arXiv subjects

Dan Aguero

Publications and source records attributed to Dan Aguero.

4 recordsLinked to original sources

arXiv2026

Concurring reduction schemes for Dirac structures

The notion of \emph{concurrence} was recently proposed as the natural compatibility relation between Dirac structures, generalizing the commutativity of two Poisson structures. We address the question of when a reduction scheme -- that is, a way to induce a Dirac structure on a quotient of a submanifold -- respects this relation. After characterizing the minimal scheme of \emph{Dirac reduction}, we prove that two concurring Dirac structures have concurring reductions whenever they share a common \emph{witness}, extending to Dirac geometry the reduction of the Marsden-Ra\cb{t}iu theorem. Two procedures for constructing such common witnesses are given, the second being the Dirac counterpart of Magri's original recipe in bihamiltonian geometry. Examples drawn from Hamiltonian actions, Dirac-Nijenhuis manifolds, and complex Dirac structures conclude the paper and illustrate our methods.

math.SG↗

arXiv2025

On the geometry of complex Poisson bivectors

We study the geometry of complex Poisson bivectors over smooth manifolds. We show that under mild regularity conditions any complex Poisson bivector has associated a complex presymplectic foliation. After that, we use techniques of Dirac geometry to provide a more concise description of this complex presymplectic foliation. Moreover, we introduce two new classes of structures: quasi-real Poisson and quasi-real Dirac structures. In the last part, we focus on the normal form of complex Poisson bivectors. Under certain regularity, we provide a normal form theorem for complex Poisson structures along certain kinds of submanifolds.

math.SG↗

arXiv2025

On Complex Lie Algebroids with Constant Real Rank

We associate a real distribution to any complex Lie algebroid that we call distribution of real elements and a new invariant that we call real rank, given by the pointwise rank of this distribution. When the real rank is constant, we obtain a real Lie algebroid inside the original complex Lie algebroid. Under another regularity condition, we associate a complex Lie subalgebroid that we call the minimal complex subalgebroid. We also provide a local splitting for complex Lie algebroids with constant real rank. In the last part, we introduce the complex matched pair of skew-algebroids; these pairs produce complex Lie algebroid structures on the complexification of a vector bundle. We use this operation to characterize all the complex Lie algebroid structures on the complexification of real vector bundles.

math.SG↗

arXiv2022

Complex Dirac structures: invariants and local structure

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.

math.DG↗