Canonical coordinates for moduli spaces of rank two irregular connections on curves.

Arata Komyo (1) , Frank Loray (2) , Masa-Hiko Saito (3) , Szilard Szabo (4)

1 Department of Mathematical Sciences, University of Hyogo
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 KobeGakuin University
4 Alfréd Rényi Institute of Mathematics

In this paper, we study a geometric counterpart of the cyclic vector which allow us to put a rank 2 meromorphic connection on a curve into a “companion” normal form. This allow us to naturally identify an open set of the moduli space of GL2-connections (with fixed generic spectral data, i.e. unramified, non resonant) with some Hilbert scheme of points on the twisted cotangent bundle of the curve. We prove that this map is symplectic, therefore providing Darboux (or canonical) coordinates on the moduli space, i.e. separation of variables. On the other hand, for SL2-connections, we give an explicit formula for the symplectic structure for a birational model given by Matsumoto. We finally detail the case of an elliptic curve with a divisor of degree 2.

Domaines: Géométrie algébrique [math.AG]

Arata Komyo, Frank Loray, Masa-Hiko Saito, Szilard Szabo. Canonical coordinates for moduli spaces of rank two irregular connections on curves.. 2023. ⟨hal-04201444⟩