Poisson structures on twistor spaces of hyperkaehler and HKT manifolds

TL;DR

This paper characterizes HKT structures via Poisson bivectors on hypercomplex manifolds and demonstrates that twistor spaces of hyperkaehler manifolds admit holomorphic Poisson structures, linking to deformations on tori and K3 surfaces.

Contribution

It extends the characterization of HKT structures to twistor spaces and shows the existence of holomorphic Poisson structures on twistor spaces of hyperkaehler manifolds.

Findings

Twistor space of hyperkaehler manifolds admits a holomorphic Poisson structure.

HKT structures are characterized by nondegenerate complex Poisson bivectors.

Connections to quaternionic and hypercomplex deformations on tori and K3 surfaces.

Abstract

We characterize HKT structure in terms of nondegenrate complex Poisson bivector on hypercomplex manifold. We extend the characterization to the twistor space. After considering the flat case in detail, we show that the twistor space of hyperkaehler manifold admits a holomorphic Poisson structure. We briefly mention the relation to quaternionic and hypercomplex deformations on tori and K3 surfaces