Let X be a compact Kähler manifold with zero first Chern class, and let L be an ample line bundle overX . The pair (X,L) is called a polarized Calabi-Yaumanifold. By Yau’s proof of the Calabi conjecture, we know such a manifold carries a unique Ricci flat metric compatable with the polarization (cf. [37]). Thus, the moduli space of such Ricci flat Kähler metrics is the moduli space of complex s...

In this article we investigate the geometry of CR-lightlike submanifolds in an indefinite Kähler product manifold. In particular, we obtain the necessary and sufficient conditions for a CR-lightlike submanifold in an indefinite Kähler product manifold to be either CR-lightlike product, or D-geodesic, or D′-geodesic. We also study totally umbilical and curvature-invariant CR-lightlike submanifol...

We prove that polarised manifolds that admit a constant scalar curvature Kähler (cscK) metric satisfy a condition we call slope semistability. That is, we define the slope µ for a projective manifold and for each of its subschemes, and show that if X is cscK then µ(Z) ≤ µ(X) for all subschemes Z. This gives many examples of manifolds with Kähler classes which do not admit cscK metrics, such as ...

A locally conformally Kähler (LCK) manifold M is one which is covered by a Kähler manifold M̃ with the deck transform group acting conformally on M̃ . If M admits a holomorphic flow, acting on M̃ conformally, it is called a Vaisman manifold. Neither the class of LCK manifolds nor that of Vaisman manifolds is stable under small deformations. We define a new class of LCK-manifolds, called LCK manifo...

We present dyonic BPS static black hole solutions for general d = 4, N = 2 supergravity theories coupled to vector and hypermultiplets. These solutions are generalisations of the spherically symmetric Majumdar-Papapetrou black hole solutions of Einstein-Maxwell gravity and are completely characterised by a set of constrained harmonic functions. In terms of the underlying special geometry, these...

A polarized Calabi-Yau manifold is a pair (X,ω) of a compact algebraic manifold X with zero first Chern class and a Kähler form ω ∈ H(X,Z). The form ω is called a polarization. Let M be the universal deformation space of (X,ω). M is smooth by the theorem of Tian [8]. By [9], we may assume that each X ′ ∈ M is a Kähler-Einstein manifold. i.e. the associated Kähler metric (g′ αβ ) is Ricci flat. ...

We prove the equivariant holomorphic Morse inequalities for a holomorphic torus action on a holomorphic vector bundle over a compact Kähler manifold when the fixed-point set is not necessarily discrete. Such inequalities bound the twisted Dolbeault cohomologies of the Kähler manifold in terms of those of the fixed-point set. We apply the inequalities to obtain relations of Hodge numbers of the ...

We prove that polarised manifolds that admit a constant scalar curvature Kähler (cscK) metric satisfy a condition we call slope semistability. That is, we define the slope µ for a projective manifold and for each of its subschemes, and show that if X is cscK then µ(Z) ≤ µ(X) for all subschemes Z. This gives many examples of manifolds with Kähler classes which do not admit cscK metrics, such as ...

We prove that polarised manifolds that admit a constant scalar curvature Kähler (cscK) metric satisfy a condition we call slope semistability. That is, we define the slope µ for a projective manifold and for each of its subschemes, and show that if X is cscK then µ(Z) ≤ µ(X) for all subschemes Z. This gives many examples of manifolds with Kähler classes which do not admit cscK metrics, such as ...