arXiv subjects

Pablo Nicolás

Publications and source records attributed to Pablo Nicolás.

4 recordsLinked to original sources

arXiv2026

Equivariant cosymplectic geometry

Cosymplectic manifolds provide a natural geometric framework for codimension-one symplectic foliations and arise throughout geometry and mathematical physics. We develop an equivariant cohomological theory for cosymplectic manifolds, studying the role of symmetry in their topology and their connections to Poisson geometry. Reinterpreting the obstruction theory of Guillemin, Miranda, and Pires, we introduce equivariant obstruction classes for group actions preserving the foliation and characterize the existence of invariant cosymplectic structures. We further show that the vanishing of the first obstruction class is equivalent to equivariant unimodularity of the associated Poisson structure, extending a classical criterion to the equivariant setting via Ginzburg's framework for equivariant Poisson cohomology. For compact cosymplectic manifolds fibering over $\mathbb{S}^1$, we construct equivariant versions of de Rham, foliated, and Poisson cohomologies, establishing formality results in each case via an equivariant Wang sequence. The key input is Kirwan's formality theorem for Hamiltonian actions on the symplectic fiber, which, through the equivariant Wang sequence, yields a complete and computable description of all three equivariant cohomology theories in terms of the monodromy action on the fiber.

math.SG↗

arXiv2026

Hamiltonian group actions in cosymplectic geometry

We develop a theory of Hamiltonian group actions on cosymplectic manifolds. These odd-dimensional manifolds combine a codimension-one symplectic foliation with a distinguished Reeb direction, and arise naturally both in stable Hamiltonian geometry and as critical hypersurfaces of $b$-symplectic manifolds. Our approach is based on a compact symplectic thickening process: every cosymplectic manifold $(M,α,β)$ gives rise to a symplectic manifold $(M\times \mathbb{S}^1,\ β+ \mathrm{d}θ\wedgeα)$. We prove that Hamiltonian cosymplectic actions lift canonically to Hamiltonian symplectic actions on this symplectic manifold. This provides a systematic bridge between equivariant symplectic geometry and the cosymplectic setting. Using this bridge, we establish cosymplectic analogues of convexity theorems for torus actions, Delzant theorem for toric actions, ABBV localization, Duistermaat-Heckman formulas, and Kirwan surjectivity. The resulting formulas are not merely formal pullbacks from the symplectic case: the Reeb direction appears explicitly through the factor $α$, and, in the mapping-torus case, the localization and volume formulas are governed by the modular period together with the equivariant geometry of the symplectic fiber. We also explain how these results apply to Hamiltonian geometry on the critical hypersurfaces of $b$-symplectic manifolds.

math.SG↗

arXiv2026

Which singular tangent bundles are isomorphic?

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.

math.DG↗

arXiv2022

Hamiltonian facets of classical gauge theories on $E$-manifolds

Manifolds with boundary, with corners, $b$-manifolds and foliations model configuration spaces for particles moving under constraints and can be described as $E$-manifolds. $E$-manifolds were introduced in [NT01] and investigated in depth in [MS20]. In this article we explore their physical facets by extending gauge theories to the $E$-category. Singularities in the configuration space of a classical particle can be described in several new scenarios unveiling their Hamiltonian aspects on an $E$-symplectic manifold. Following the scheme inaugurated in [Wei78], we show the existence of a universal model for a particle interacting with an $E$-gauge field. In addition, we generalize the description of phase spaces in Yang-Mills theory as Poisson manifolds and their minimal coupling procedure, as shown in [Mon86], for base manifolds endowed with an $E$-structure. In particular, the reduction at coadjoint orbits and the shifting trick are extended to this framework. We show that Wong's equations, which describe the interaction of a particle with a Yang-Mills field, become Hamiltonian in the $E$-setting. We formulate the electromagnetic gauge in a Minkowski space relating it to the proper time foliation and we see that our main theorem describes the minimal coupling in physical models such as the compactified black hole.

math-ph↗