This paper concerns the interplay of the fundamental group and flat connections on smooth quasi-projective varieties.
Indeed, over the complex numbers, a theorem of Malcev and Grothendieck asserts that the étale fundamental group \(\pi_1^{\text{ét}}(X)\) controls the regular singular flat connections, i.e. the regular singular \(\mathcal O_X\)-coherent \(\mathcal D_X\)-modules. In particular, there are non if \(\pi_1^{\text{ét}}(X)=0\). The proof makes substantial use of the underlying ground field as it builds on the topological fundamental group (which is finitely generated) and a certain specialization argument.
In positive characteristic, these tools are not at our disposal, but a similar result, due to Esnault and Mehta, holds true for projective varieties: if \(\pi_1^{\text{ét}}(X)=0\), then there are no non-trivial \(\mathcal O_X\)-coherent \(\mathcal D_X\)-modules. (Indeed, the profinite completion of the category of the latter is exactly \(\pi_1^{\text{ét}}(X)\), as proven by dos Santos.)
For a quasi-projective variety \(X\), one is thus led to consider the following refined problems (based on work of Kindler):
1. Does \(\pi_1^{\text{ét}}(X)=0\) imply the triviality of all \(\mathcal O_X\)-coherent \(\mathcal D_X\)-modules?
2. Does \(\pi_1^{\text{ét, tame}}(X)=0\) imply the triviality of all regular singular \(\mathcal O_X\)-coherent \(\mathcal D_X\)-modules?
The paper’s main result asserts that the first question has an affirmative answer if the ground field is algebraically closed and the quasi-projective variety \(X\) admits a normal compactification with boundary of codimension at least 2. Here the first assumption is related to specialization arguments (and to a theorem of Hrushovski) while the second gives rise to strong boundedness structures (notably on families of sheaves).
A crucial input consists in the Lefschetz theorem for stratified bundles which rests on recent work of Bost as laid out in an extended appendix.