Moduli space of rank one logarithmic connections over a compact Riemann surface

Abstract

Let X be a compact Riemann surface of genus g ≥ 3. We consider the moduli space of holomorphic connections over X and the moduli space of logarithmic connections singular over a finite subset of X with fixed residues. We determine the Chow group of these moduli spaces. We compute the global sections of the sheaves of differential operators on ample line bundles and their symmetric powers over these moduli spaces, and show that they are constant under certain condition. We show the Torelli type theorem for the moduli space of logarithmic connections. We also describe the rational connectedness of these moduli spaces. Contents 1. Introduction and statements of the results 1 2. Preliminaries 4 2.1. Moduli spaces of holomorphic and logarithmic connections 4 3. Chow group of the moduli spaces 9 4. Differential operators on the moduli spaces 14 5. Torelli type theorem for the moduli spaces 21 6. Rational connectedness of the moduli spaces 28 References 29

Moduli space of logarithmic connections singular over a finite subset of a compact Riemann surface

“…In [5], the moduli space of rank n logarithmic connections singular exactly over one point has been considered and several properties, like algebraic functions, compactification and computation of Picard group have been studied. Also, the moduli space of rank one logarithmic connections singular over finitely many points with fixed residues has been considered in [15], [16], and it is proved that it has a natural symplectic structure and there are no non-constant algebraic functions on it.…”