_1e\_1+\ldots+f\_n e\_n$ is a section of $E$, which is not horizontal, then some $f\_i\in \mathcal{O}\_{X,x}$ is nonconstant, say $f\_1$. One finds a differential operator $\partial$, such that $\partial(f\_1)\in \mathcal{O}\_{X,x}^\times$, and concludes that $E$ intersects the $\mathcal{O}\_X$-span of $e\_2,\ldots, e\_n$ nontrivially. Hence, either $e\_1\in E$, or $E\subset \bigoplus\_{i\geq 2} e\_i\mathcal{O}\_X$, and in both cases we are done.


