Which singular tangent bundles are isomorphic?

Abstract

Logarithmic and $b$-tangent bundles provide a versatile framework for addressing singularities in geometry. Introduced by Deligne and Melrose, these modified bundles resolve singularities by reframing singular vector fields as well-behaved sections of these singular bundles. This approach has gained significant attention in symplectic geometry, particularly through its applications to the study of Poisson manifolds that are symplectic away from a hypersurface ($b^m$-symplectic forms). In this article, we investigate the conditions under which these singular tangent bundles are isomorphic to the tangent bundle or other singular bundles, analyzing in detail the low-dimensional case and the case of spheres. We also examine the existence of geometric structures in light of these conditions. Furthermore, we establish a Poincaré-Hopf theorem for the $b^m$-tangent bundle, offering new insights into the interplay between singular structures and topological invariants.

33 pages. Major changes in section 7. The statement and proof of theorem 7.5 have been corrected, and applications to the existence of geometric structures have been added. Final version published in J. London Math. Soc