# On the Hodge structures of compact hyperk\"ahler manifolds

## TL;DR

This paper explains how the Hodge structure on the second cohomology of a compact hyperk"ahler manifold uniquely determines the Hodge structures on all higher cohomology groups, clarifying a well-known but hard-to-find result.

## Contribution

It provides a clear account and proof of a folklore result linking the second cohomology's Hodge structure to all higher cohomologies in hyperk"ahler manifolds.

## Key findings

- Hodge structure on second cohomology determines all higher cohomology Hodge structures
- Clarification of the proof and statement of this folklore result
- Addresses the difficulty in locating this result in existing literature

## Abstract

The purpose of this note is to give an account of a well-known folklore result: the Hodge structure on the second cohomology of a compact hyperk\"ahler manifold uniquely determines Hodge structures on all higher cohomology groups. We discuss the precise statement and its proof, which are somewhat difficult to locate in the literature.