We survey briefly the definition of the Rozansky-Witten invariants, and review their relevance to the study of compact hyperkähler manifolds. We consider how various generalisations of the invariants might prove useful for the study of non-compact hyperkähler manifolds, of quaternionic-Kähler manifolds, and of relations between hyperkähler manifolds and Lie algebras. The paper concludes with a ...

We obtain an embedding theorem for compact strongly pseudoconvex CR manifolds which are bounadries of some complete Hermitian manifolds. We use this to compactify some negatively curved Kähler manifolds with compact strongly pseudoconvex boundary. An embedding theorem for Sasakian manifolds is also derived.

This article is concerned with conformal harmonic maps f : Σ → M from a closed connected surface Σ into a compact Riemannian manifoldM of dimension at least four, studied via a perturbative approach based upon the α-energy of Sacks and Uhlenbeck. It gives an estimate on the rate of growth of energy density in the bubbles of minimax α-energy critical points as α→ 1, when the bubbles are at a dis...

The paradigmatic result in symplectic toric geometry is the paper of Delzant that classifies compact connected symplectic manifolds with effective completely integrable torus actions, the so called (compact) symplectic toric manifolds. The moment map induces an embedding of the quotient of the manifold by the torus action into the dual of the Lie algebra of the torus; its image is a simple unim...

We show a number of applications to geometry of the study of cohomology algebras of various kinds of manifolds. The main tool is Hodge theory, and we use it to show that projective complex manifolds are more restricted topologically than compact Kähler manifolds. We also make explicit numerous constraints satisfied by cohomology algebras of compact Kähler manifolds, making them very non generic...

This paper extends spline methods to compact Riemannian manifolds in an rkhs setting. The approach is to use the mathematical framework of rkhs, along with integrating spectral geometry associated with compact Riemannian manifolds. This combination aarmatively answers a conjecture made by Wahba (1981) that spline interpolation and smoothing available for the 2{sphere can be generalized to compa...

The so-called Kodaira problem left open by this result asked whether more generally any compact Kähker manifold can be deformed to a projective complex manifold. Recently, we solved negatively this question by constructing, in any dimension n ≥ 4, examples of compact Kähler manifolds, which do not deform to projective complex manifolds, as a consequence of the following stronger statement conce...

In this paper I construct, using off the shelf components, a compact symplectic manifold with a non-trivial Hamiltonian circle action that admits no Kaehler structure. The non-triviality of the action is guaranteed by the existence of an isolated fixed point. The motivation for this work comes from the program of classification of Hamiltonian group actions. The Audin-Ahara-Hattori-Karshon class...

A new class of compact Kähler manifolds, called special, is defined: they are the ones having no fibration with base an orbifold of general type, the orbifold structure on the base being defined by the multiple fibres of the fibration (see §1). The special manifolds are in many respect higher-dimensional generalisations of rational and elliptic curves. For example, we show that being rationally...