Normal forms for rank two linear irregular differential equations and moduli spaces
Abstract
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.
Origin : Files produced by the author(s)
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