On the type of generalized hypercomplex structures

TL;DR

This paper explores the classification and properties of generalized hypercomplex structures within generalized geometry, revealing new examples on tori and surfaces, and analyzing their types and associated twistor spaces.

Contribution

It introduces new examples of generalized hypercomplex structures on tori and surfaces, and studies their types and twistor spaces, expanding understanding of their geometric properties.

Findings

Existence of generalized hypercomplex structures on 4n-dimensional tori without maximal complex type.

The Kodaira-Thurston surface admits a generalized hypercomplex structure with all structures of type 1.

Analysis of the types of generalized complex structures and their twistor spaces.

Abstract

The generalized hypercomplex structures defined within the framework of generalized geometry include hypercomplex and holomorphic symplectic structures as particular cases. They have a $S^2$-family of generalized complex structures, and in this paper we study the types of these structures and the corresponding twistor space. We show that there are generalized hypercomplex structures on the $4n$-dimensional tori, which do not contain a structure of maximal (complex) type. Moreover, we show that the Kodaira-Thurston surface which has a holomorphic symplectic structure, admits also a generalized hypercomplex structure in which all generalized complex structures are of type $1$.