# Cohomologies on hypercomplex manifolds

## TL;DR

This paper reviews cohomological properties of complex and hypercomplex manifolds, highlighting differences and similarities, especially focusing on compact cases and specific holonomy conditions.

## Contribution

It provides a comparative analysis of cohomologies in complex and hypercomplex manifolds, emphasizing the case of 8-dimensional hypercomplex manifolds with special holonomy.

## Key findings

- Differences between cohomological aspects of complex and hypercomplex manifolds
- Similarities between compact complex surfaces and certain hypercomplex manifolds
- Insights into the structure of hypercomplex manifolds with Obata connection in SL(2,H)

## Abstract

We review some cohomological aspects of complex and hypercomplex manifolds and underline the differences between both realms. Furthermore, we try to highlight the similarities between compact complex surfaces on one hand and compact hypercomplex manifolds of real dimension 8 with holonomy of the Obata connection in SL(2,H) on the other hand.