Homogeneous Contact Manifolds Research Articles

Homogeneous Contact Manifolds and Resolutions of Calabi–Yau Cones

In the present work we provide a constructive method to describe contact structures on compact homogeneous contact manifolds. The main feature of our approach is to describe the Cartan-Ehresmann connection (gauge field) for principal circle bundles over complex flag manifolds by using elements of representation theory of simple Lie algebras. This description allows us to compute explicitly the expression of the contact form for any Boothby-Wang fibration over complex flag manifolds as well as their induced homogeneous Sasaki-Einstein structures. As an application of our results we use the Cartan-Remmert reduction and the Calabi ansatz technique to provide many explicit examples of crepant resolutions of Calabi-Yau cones with certain homogeneous Sasaki-Einstein manifolds realized as links of isolated hypersurface singularities.

Lowest degree invariant second-order PDEs over rational homogeneous contact manifolds

For each simple Lie algebra [Formula: see text] (excluding, for trivial reasons, type [Formula: see text]), we find the lowest possible degree of an invariant second-order PDE over the adjoint variety in [Formula: see text], a homogeneous contact manifold. Here a PDE [Formula: see text] has degree [Formula: see text] if [Formula: see text] is a polynomial of degree [Formula: see text] in the minors of [Formula: see text], with coefficient functions of the contact coordinate [Formula: see text], [Formula: see text], [Formula: see text] (e.g., Monge–Ampère equations have degree 1). For [Formula: see text] of type [Formula: see text] or [Formula: see text], we show that this gives all invariant second-order PDEs. For [Formula: see text] of types [Formula: see text] and [Formula: see text], we provide an explicit formula for the lowest-degree invariant second-order PDEs. For [Formula: see text] of types [Formula: see text] and [Formula: see text], we prove uniqueness of the lowest-degree invariant second-order PDE; we also conjecture that uniqueness holds in type [Formula: see text].

FANO MANIFOLDS WITH GEOMETRIC STRUCTURES MODELED AFTER HOMOGENEOUS CONTACT MANIFOLDS

In this paper we present a study on geometric structures modeled after homogeneous contact manifolds and show that on Fano manifolds these geometric structures are locally isomorphic to the standard geometric structures on the model spaces. This conclusion is analogous to those of [13, 7]. We expect that this work will help prove the conjecture that a compact quaternionic Kähler manifold of positive scalar curvature is a quaternionic symmetric space [21].

Rigidity of homogeneous contact manifolds under Fano deformation.

Rigidity of homogeneous contact manifolds under Fano deformation.

Five-dimensional homogeneous contact manifolds and related problems

Five-dimensional homogeneous contact manifolds and related problems

Homogeneous contact manifolds and almost Finsler manifolds

Homogeneous contact manifolds and almost Finsler manifolds