complex geometry

# Contents

## Idea

A generalization of the notion of Calabi-Yau manifold in the context of generalized complex geometry.

## Definition

### In terms of $G$-structure

For $X$ a $2n$-dimensional smooth manifold, a generalized complex structure on $X$ is a reduction of the structure group of the generalized tangent bundle $T X \oplus T^* X$ along the inclusion

$U(n,n) \hookrightarrow O(2n,2n)$

into the Narain group.

Recall that for $X$ an ordinary compact complex manifold of real dimension $2n$, a Calabi-Yau manifold structure on $X$ is a reduction of the structure group along the inclusion $SU(n) \hookrightarrow U(n)$ of the special unitary group into the unitary group.

A generalized Calabi-Yau structure on a generalized complex manifold $X$ is a further reduction of the structure group along

$SU(n,n) \hookrightarrow U(n,n) \hookrightarrow O(2n,2n) \,.$

## References

The notion was introduced in

The role of generalized CY-manifolds as (factors of) target spaces in string theory is discussed for instance in

Last revised on December 14, 2012 at 07:37:41. See the history of this page for a list of all contributions to it.