# On the Cohomology of Moduli Space of Parabolic Connections

## TL;DR

This paper computes the cohomology of the moduli space of rank 2 logarithmic connections on the punctured projective line with five poles, extending Geometric Langlands results to this specific case.

## Contribution

It provides the first cohomology computation for this moduli space, enabling extension of Geometric Langlands Correspondence to five-pole connections.

## Key findings

- Cohomology of the moduli space explicitly computed
- Extension of Geometric Langlands Correspondence demonstrated
- New insights into parabolic connection moduli spaces obtained

## Abstract

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 $\mathbb{P}^1$.