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Rui Loja Fernandes

Publications and source records attributed to Rui Loja Fernandes.

At least 19 recordsLinked to original sources
arXiv2026

Kähler metrics and toric Lagrangian fibrations

We extend the Abreu-Guillemin theory of invariant Kähler metrics from toric symplectic manifolds to any symplectic manifold admitting a toric action of a symplectic torus bundle. We show that these are precisely the symplectic manifolds admitting a Lagrangian fibration with elliptic singularities. The base of such a toric Lagrangian fibration is a codimension 0 submanifold with corners of an integral affine manifold, called a Delzant subspace. This concept generalizes the Delzant polytope associated with a compact symplectic toric manifold. Given a Delzant subspace of finite type, we provide a Delzant-type construction of a Lagrangian fibration with the moment image being the specified Delzant subspace. We establish a 1:1 correspondence between invariant Kähler metrics and a pair consisting of an elliptic connection on the total space of the fibration and a hybrid $b$-metric on the base Delzant subspace, both with specified residues over the facets. Finally, we characterize extremal invariant Kähler metrics as those whose scalar curvature descends to an affine function on the base integral affine manifold. We show that this provides a method for finding and constructing extremal Kähler metrics.

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arXiv2025

Poisson Manifolds of Compact Types with an Appendix by Joshua Mundinger

We develop the theory of Poisson and Dirac manifolds of compact types, a broad generalization in Poisson and Dirac geometry of compact Lie algebras and Lie groups. We establish key structural results, including local normal forms, canonical stratifications, and a Weyl type resolution, which provides a way to resolve the singularities of the original structure. These tools allow us to show that the leaf space of such manifolds is an integral affine orbifold and to define their Weyl group. This group is a Coxeter group acting on the orbifold universal cover of the leaf space by integral affine transformations, and one can associate to it Weyl chambers, reflection hyperplanes, etc. We further develop a Duistermaat-Heckman theory for Poisson manifolds of s-proper type, proving the linear variation of cohomology of leafwise symplectic form and establishing a Weyl integration formula. As an application, we show that every Poisson manifold of compact type is necessarily regular. We conclude the paper with a list of open problems.

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arXiv2023

Poisson geometry around Poisson submanifolds

We construct a first order local model for Poisson manifolds around a large class of Poisson submanifolds and we give conditions under which this model is a local normal form. The resulting linearization theorem includes as special cases all the known linearization theorems for fixed points and symplectic leaves. The symplectic groupoid version of these results gives a solution to the groupoid coisotropic embedding problem.

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arXiv2023

Cosymplectic groupoids

A cosymplectic groupoid is a Lie groupoid with a multiplicative cosymplectic structure. We provide several structural results for cosymplectic groupoids and we discuss the relationship between cosymplectic groupoids, Poisson groupoids of corank 1, and oversymplectic groupoids of corank 1.

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arXiv2023

Multiplicative Ehresmann connections

We develop the theory of multiplicative Ehresmann connections for Lie groupoid submersions covering the identity, as well as their infinitesimal counterparts. We construct obstructions to the existence of such connections, and we prove existence for several interesting classes of Lie groupoids and Lie algebroids, including all proper Lie groupoids. We show that many notions from the theory of principal bundle connections have analogues in this general setup, including connections 1-forms, curvature 2-forms, Bianchi identity, etc. In [17] we provide a non-trivial application of the results obtained here to construct local models in Poisson geometry and to obtain linearization results around Poisson submanifolds.

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arXiv2022

The Global Solutions to Cartan's Realization Problem

We introduce a systematic method to solve a type of Cartan's realization problem. Our method builds upon a new theory of Lie algebroids and Lie groupoids with structure group and connection. This approach allows to find local as well as complete solutions, their symmetries, and to determine the moduli spaces of local and complete solutions. We apply our method to the problem of classification of extremal Kähler metrics on surfaces.

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arXiv2021

The Classifying Lie Algebroid of a Geometric Structure II: G-structures with connection

Given a G-structure with connection satisfying a regularity assumption we associate to it a classifying Lie algebroid. This algebroid contains all the information about the equivalence problem and is an example of a G-structure Lie algebroid. We discuss the properties of this algebroid, the G-structure groupoids integrating it and the relationship with Cartan's realization problem.

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arXiv2019

Associativity and Integrability

We provide a complete solution to the problem of extending a local Lie groupoid to a global Lie groupoid. First, we show that the classical Mal'cev's theorem, which characterizes local Lie groups that can be extended to global Lie groups, also holds in the groupoid setting. Next, we describe a construction that can be used to obtain any local Lie groupoid with integrable algebroid. Last, our main result establishes a precise relationship between the integrability of a Lie algebroid and the failure in associativity of a local integration. We give a simplicial interpretation of this result showing that the monodromy groups of a Lie algebroid manifest themselves combinatorially in a local integration, as a lack of associativity.

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arXiv2019

Regular Poisson manifolds of compact types (PMCT 2)

This is the second paper of a series dedicated to the study of Poisson structures of compact types (PMCTs). In this paper, we focus on regular PMCTs, exhibiting a rich transverse geometry. We show that their leaf spaces are integral affine orbifolds. We prove that the cohomology class of the leafwise symplectic form varies linearly and that there is a distinguished polynomial function describing the leafwise sympletic volume. The leaf space of a PMCT carries a natural Duistermaat-Heckman measure and a Weyl type integration formula holds. We introduce the notion of a symplectic gerbe, and we show that they obstruct realizing PMCTs as the base of a symplectic complete isotropic fibration (a.k.a. a non-commutative integrable system).

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arXiv2019

Genus Integration, Abelianization and Extended Monodromy

Given a Lie algebroid we discuss the existence of a smooth abelian integration of its abelianization. We show that the obstructions are related to the extended monodromy groups introduced recently in \cite{CFMb}. We also show that this groupoid can be obtained by a path-space construction, similar to the Weinstein groupoid of \cite{CF1}, but where the underlying homotopies are now supported in surfaces with arbitrary genus. As an application, we show that the prequantization condition for a (possibly non-simply connected) manifold is equivalent to the smoothness of an abelian integration. Our results can be interpreted as a generalization of the classical Hurewicz theorem.

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arXiv2019

On deformations of compact foliations

We combine classic stability results for foliations with recent results on deformations of Lie groupoids and Lie algebroids to provide a cohomological characterization for rigidity of compact foliations on compact manifolds.

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arXiv2018

Riemannian metrics on differentiable stacks

We study Riemannian metrics on Lie groupoids in the relative setting. We show that any split fibration between proper groupoids can be made Riemannian, and we use these metrics to linearize proper groupoid fibrations. As an application, we derive rigidity theorems for Lie groupoids, which unify, simplify and improve similar results for classic geometries. Then we establish the Morita invariance for our metrics, introduce a notion for metrics on stacks, and use them to construct stacky tubular neighborhoods and to prove a stacky Ehresmann theorem.

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arXiv2016

Picard groups of Poisson manifolds

For a Poisson manifold $M$ we develop systematic methods to compute its Picard group $Pic(M)$, i.e., its group of self Morita equivalences. We establish a precise relationship between $Pic(M)$ and the group of gauge transformations up to Poisson diffeomorphisms showing, in particular, that their connected components of the identity coincide; this allows us to introduce the Picard Lie algebra of $M$ and to study its basic properties. Our methods lead, in particular, to the proof of a conjecture from [BW04] stating that for any compact simple Lie algebra $\mathfrak{g}$ the group $Pic(\mathfrak{g}^*)$ concides with the group of outer automorphisms of $\mathfrak{g}$.

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arXiv2016

Poisson Manifolds of Compact Types (PMCT 1)

This is the first in a series of papers dedicated to the study of Poisson manifolds of compact types (PMCTs). This notion encompasses several classes of Poisson manifolds defined via properties of their symplectic integrations. In this first paper we establish some fundamental properties of PMCTs, which already show that they are the analogues of compact symplectic manifolds, thus placing them in a prominent position among all Poisson manifolds. For instance, their Poisson cohomology behaves very much like the de Rham cohomology of compact symplectic manifolds (Hodge decomposition, non-degenerate Poincaré duality pairing, etc.) and the Moser trick can be adapted to PMCTs. More important, we find unexpected connections between PMCTs and Symplectic Topology: PMCTs are related with the theory of Lagrangian fibrations and we exhibit a construction of a nontrivial PMCT related to a classical question on the topology of the orbits of a free symplectic circle action. In subsequent papers, we will establish deep connections between PMCTs and integral affine geometry, Hamiltonian $G$-spaces, foliation theory, Lie theory and symplectic gerbes.

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arXiv2015

Integration of Coupling Dirac Structures

Coupling Dirac structures are Dirac structures defined on the total space of a fibration, generalizing hamiltonian fibrations from symplectic geometry, where one replaces the symplectic structure on the fibers by a Poisson structure. We study the associated Poisson gauge theory, in order to describe the presymplectic groupoid integrating coupling Dirac structures. We find the obstructions to integrability and we give explicit geometric descriptions of the integration.

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