Linear generalised complex structures

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 $TM\oplus E^{\ast }$ and $\mathbb{C}$-Lie algebroid structure on $TM\oplus E^{\ast }$. Generalised complex Lie algebroids (or Glanon algebroids) are then studied in this context, and expressed as a pair of complex conjugated Lie bialgebroids.

36 pages