Spheres, Kähler geometry and the Hunter–Saxton system

Lectures on Kähler Geometry

On Nearly Kähler Geometry *

We consider complete nearly Kähler manifolds with the canonical Hermitian connection. We prove some metric properties of strict nearly Kähler manifolds and give a sufficient condition for the reducibility of the canonical Hermitian connection. A holonomic condition for a nearly Kähler manifold to be a twistor space over a quaternionic Kähler manifold is given. This enables us to give classifica...

Recent Progress in Kähler Geometry ∗

In recent years, there are many progress made in Kähler geometry. In particular, the topics related to the problems of the existence and uniqueness of extremal Kähler metrics, as well as obstructions to the existence of such metrics in general Kähler manifold. In this talk, we will report some recent developments in this direction. In particular, we will discuss the progress recently obtained i...

Torsion in Almost Kähler Geometry *

We study almost Kähler manifolds whose curvature tensor satisfies the second curvature condition of Gray (shortly AK 2). This condition is interpreted in terms of the first canonical Hermitian connection. It turns out that this condition forces the torsion of this connection to be parallel in directions orthogonal to the Kähler nullity of the almost complex structure. We prove a local structure...

Tensor supermultiplets and toric quaternion - Kähler geometry †

We review the relation between 4n-dimensional quaternion-Kähler metrics with n + 1 abelian isometries and superconformal theories of n + 1 tensor supermultiplets. As an application we construct the class of eight-dimensional quaternion-Kähler metrics with three abelian isometries in terms of a single function obeying a set of linear second-order partial differential equations. Talk given by F.S...