Normal forms for rank two linear irregular differential equations and moduli spaces

Karamoko Diarra (1) , Frank Loray (2)

1 USTTB - Université des Sciences, des Techniques et des Technologies de Bamako
2 IRMAR - Institut de Recherche Mathématique de Rennes

We provide a unique normal form for rank two irregular connections on the Riemann sphere. In fact, we provide a birational model where we introduce apparent singular points and where the bundle has a fixed Birkhoff-Grothendieck decomposition. The essential poles and the apparent poles provide two parabolic structures. The first one only depend on the formal type of the singular points. The latter one determine the connection (accessory parameters). As a consequence, an open set of the corresponding moduli space of connections is canonically identified with an open set of some Hilbert scheme of points on the explicit blow-up of some Hirzebruch surface. This generalizes to the irregular case a description due to Oblezin, and Saito-Szabo in the logarithmic case. This approach is also very close to the work of Dubrovin-Mazzocco with the cyclic vector.

Domaines: Analyse classique [math.CA]

Karamoko Diarra, Frank Loray. Normal forms for rank two linear irregular differential equations and moduli spaces. Periodica Mathematica Hungarica, 2022, 84, pp.303-320. ⟨10.1007/s10998-021-00408-8⟩. ⟨hal-02183978v3⟩