Holomorphic Jacobi manifolds and holomorphic contact groupoids

Abstract

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.

Homogeneous G-structures

“…where "objects" refers to the basic building blocks -functions, vector fields, differential forms, etc. Let us explain this in slightly more detail (and we refer the reader to Section 2 of [11] for further details). The role that functions on M have in usual geometry is played by sections of L ("Atiyah functions").…”

“…Next, for a (1, 1)-tensor K : T M → T M, we denote by K † : T M → T M its -adjoint (1, 1)tensor. Then, from(X , Y )u = I(G(∇ |L | X , ∇ |L | Y )) = G(∇ D I ∇ |L | X , ∇ |L | Y ) + G(∇ |L | X , ∇ D I ∇ |L | Y ) 0 = ∇ |L | X (G(I, ∇ |L | Y )) = G(∇ D A † )(X ) = (B + B † )(X ) = X , (A − A † )(X ) = (B − B † )(X ) = (i X dη) ♯ ,(6 11).…”

Contact geometry and thermodynamics of black holes in AdS spacetimes

“…A contact manifold (M, η) has naturally associated with it, a Jacobi structure [74,75] which can be defined by a bilinear map, {., .} :…”

The Local Structure of Generalized Contact Bundles

“…In turn, such line bundles are intrinsic models for so called normal almost contact manifolds [7]. In our opinion, generalized contact bundles have an advantage over previous proposals of a generalized contact geometry: they have a firm conceptual basis in the so called homogenization scheme [40], which is, in essence, a dictionary from contact and related geometries to symplectic and related geometries. In principle, applying the dictionary is straightforward: it is enough to replace functions on a manifold M with sections of a line bundle L → M , vector fields over M with derivations of L, etc.…”

“…In principle, applying the dictionary is straightforward: it is enough to replace functions on a manifold M with sections of a line bundle L → M , vector fields over M with derivations of L, etc. In practice, applying the dictionary can be actually challenging, and may lead to interesting new features [19,36,37,24,25,9,34,33,39,40].…”