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On the type of generalized hypercomplex structures

Anna Fino, Gueo Grantcharov

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 $1$.

On the type of generalized hypercomplex structures

Abstract

The generalized hypercomplex structures defined within the framework of generalized geometry include hypercomplex and holomorphic symplectic structures as particular cases. They have a -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 -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 .

Paper Structure

This paper contains 5 sections, 10 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on linear generalized complex structures
  3. Linear generalized hypercomplex structures
  4. Generalized hypercomplex manifolds and their twistor spaces
  5. Examples on tori and Kodaira-Thurston surface

Key Result

Lemma 3.1

Let ${\mathcal{I}}_1 \neq \pm{\mathcal{I}}_2$ be two generalized complex structures on $V\oplus V^*$. Then they determine a generalized hypercomplex structure iff there is $p, |p|<1$, such that

Theorems & Definitions (21)

  • Example 2.1
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 11 more