Almost Lagrangian Obstruction

The aim of this paper is to describe the obstruction for an almost Lagrangian fibration to be Lagrangian, a problem which is central to the classification of Lagrangian fibrations and, more generally, to understanding the obstructions to carry out surgery of integrable systems, an idea introduced by Zung. It is shown that this obstruction (namely, the homomorphism of Dazord and Delzant) is related to the cup product in cohomology with local coefficients on the base space B of the fibration. The map is described explicitly and some examples are calculated, thus providing the first examples of non-trivial Lagrangian obstructions.Comment: 17 pages, to appear in Diff. Geom. App

Twisted isotropic realisations of twisted Poisson structures

Motivated by the recent connection between nonholonomic integrable systems and twisted Poisson manifolds made in \cite{balseiro_garcia_naranjo}, this paper investigates the global theory of integrable Hamiltonian systems on almost symplectic manifolds as an initial step to understand Hamiltonian integrability on twisted Poisson (and Dirac) manifolds. Non-commutative integrable Hamiltonian systems on almost symplectic manifolds were first defined in \cite{fasso_sansonetto}, which proved existence of local generalised action-angle coordinates in the spirit of the Liouville-Arnol'd theorem. In analogy with their symplectic counterpart, these systems can be described globally by twisted isotropic realisations of twisted Poisson manifolds, a special case of symplectic realisations of twisted Dirac structures considered in \cite{bursztyn_crainic_weinstein_zhu}. This paper classifies twisted isotropic realisations up to smooth isomorphism and provides a cohomological obstruction to the construction of these objects, generalising the main results of \cite{daz_delz}.Comment: 20 page

Contact Isotropic Realisations of Jacobi Manifolds via Spencer Operators

Motivated by the importance of symplectic isotropic realisations in the study of Poisson manifolds, this paper investigates the local and global theory of contact isotropic realisations of Jacobi manifolds, which are those of minimal dimension. These arise naturally when considering multiplicity-free actions in contact geometry, as shown in this paper. The main results concern a classification of these realisations up to a suitable notion of isomorphism, as well as establishing a relation between the existence of symplectic and contact isotropic realisations for Poisson manifolds. The main tool is the classical Spencer operator which is related to Jacobi structures via their associated Lie algebroid, which allows to generalise previous results as well as providing more conceptual proofs for existing ones

Optimal Contract Design with Unilateral Market Option

Contrary to previous literature, we show that the assignment of authority decision matters in optimal contract design with bilateral specific self-investments. This is the case when we enlarge the set of the states of nature, to explicitly consider the event that a party's market option turns out to be binding ex-post. We show that, under this setting, simple contracts protected by specific performance remedies may generate hold-up and thus parties' incentives to under-invest. However, investment efficiency is enhanced when authority is assigned to the party with ex-post binding market option. Our results suggest a neglected rationale for vertical integration as a remedy against hold-upincomplete contracts, outside options, mechanism design