Simply connected varieties in characteristic \(p>0\) (Q2797471)

This paper concerns the interplay of the fundamental group and flat connections on smooth quasi-projective varieties.NEWLINENEWLINEIndeed, 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.NEWLINENEWLINEIn 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.)NEWLINENEWLINEFor a quasi-projective variety \(X\), one is thus led to consider the following refined problems (based on work of Kindler):NEWLINENEWLINE1. Does \(\pi_1^{\text{ét}}(X)=0\) imply the triviality of all \(\mathcal O_X\)-coherent \(\mathcal D_X\)-modules?NEWLINENEWLINE2. Does \(\pi_1^{\text{ét, tame}}(X)=0\) imply the triviality of all regular singular \(\mathcal O_X\)-coherent \(\mathcal D_X\)-modules?NEWLINENEWLINEThe 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).NEWLINENEWLINEA crucial input consists in the Lefschetz theorem for stratified bundles which rests on recent work of Bost as laid out in an extended appendix.