Rank one micro-locally free \(\hat {\mathcal E}\)-modules and non-integrable connexions in dimension two (Q2721645)

It is proved by A. D'Agnolo and P. Shapira that every micro-locally free \(\widehat{\mathcal E}\)-module of rank 1 is of the form \(\widehat{\mathcal E} \otimes_{\pi^{-1} {\mathcal O}}\pi^{-1} {\mathcal L}\) with a line bundle \({\mathcal L}\) on \(X\) for a complex manifold \(X\) of dimension \(\geq 3\).NEWLINENEWLINENEWLINEThe purpose of this paper is to show that the above theorem is not valid for a complex manifold \(X\) of dimension \(\leq 2\) by determining the structure of the micro-locally free \(\widehat {\mathcal E}\)- and \({\mathcal D}\)-modules of rank one. In order to do that, the author proves two theorems about categorical equivalence: one is the category of micro-locally free \(\widehat {\mathcal E}\)-modules of rank 1 and the other is that of micro-locally free \({\mathcal D}\)-modules of rank 1 without torsion.