arXiv subjects

Alfonso G. Tortorella

Publications and source records attributed to Alfonso G. Tortorella.

9 recordsLinked to original sources

arXiv2025

Simultaneous Deformations of Symplectic Forms and Lagrangian Submanifolds

Given a compact symplectic manifold $(M,ω)$ and a compact Lagrangian submanifold $L\subset(M,ω)$, we describe small deformations of the pair $(ω,L)$ modulo the action by isotopies. We show that the resulting moduli space can be identified with an open neighborhood of the origin in the second relative de Rham cohomology group $H^2(M,L)$. This implies in particular that the moduli space is smooth and finite dimensional.

math.SG↗

arXiv2024

A rigidity result for coisotropic submanifolds in contact geometry

We study coisotropic deformations of a compact regular coisotropic submanifold $C$ in a contact manifold $(M,ξ)$. Our main result states that $C$ is rigid among nearby coisotropic submanifolds whose characteristic foliation is diffeomorphic to that of $C$. When combined with a classical rigidity result for foliations, this yields conditions under which $C$ is rigid among all nearby coisotropic submanifolds.

math.SG↗

arXiv2024

Shifted Contact Structures on Differentiable Stacks

We define \emph{$0$-shifted} and \emph{$+1$-shifted contact structures} on differentiable stacks, thus laying the foundations of \emph{shifted Contact Geometry}. As a side result we show that the kernel of a multiplicative $1$-form on a Lie groupoid (might not exist as a Lie groupoid but it) always exists as a differentiable stack, and it is naturally equipped with a stacky version of the curvature of a distribution. Contact structures on orbifolds provide examples of $0$-shifted contact structures, while prequantum bundles over $+1$-shifted symplectic groupoids provide examples of $+1$-shifted contact structures. Our shifted contact structures are related to shifted symplectic structures via a Symplectic-to-Contact Dictionary.

math.DG↗

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

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↗

arXiv2020

Homogeneous G-structures

The theory of $G$-structures provides us with a unified framework for a large class of geometric structures, including symplectic, complex and Riemannian structures, as well as foliations and many others. Surprisingly, contact geometry - the "odd-dimensional counterpart" of symplectic geometry - does not fit naturally into this picture. In this paper, we introduce the notion of a homogeneous $G$-structure, which encompasses contact structures, as well as some other interesting examples that appear in the literature.

math.DG↗

arXiv2019

Deformations of Coisotropic Submanifolds in Jacobi Manifolds

In this paper, we attach an $L_\infty$-algebra to any coisotropic submanifold in a Jacobi manifold. Our construction generalizes and unifies analogous constructions by Oh-Park (symplectic case), Cattaneo-Felder (Poisson case), Lê-Oh (locally conformal symplectic case). As a new special case, we attach an $L_\infty$-algebra to any coisotropic submanifold in a contact manifold. The $L_\infty$-algebra of a coisotropic submanifold $S$ governs the (formal) deformation problem of $S$.

math.DG↗

arXiv2017

Jacobi bundles and the BFV-complex

We extend the construction of the BFV-complex of a coisotropic submanifold from the Poisson setting to the Jacobi setting. In particular, our construction applies in the contact and l.c.s. settings. The BFV-complex of a coisotropic submanifold $S$ controls the coisotropic deformation problem of $S$ under both Hamiltonian and Jacobi equivalence.

math.DG↗

arXiv2016

Kirillov structures up to homotopy

We present the notion of higher Kirillov brackets on the sections of an even line bundle over a supermanifold. When the line bundle is trivial we shall speak of higher Jacobi brackets. These brackets are understood furnishing the module of sections with an $L_{\infty}$-algebra, which we refer to as a homotopy Kirillov algebra. We are then to higher Kirillov algebroids as higher generalisations of Jacobi algebroids. Furthermore, we show how to associate a higher Kirillov algebroid and a homotopy BV-algebra with every higher Kirillov manifold. In short, we construct homotopy versions of some of the well-known theorems related to Kirillov's local Lie algebras.

math.DG↗