  Let $X$ be a compact Riemann surface of genus $g \geq 3$. 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.
