arXiv research

Study non-formal pseudo-differential operators over formal ones.

Compact non-formal $G_2$ manifold with $b_1=1$.

We recall the construction of non-formal deformation quantization of the Poincare Group ISO(1,1) on its coadjoint orbit and exhibit the associated non-formal star-exponentials.

We prove the formality and the evenness of odd-degree Betti numbers for compact Kähler orbifolds, by adapting the classical proofs for Kähler manifolds. As a consequence, we obtain examples of symplectic orbifolds not admitting any Kähler orbifold structure. We also review the known examples of non-formal simply connec…

We construct examples of non-formal simply connected and compact oriented manifolds of any dimension bigger or equal to 7.

New examples of non-formal Sasaki-Einstein 7-manifolds and their submanifolds found.

In previous work, we introduced a natural $\mathcal{A}_{\infty}$-structure on the $\mathrm{Pin}(2)$-monopole Floer chain complex of a closed, oriented three-manifold $Y$, and showed that it is non-formal in the simplest case in which $Y$ is the three-sphere $S^3$. In this paper, we provide explicit descriptions of seve…

New hierarchies derived from KP hierarchy using non-formal operators and Yang-Mills action.

Several large classes of homogeneous spaces are known to be formal---in the sense of Rational Homotopy Theory. However, it seems that far fewer examples of non-formal homogeneous spaces are known. In this article we provide several construction principles and characterisations for non-formal homogeneous spaces, which w…

Non-formal G2 manifold found with holonomy.

We establish a rigorous link between infinite-dimensional regular Frölicher Lie groups built out of non-formal pseudodifferential operators and the Kadomtsev-Petviashvili hierarchy. We introduce a version of the Kadomtsev-Petviashvili hierarchy on a regular Frölicher Lie group of series of non-formal odd-class pseudodi…

We show that, for any $k\geq 1$, there exist non-formal compact orientable $(k-1)$-connected $n$-manifolds with $k$-th Betti number $b_k=b\geq 0$ if and only if $n\geq \max \{4k-1, 4k+3-2b\}$.

Quantizes functions on Kähler manifolds without formal deformation.

We show that there exist non-formal compact oriented manifolds of dimension $n$ and with first Betti number $b_1=b\geq 0$ if and only if $n\geq 3$ and $b\geq 2$, or $n\geq (7-2b)$ and $0\leq b\leq 2$. Moreover, we present explicit examples for each one of these cases.

We investigate some topological properties, in particular formality, of compact Sasakian manifolds. Answering some questions raised by Boyer and Galicki, we prove that all higher (than three) Massey products on any compact Sasakian manifold vanish. Hence, higher Massey products do obstruct Sasakian structures. Using th…

In this work we analyze the behavior of Massey products of closed manifolds under the blow-up construction. The results obtained in the article are applied to the problem of constructing closed symplectic non-formal manifolds. The proofs use Thom spaces as an important technical tool. This application of Thom spaces is…

We study the formality of the mapping torus of an orientation-preserving diffeomorphism of a manifold. In particular, we give conditions under which a mapping torus has a non-zero Massey product. As an application we prove that there are non-formal compact co-symplectic manifolds of dimension $m$ and with first Betti n…

We study the formality of the total space of principal SU(2) and SO(3)-bundles over a Wolf space, that is a symmetric positive quaternionic Kähker manifold. We apply this to conclude that all the 3-Sasakian homogeneous spaces are formal. We also determine the principal SU(2) and SO(3)-bundles over the Wolf spaces whose…

We construct closed $(k-1)$-connected manifolds of dimensions $\ge 4k-1$ that possess non-trivial rational Massey triple products. We also construct examples of manifolds $M$ such that all the cup-products of elements of $H^k(M)$ vanish, while the group $H^{3k-1}(M;\Q)$ is generated by Massey products: such examples ar…

We review recent works concerning deformation quantization of abelian supergroups. Indeed, we expose the construction of an induced representation of the Heisenberg supergroup and an associated pseudodifferential calculus by using Kirillov's orbits method. Then, a star-product is built on the abelian supergroup R^{m|n}…

Research covers geometry, analysis, and integration on infinite-dimensional spaces.

We construct a non-formal deformation machinery for the actions of the Heisenberg supergroup analogue to the one developed by M. Rieffel for the actions of R^d. However, the method used here differs from Rieffel's one: we obtain a Universal Deformation Formula for the actions of R^{m|n} as a byproduct of Weyl ordered K…

The paper classifies quantizable functions and explores symmetry in quantization methods.

In this thesis, we study the deformation problem of coisotropic submanifolds in Jacobi manifolds. In particular we attach two algebraic invariants to any coisotropic submanifold $S$ in a Jacobi manifold, namely the $L_\infty[1]$-algebra and the BFV-complex of $S$. Our construction generalizes and unifies analogous cons…

Upper bounds on map degrees for various manifold types.

This paper connects symplectic and Kähler manifolds via brane quantization.

Mathematical framework for brane quantization using SYZ mirror symmetry.

The paper introduces new structures for left-symmetric algebroids.

We give a notion of compatibility between a Riemannian structure and a Jacobi structure. We prove that in case of fundamental examples of Jacobi structures : Poisson structures, contact structures and locally conformally symplectic structures, we get respectively Riemann-Poisson structures in the sense of M. Boucetta, …

We give a notion of compatibility between a Riemannian metric and a Jacobi structure. We prove that in case of Poisson structures, contact structures and locally conformally symplectic structures, fundamental examples of Jacobi structures, we get respectively Riemann-Poisson structures in the sense of M. Boucetta, $\fr…

Defines structure constants for specific geometric structures on Lie groups.

Study on $G_2^*$ structures and almost para-contact structures in 7D.

Defines a new Poisson structure for generalized Sasakian spaces.

In a preceding paper we introduced a notion of compatibility between a Jacobi structure and a Riemannian structure on a smooth manifold. We proved that in the case of fundamental examples of Jacobi structures : Poisson structures, contact structures and locally conformally symplectic structures, we get respectively Rie…

Study on types of generalized hypercomplex structures on tori and Kodaira-Thurston surface.

Classifies complex Dirac structures with invariants and local structure.

Extends corner structure study to general case, constructs normal Trans-Sasakian structures.

New metric structures generalize Sasakian and cosymplectic structures, proving rigidity and finding conditions.

3D projective structures can be metrized with conformal structures.

Hypersymplectic structures with torsion on Lie algebroids are investigated. We show that each hypersymplectic structure with torsion on a Lie algebroid determines three Nijenhuis morphisms. From a contravariant point of view, these structures are twisted Poisson structures. We prove the existence of a one-to-one corres…

Study equivalence between Hessian and Born structures on tangent bundles.

Introduces compatibility between Dirac structures and Nijenhuis tensors.

We introduce generalized almost contact structures which admit the $B$-field transformations on odd dimensional manifolds. We provide definition of generalized Sasakain structures from the view point of the generalized almost contact structures. We obtain a generalized Sasakian structure on a non-compact manifold which…

Introduces semi-abelian generalized complex structures.

Introduces VB-structures for geometric objects on manifolds.

The paper studies nilpotent structures in oriented neutral vector bundles and neutral hyperKähler structures.

Spin-structures on real Bott manifolds with Kähler structure are characterized.

Expanding on previous work, this note generalizes geometric structures results.