Linear generalised complex structures

TL;DR

This paper explores linear generalized complex structures on vector bundles, linking them to complex Lie algebroid structures and introducing generalized complex Lie algebroids as conjugate pairs of Lie bialgebroids.

Contribution

It characterizes linear generalized complex structures via complex multiplication and Lie algebroid structures, extending generalized geometry to vector bundles.

Findings

Equivalence of linear generalized complex structures to complex multiplication and Lie algebroid structures.

Introduction of generalized complex Lie algebroids as conjugate pairs of Lie bialgebroids.

Framework for studying generalized complex structures in the context of vector bundles.

Abstract

This paper studies linear generalised complex structures over vector bundles, as a generalised geometry version of holomorphic vector bundles. In an adapted linear splitting, a linear generalised complex structure on a vector bundle $E\to M$ is equivalent to a $\mathbb C$-multiplication $j$ in the fibers of $TM\oplus E^*$ and $\mathbb C$-Lie algebroid structure on $TM\oplus E^*$. Generalised complex Lie algebroids (or Glanon algebroids) are then studied in this context, and expressed as a pair of complex conjugated Lie bialgebroids.