Cohomology of vector bundles on the moduli space of parabolic connections on ${\mathbb{P}}^1$ minus $5$ points

We study the moduli space of parabolic connections of rank two on the complex projective line ${\mathbb{P}}^1$ minus five points with fixed spectral data. This paper aims to compute the cohomology of the structure sheaf and a certain vector bundle on this space. We use this computation to extend the results of Arinkin, which proved a specific Geometric Langlands Correspondence to the case where these connections have five simple poles on ${\mathbb{P}}^1$. Moreover, we give an explicit geometric description of the compactification of this moduli space.