arXiv subjects
Publications and source records attributed to Andreas Schüßler.
Lie algebroids, singular foliations, and Dirac structures are closely related objects. We examine the relation between their pullbacks under maps satisfying a constant rank or transversality assumption. A special case is given by blowdown maps. In that case, we also establish the relation between the blowup of a Lie algebroid and its singular foliation.
Given a real, twisted Dirac structure $L$ on a smooth manifold $M$, and a closed embedded submanifold $N\subseteq M$ of codimension $>1$, we characterise when $L$ lifts to a smooth, twisted Dirac structure on the real projective blowup of $M$ along $N$. This holds precisely when $N$ is either a submanifold transverse to $L$ (with no further restrictions) or a submanifold invariant for $L$, for which the Lie algebras transverse to $N$ have all of the same constant height $k\geq 0$. We also classify Lie algebras satisfying this Lie-theoretic property. We recover a theorem of Polishchuk, which establishes that a Poisson structure lifts to a Poisson structure on the blowup of a submanifold exactly when the submanifold is invariant and the transverse Lie algebras have constant height $k=0$.
We study a spectral sequence approximating Lie algebroid cohomology associated to a Lie subalgebroid. This is a simultaneous generalisation of several classical constructions in differential geometry, including the Leray-Serre spectral sequence for de Rham cohomology associated to a fibration, the Hochschild-Serre spectral sequence for Lie algebras, and the Mackenzie spectral sequence for Lie algebroid extensions. We show that, for wide Lie subalgebroids, the spectral sequence converges to the Lie algebroid cohomology, and that, for Lie subalgebroids over proper submanifolds, the spectral sequence converges to the formal Lie algebroid cohomology. We discuss applications and recover several constructions in Poisson geometry in which this spectral sequence has appeared naturally in the literature.
We study the blow-down map in cohomology in the context of real projective blowups of Lie algebroids. Using the blow-down map in cohomology we compute the Lie algebroid cohomology of the blowup of transversals of arbitrary codimension, generalising the Mazzeo-Melrose theorem on b-cohomology. To prove the result we develop a Gysin sequence for Lie algebroids. As another example we use the developed tools to compute the Lie algebroid cohomology of the action Lie algebroid $\mathfrak{so}(3)\ltimes \mathbb{R}^3$, a result known in Poisson geometry literature. Moreover, we use similar techniques to compute the de Rham cohomology of real projective blowups.