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Aldo Witte

Publications and source records attributed to Aldo Witte.

8 recordsLinked to original sources
arXiv2026

Polynomial degeneration and the Poisson geometry of truncated polynomials

We develop a formalism for studying geometric structures that degenerate to polynomial order along a hypersurface $W \subset M$. We then demonstrate it in the study of Poisson geometry, where it leads to methods for constructing generically symplectic Poisson structures with non-trivial symplectic variation along their degeneracy locus. This is in contrast to $b/\log$-symplectic and $b^k$-symplectic structures, where this variation always vanishes. Our main insight is that the higher residue data along the hypersurface is controlled by a group of transverse diffeomorphisms, which in our case is the group $G_k$ of degree-$k$ truncated polynomials. We show that the symplectic variation of our Poisson structures is determined by the obstruction to lifting a $G_k$-representation of the fundamental group $\pi_1(W)$ to $G_{k+1}$, and we construct maps from a $G_k$-character variety into the moduli space of Poisson structures, with the variation detecting the non-triviality of the resulting families.

math.DG↗
arXiv2025

$b^k$-algebroids and the variety of foliation jets

We introduce and classify singular foliations of $b^{k+1}$-type, which formalize the properties of vector fields that are tangent to a submanifold $W \subset M$ to order $k$. When $W$ is a hypersurface, these structures are Lie algebroids generalizing the $b^{k+1}$-tangent bundles introduced by Scott. We prove that singular foliations of $b^{k+1}$-type are encoded by $k$-th order foliations: jets of distributions that are involutive up to order $k$, equivalently described as foliations on the $k$-th order neighborhood of $W$. Using this encoding, we construct topological groupoids of $k$-th order foliations and employ the holonomy invariant to show that these groupoids fiber over certain character stacks, yielding Riemann-Hilbert style classifications up to local isomorphism and isotopy. We also study the problem of extending a $k$-th order foliation to a $(k+1)$-st order foliation. We prove that this is obstructed by a characteristic class that arises as a section of a vector bundle over the relevant character stack.

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arXiv2023

Jets of foliations and $b^k$-algebroids

In this article, we introduce and study singular foliations of $b^k$-type. These singular foliations formalize the properties of vector fields that are tangent to order $k$ along a submanifold $W \subset M$. Our first result is a classification of these foliations, relating them to geometric structures defined in a formal neighborhood of the submanifold, such as jets of distributions that are involutive up to order $k-1$. When $W$ is a hypersurface, singular foliations of $b^k$-type are Lie algebroids. In this particular case, they are generalizations of the $b^k$-tangent bundles introduced by Scott. Indeed, they are always locally isomorphic to $b^k$-tangent bundles, but globally such an isomorphism is obstructed by a holonomy invariant. Our second main result is a Riemann-Hilbert-style classification of singular foliations of $b^k$-type in terms of holonomy representations. In this paper, we study singular foliations of $b^k$-type from several different perspectives. In particular: (1) We study the problem of extending a $k$-th-order foliation to a $(k+1)$-th order foliation and prove that this is obstructed by a characteristic class. (2) When $W$ is a hypersurface, we give a detailed study of algebroid differential forms and extend Scott's calculation of the cohomology. (3) We study algebroid symplectic forms in terms of the geometric structures induced on $W$. In particular, we find that there is a close relationship between the above obstruction class for extensions and the symplectic variation of the symplectic foliation induced on $W$.

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arXiv2022

Non-principal T-duality, generalized complex geometry and blow-ups

We extend the notion of T-duality to manifolds endowed with non-principal torus actions. The singularities of the torus action are controlled by a certain Lie algebroid, called the elliptic tangent bundle. Using this Lie algebroid, we explain how certain invariant generalized complex structures can be transported via T-duality. Along the way, we use the elliptic tangent bundle to define connections for these torus action, and give new insight to the classification of such actions by Haefliger-Salem.

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arXiv2022

Regularisation of Lie algebroids and Applications

We describe a procedure, called regularisation, that allows us to study geometric structures on Lie algebroids via foliated geometric structures on a manifold of higher dimension. This procedure applies to various classes of Lie algebroids; namely, those whose singularities are of b^k, complex-log, or elliptic type, possibly with self-crossings. One of our main applications is a proof of the Weinstein conjecture for overtwisted b^k-contact structures. This was proven by Miranda-Oms using a certain technical hypothesis. Our approach avoids this assumption by reducing the proof to the foliated setting. As a by-product, we also prove the Weinstein conjecture for other Lie algebroids. Along the way we also introduce tangent distributions, i.e. subbundles of Lie algebroids, as interesting objects of study and present a number of constructions for them.

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arXiv2020

Fibrations in semi-toric and generalized complex geometry

This paper studies the interplay between self-crossing boundary Lefschetz fibrations and generalized complex structures. We show that these fibrations arise from the moment maps in semi-toric geometry and use them to construct self-crossing stable generalized complex four-manifolds using Gompf--Thurston methods for Lie algebroids. These results bring forth further structure on several previously known examples of generalized complex manifolds. We moreover show that these fibrations are compatible with taking connected sums, and use this to prove a singularity trade result between two types of singularities occurring in these fibrations.

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arXiv2020

Self-crossing stable generalized complex structures

We extend the notion of (smooth) stable generalized complex structures to allow for an anticanonical section with normal self-crossing singularities. This weakening not only allows for a number of natural examples in higher dimensions but also sheds some light into the smooth case in dimension four. We show that in four dimensions there is a natural connected sum operation for these structures as well as a smoothing operation which changes a self-crossing stable generalized complex structure into a smooth stable generalized complex structure on the same manifold. This allows us to construct large families of stable generalized complex manifolds.

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