Integration of Conformal Jacobi Fibrations and Prequantization of Poisson Fibrations

arXiv: Differential Geometry, 2017

This is the second part of a series of two papers dedicated to a systematic study of holomorphic Jacobi structures. In the first part, we introduced and study the concept of a holomorphic Jacobi manifold in a very natural way as well as various tools. In the present paper, we solve the integration problem for holomorphic Jacobi manifolds by proving that they integrate to complex 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 complex contact groupoids with homogeneous complex symplectic groupoids.

Eprint Arxiv Math 0301108, 2003

Locally conformal symplectic (l.c.s.) groupoids are introduced as a generalization of symplectic groupoids. We obtain some examples and we prove that l.c.s. groupoids are examples of Jacobi groupoids in the sense of [5]. Finally, we describe the Lie algebroid of a l.c.s. groupoid.

Mathematische Zeitschrift, 2019

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.

Journal of Physics A: Mathematical and General, 2006

We study precontact groupoids whose infinitesimal counterparts are Dirac-Jacobi structures. These geometric objects generalize contact groupoids. We also explain the relationship between precontact groupoids and homogeneous presymplectic groupoids. Finally, we present some examples of precontact groupoids.

International Journal of Mathematics and Mathematical Sciences, 2005

We prove that a closed 3-dimensional manifold is a torus bundle over the circle if and only if it carries a closed nonsingular 1-form which is linearly deformable into contact forms.

Journal of Geometry and Physics, 2003

Jacobi groupoids are introduced as a generalization of Poisson and contact groupoids and it is proved that generalized Lie bialgebroids are the infinitesimal invariants of Jacobi groupoids. Several examples are discussed.

Cornell University - arXiv, 2015

We give the parallelism between locally conformal symplectic manifolds and contact manifolds. We also give the generalization of exact contact manifolds.

Commentarii Mathematici Helvetici, 2002

We show that the dω-cohomology is isomorphic to a conformally invariant usual de Rham cohomology of an appropriate cover. We also prove a Moser theorem for locally conformal symplectic (lcs) forms. We point out a connection between lcs geometry and contact geometry. Finally, we show the connections between first kind, second kind, essential, inessential, local, and global conformal symplectic structures through several invariants.

Geometriae Dedicata, 2013

We prove h-principle for locally conformal symplectic foliations and contact foliations on open manifolds. We then interpret the results in terms of regular Jacobi structures on manifolds.

Differential Geometry and its Applications, 2004