Removable Presymplectic Singularities and the Local Splitting of Dirac Structures

Abstract

We call a singularity of a presymplectic form ω removable in its graph if its graph extends to a smooth Dirac structure over the singularity. An example for this is the symplectic form of a magnetic monopole. A criterion for the removability of singularities is given in terms of regularizing functions for pure spinors. All removable singularities are poles in the sense that the norm of ω is not locally bounded. The points at which removable singularities occur are the non-regular points of the Dirac structure for which we prove a general splitting theorem: Locally, every Dirac structure is the gauge transform of the product of a tangent bundle and the graph of a Poisson structure. This implies that in a neighborhood of a removable singularity ω can be split into a non-singular presymplectic form and a singular presymplectic form which is the partial inverse of a Poisson bivector that vanishes at the singularity. An interesting class of examples is given by log-Dirac structures which generalize logsymplectic structures. The analogous notion of removable singularities of Poisson structures is also studied.

On dual pairs in Dirac geometry

“…The Poisson geometric version of this result appeared in [21]; a local version for Dirac structures occurs in [5]; a version for generalized complex structures was described in [3]. Moreover, the same statement as the one below, but with a different proof, appeared in [13]:…”

Splitting theorems for Poisson and related structures

“…Equivalently, its local flow is the differential T Φ s of the local flow Φ s of X. If X is tangent to N , then the infinitesimal version of the identification ν(T Φ s ) = T (ν(Φ s )) shows that (9) ν(X T ) = ν(X) T as vector fields on ν(T M, T N ) = T ν(M, N ).…”

“…(a) Weinstein's splitting theorem [41] for Poisson manifolds (M, π), which asserts the existence of a neighborhood of m that is Poisson diffeomorphic to a product (S, π S ) × (N, π N ), where π S is non-degenerate while π N vanishes at m; (b) the splitting theorem for Dirac manifolds [13], obtained by Blohmann [9] (see also for related results); (c) the splitting theorem for Lie algebroids E → M , due to Dufour [16], Fernandes [19], and Weinstein [42], which gives an isomorphism near m with a product of Lie algebroids T S × F , where the anchor of the Lie algebroid F → N vanishes at m; (d) the splitting theorem for generalized complex manifolds [23], due to Abouzaid-Boyarchenko [1], which shows that up to a B-field transform, any generalized complex manifold is locally a product S × N of generalized complex manifolds, where S is 'of symplectic type' and N is 'of complex type' at m. In this article, we develop a novel approach towards splitting theorems, which allows us to generalize them in various directions and to new contexts. Rather than taking N to be 'small', we will allow transverse submanifolds N ֒→ M that may be quite large.…”

The Homology Class of a Poisson Transversal

“…The above results imply that Corollary 3 and Theorem 3 extend directly to the Dirac setting: The Dirac analogue of log-symplectic structures was recently introduced in [2,Definition 4.11] under the name of log-Dirac structures, and we believe that also the result of section 8 generalize to this setting. However, the theory of these structures has not been developed yet, and so we do not investigate this here.…”