Generalized complex submanifolds

James Barton and Mathieu Stiénon

Vol. 236 (2008), No. 1, 23–44

DOI: 10.2140/pjm.2008.236.23

Abstract

We introduce the notion of twisted generalized complex submanifolds and describe an equivalent characterization in terms of Poisson–Dirac submanifolds. Our characterization recovers a result of Vaisman (2007). An equivalent characterization is also given in terms of spinors. As a consequence, we show that the fixed locus of an involution preserving a twisted generalized complex structure is a twisted generalized complex submanifold. We also prove that a twisted generalized complex manifold has a natural Poisson structure. We also discuss generalized Kähler submanifolds.

Keywords

generalized complex geometry, Poisson bivector, Poisson–Dirac submanifold

Mathematical Subject Classification 2000

Primary: 53C56, 53D17, 53D35

Milestones

Received: 12 August 2007

Revised: 21 January 2008

Accepted: 23 January 2008

Published: 1 May 2008