Complex Dirac structures: invariants and local structure

TL;DR

This paper investigates complex Dirac structures, introducing invariants for classification, and establishes local structure theorems, thereby advancing understanding of their geometric properties and underlying real structures.

Contribution

It introduces the order and type invariants for complex Dirac structures and provides classification and local structure theorems based on these invariants.

Findings

Classification of complex Dirac structures using invariants.

Existence of an underlying real Dirac structure for constant order.

A splitting theorem describing local structure for constant real index and order.

Abstract

We study complex Dirac structures, that is, Dirac structures in the complexified generalized tangent bundle. These include presymplectic foliations, transverse holomorphic structures, CR-related geometries and generalized complex structures. We introduce two invariants, the order and the (normalized) type. We show that, together with the real index, they allow us to obtain a pointwise classification of complex Dirac structures. For constant order, we prove the existence of an underlying real Dirac structure, which generalizes the Poisson structure associated to a generalized complex structure. For constant real index and order, we prove a splitting theorem, which gives a local description in terms of a presymplectic leaf and a small transversal.