  We study the moduli space of logarithmic connections of rank $2$ on
$\mathbb{P}^1 \setminus \{ t_1, \dots, t_5 \}$ with fixed spectral data. The
aim of this paper is to compute the cohomology of such space, and this
computation will be used to extend the results of Geometric Langlands
Correspondence due to D. Arinkin to the case where this type of connections
have five simple poles on $\mathbb{P}^1$.
