Deformations of Coisotropic Submanifolds in Jacobi Manifolds

Lê, Hông Vân; Oh, Yong─Geun; Tortorella, Alfonso Giuseppe; Vitagliano, Luca

doi:10.48550/arxiv.1410.8446

Cited by 6 publications

(18 citation statements)

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“…(1) S is a regular coisotropic submanifold of (M, C), (2) Remark 3.3. Every regular coisotropic submanifold inherits a precontact structure.…”

Section: Precontact Geometry As Presymplectic Geometry On the Atiyah ...mentioning

confidence: 99%

“…4.8]). See also [2] for details about the L ∞ -algebra attached to a coisotropic submanifold, in the more general setting of Jacobi manifolds, and its role in the coisotropic deformation problem.…”

Section: Precontact Geometry As Presymplectic Geometry On the Atiyah ...mentioning

confidence: 99%

“…The infinitesimal generator of ϕ t is the smooth Recall that ∆ λt := (̟ ♭ ) −1 (d D λ t ) = J ♯ (j 1 λ t ) = { λ t , −} is a time-dependent Hamiltonian derivation of the Jacobi bundle (L, J) (cf., e.g., [2,3,8]). Hence it generates a smooth 1-parameter family of local automorphisms ψ t of the Jacobi bundle (L, J) which covers the contact isotopy ψ t of (M, C) generated by the time dependent contact vector field X λt = σ(∆ λt ).…”

Section: Proposition 42 (Contact Reduction)mentioning

confidence: 99%

“…locally their smooth Legendrian deformations are induced by contact isotopies. Moreover their deformation problem under contact equivalence is controlled by an acyclic dg-space, so that it is unobstructed with discrete local moduli space (cf., e.g., [2,Section 6]). As recently shown in [2], in the contact setting as well, the coisotropic deformation problem is controlled by an L ∞ -algebra, rather than an acyclic dg-space.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover their deformation problem under contact equivalence is controlled by an acyclic dg-space, so that it is unobstructed with discrete local moduli space (cf., e.g., [2,Section 6]). As recently shown in [2], in the contact setting as well, the coisotropic deformation problem is controlled by an L ∞ -algebra, rather than an acyclic dg-space. In this note we will explicitly construct a first example, in the contact setting, of coisotropic submanifold whose deformation problem is obstructed (Section 3.1).…”

Section: Introductionmentioning

confidence: 99%

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Rigidity of integral coisotropic submanifolds of contact manifolds

Tortorella

2017

Lett Math Phys

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Unlike Legendrian submanifolds, the deformation problem of coisotropic submanifolds can be obstructed. Starting from this observation, we single out in the contact setting the special class of integral coisotropic submanifolds as the direct generalization of Legendrian submanifolds for what concerns deformation and moduli theory. Indeed, being integral coisotropic is proved to be a rigid condition, and moreover the integral coisotropic deformation problem is unobstructed with discrete moduli space.Comment: 10 pages, exposition slightly changed, mathematical content unchanged, references added. Comments welcome

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“…(1) S is a regular coisotropic submanifold of (M, C), (2) Remark 3.3. Every regular coisotropic submanifold inherits a precontact structure.…”

Section: Precontact Geometry As Presymplectic Geometry On the Atiyah ...mentioning

confidence: 99%

Section: Precontact Geometry As Presymplectic Geometry On the Atiyah ...mentioning

confidence: 99%

Section: Proposition 42 (Contact Reduction)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Rigidity of integral coisotropic submanifolds of contact manifolds

Tortorella

2017

Lett Math Phys

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Holomorphic Jacobi manifolds and holomorphic contact groupoids

Vitagliano

Wade

2019

Math. Z.

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This paper belongs to a series of works aiming at exploring generalized (complex) geometry in odd dimensions. Holomorphic Jacobi manifolds were introduced and studied by the authors in a separate paper as special cases of generalized contact bundles. In fact, generalized contact bundles are nothing but odd dimensional analogues of generalized complex manifolds. In the present paper, we solve the integration problem for holomorphic Jacobi manifolds by proving that they integrate to holomorphic contact groupoids. A crucial tool in our proof is what we call the homogenization scheme, which allows us to identify holomorphic Jacobi manifolds with homogeneous holomorphic Poisson manifolds and holomorphic contact groupoids with homogeneous complex symplectic groupoids. MSC: Primary 53Dxx; Secondary 53D10, 53D15, 53D17, 53D18, 53D35.

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Kirillov structures up to homotopy

Bruce

Tortorella

2016

Differential Geometry and its Applications

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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∞-algebra, which we refer to as a homotopy Kirillov algebra. We are then led 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.MSC (2010):16E45, 53D17, 58A50, 58E40.

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Deformations of Coisotropic Submanifolds in Jacobi Manifolds

Cited by 6 publications

References 0 publications

Rigidity of integral coisotropic submanifolds of contact manifolds

Rigidity of integral coisotropic submanifolds of contact manifolds

Holomorphic Jacobi manifolds and holomorphic contact groupoids

Kirillov structures up to homotopy

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