# Kuga-Satake construction and cohomology of hyperkahler manifolds

## TL;DR

This paper extends the Kuga-Satake construction to hyperkahler manifolds, embedding their cohomology into that of a torus while respecting Hodge structures and symmetries, revealing new geometric and algebraic relations.

## Contribution

It constructs a new embedding of hyperkahler manifold cohomology into a torus's cohomology, compatible with Hodge structures, pairings, and Lie algebra actions, generalizing previous Kuga-Satake results.

## Key findings

- Established an embedding of H^*(M,C) into H^{*+l}(T,C)
- Demonstrated compatibility with Hodge structures and Poincare pairing
- Showed the embedding respects Lefschetz sl(2)-actions

## Abstract

Let M be a simple hyperkahler manifold. Kuga-Satake construction gives an embedding of H^2(M,C) into the second cohomology of a torus, compatible with the Hodge structure. We construct a torus T and an embedding of the graded cohomology space H^*(M,C) \to H^{*+l}(T,C) for some l, which is compatible with the Hodge structures and the Poincare pairing. Moreover, this embedding is compatible with an action of the Lie algebra generated by all Lefschetz sl(2)-triples on M.