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Marco Zambon

Publications and source records attributed to Marco Zambon.

At least 19 recordsLinked to original sources
arXiv2026

On the Dirac complement problem

The existence of a Dirac complement for a given Dirac structure is a central question in the structure theory of Courant algebroids and the deformation theory of Dirac structures. We study this problem in detail, proving the unobstructedness of lagrangian or local Dirac complements and providing examples that show the complexity of this question. We introduce a cohomology class whose nonvanishing prevents the existence of a Dirac complement and apply it to several families of examples. On the other hand, by using Lie-theoretical techniques, we prove that, for a Lie algebra $\mathfrak{g}$ endowed with a definite form, the diagonal in $\mathfrak{g} \oplus \bar{\mathfrak{g}}$ does not admit a complement unless $\mathfrak{g}$ is abelian. This includes real compact semisimple Lie algebras with their Killing form.

math.SG↗
arXiv2025

Coisotropic branes in symplectic manifolds

A brane in a symplectic manifold is a coisotropic submanifold $Y$ endowed with a compatible closed 2-form $F$, which together induce a transverse complex structure. For a specific class of branes we give an explicit description of branes nearby a given one, and for arbitrary branes we describe the infinitesimal deformations and provide an associated cochain complex. As an application, we determine to what extent coisotropic submanifolds near a given brane admit brane structures.

math.SG↗
arXiv2025

The Linearizability of Singular Foliations Is a Morita Invariant

Hausdorff Morita equivalence is an equivalence relation on singular foliations, which induces a bijection between their leaves. Our main statement is that linearizability along a leaf is invariant under Hausdorff Morita equivalence. The proof relies on a characterization of tubular neighborhood embeddings using Euler-like vector fields.

math.DG↗
arXiv2025

Blowups of Dirac structures

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 $k=0$.

math.SG↗
arXiv2025

Moduli spaces of spacefilling branes in symplectic 4-manifolds

On a symplectic manifold $(M, \omega)$, a spacefilling brane structure is a closed 2-form $F$ which determines a complex structure, with respect to which $F +i\omega$ is holomorphic symplectic. For holomorphic symplectic compact K\"ahler 4-manifolds, we show that the moduli space of spacefilling branes is smooth, and determine its dimension. The proof relies on the local Torelli theorem for K3 surfaces and tori.

math.SG↗
arXiv2023

Stability of fixed points of Dirac structures

Given an $L_{\infty}$-algebra $V$ and an $L_{\infty}$-subalgebra $W$, we give sufficient conditions for all small Maurer-Cartan elements of $V$ to be equivalent to Maurer-Cartan elements lying in $W$. As an application, we obtain a stability criterion for fixed points of a Dirac structure (for instance a twisted Poisson structure), i.e. points where the corresponding leaf is zero-dimensional. The criterion guarantees that any nearby Dirac structure also has a fixed point.

math.SG↗
arXiv2023

Graded geometry and generalized reduction

We present general reduction procedures for Courant, Dirac and generalized complex structures, in particular when a group of symmetries is acting. We do so by taking the graded symplectic viewpoint on Courant algebroids and carrying out graded symplectic reduction, both in the coisotropic and hamiltonian settings. Specializing the latter to the exact case, we recover in a systematic way the reduction schemes of Bursztyn-Cavalcanti-Gualtieri.

math.SG↗
arXiv2023

Deformations of Symplectic Foliations: algebraic aspects

In the companion paper arXiv:2110.05298, we developed the deformation theory of symplectic foliations, focusing on geometric aspects. Here, we address some algebraic questions that arose naturally. We show that the $L_{\infty}$-algebra constructed there is independent of the choices made, and we prove that the gauge equivalence of Maurer-Cartan elements corresponds to the equivalence by isotopies of symplectic foliations.

math.SG↗
arXiv2022

Observables on multisymplectic manifolds and higher Courant algebroids

Let $\omega$ be a closed, non-degenerate differential form of arbitrary degree. Associated to it there are an $L_{\infty}$-algebra of observables, and an $L_{\infty}$-algebra of sections of the higher Courant algebroid twisted by $\omega$. Our main result is the existence of an $L_{\infty}$-embedding of the former into the latter. We display explicit formulae for the embedding, involving the Bernoulli numbers. When $\omega$ is an integral symplectic form, the embedding can be realized geometrically via the prequantization construction, and when $\omega$ is a 3-form the embedding was found by Rogers in 2010. Further, in the presence of homotopy moment maps, we show that the embedding is compatible with gauge transformations.

math.SG↗
arXiv2022

Deformations of Symplectic Foliations

We develop the deformation theory of symplectic foliations, i.e. regular foliations equipped with a leafwise symplectic form. The main result of this paper is that each symplectic foliation has an attached $L_\infty$-algebra controlling its deformation problem. Indeed, viewing symplectic foliations as regular Poisson structures, we establish a one-to-one correspondence between the small deformations of a given symplectic foliation and the Maurer-Cartan elements of the associated $L_\infty$-algebra. Using this, we show that infinitesimal deformations of symplectic foliations can be obstructed. Further, we relate symplectic foliations with foliations on one side and with (arbitrary) Poisson structures on the other, showing that obstructed infinitesimal deformations of the former may give rise to unobstructed infinitesimal deformations of the latter.

math.SG↗
arXiv2021

Deformations of Lagrangian submanifolds in log-symplectic manifolds

This paper is devoted to deformations of Lagrangian submanifolds contained in the singular locus of a log-symplectic manifold. We prove a normal form result for the log-symplectic structure around such a Lagrangian, which we use to extract algebraic and geometric information about the Lagrangian deformations. We show that the deformation problem is governed by a DGLA, we discuss whether the Lagrangian admits deformations not contained in the singular locus, and we give precise criteria for unobstructedness of first order deformations. We also address equivalences of deformations, showing that the gauge equivalence relation of the DGLA corresponds with the geometric notion of equivalence by Hamiltonian isotopies. We discuss the corresponding moduli space, and we prove a rigidity statement for the more flexible equivalence relation by Poisson isotopies.

math.SG↗
arXiv2021

Singular subalgebroids

We introduce singular subalgebroids of an integrable Lie algebroid, extending the notion of Lie subalgebroid by dropping the constant rank requirement. We lay the bases of a Lie theory for singular subalgebroids: we construct the associated holonomy groupoids, adapting the procedure of Androulidakis-Skandalis for singular foliations, in a way that keeps track of the choice of Lie groupoid integrating the ambient Lie algebroid. The holonomy groupoids are topological groupoids, and are suitable for noncommutative geometry as they allow for the construction of the associated convolution algebras. Further we carry out the construction for morphisms in a functorial way.

math.DG↗
arXiv2021

Holonomy transformations for Lie subalgebroids

Given a foliation, there is a well-known notion of holonomy, which can be understood as an action that differentiates to the Bott connection on the normal bundle. We present an analogous notion for Lie subalgebroids, consisting of an effective action of the minimal integration of the Lie subalgebroid, and provide an explicit description in terms of conjugation by bisections. The construction is done in such a way that it easily extends to singular subalgebroids, which provide our main motivation.

math.DG↗
arXiv2020

Gauge equivalences for foliations and pre-symplectic structures

We consider the deformation theory of two kinds of geometric objects: foliations on one hand, pre-symplectic forms on the other. For each of them, we prove that the geometric notion of equivalence given by isotopies agrees with the algebraic notion of gauge equivalence obtained from the $L_{\infty}$-algebras governing these deformation problems.

math.DG↗
arXiv2020

Integration of singular subalgebroids by diffeological groupoids

We establish an integration theory for singular subalgebroids, by diffeological groupoids. To do so, we single out a class of diffeological groupoids satisfying specific properties, and we introduce a differentiation-integration procedure under which they correspond to singular subalgebroids. Our definition of integration distinguishes the holonomy groupoid from the graph, although both differentiate to the original singular subalgebroid: the holonomy groupoid satisfies a certain submersive property, while the graph does not.

math.DG↗
arXiv2020

Quotients of singular foliations and Lie 2-group actions

Every singular foliation has an associated topological groupoid, called holonomy groupoid (see arXiv:math/0612370). In this note we exhibit some functorial properties of this assignment: if a foliated manifold $(M,\mathcal{F}_M)$ is the quotient of a foliated manifold $(P,\mathcal{F}_P)$ along a surjective submersion with connected fibers, then the same is true for the corresponding holonomy groupoids. For quotients by a Lie group action, an analog statement holds under suitable assumptions, yielding a Lie 2-group action on the holonomy groupoid.

math.DG↗
arXiv2020

Lie 2-algebra moment maps in multisymplectic geometry

Consider a closed non-degenerate 3-form $ω$ with an infinitesimal action of a Lie algebra $\mathfrak{g}$. Motivated by the fact that the observables associated to $ω$ form a Lie 2-algebra, we introduce homotopy moment maps defined on a Lie 2-algebra rather than just on the Lie algebra $\mathfrak{g}$. We formulate existence criteria and provide a construction for such homotopy moment maps, by characterizing them in terms of cohomology.

math.DG↗