We investigate the limiting behavior of the unnormalized Kähler-Ricci flow on a Kähler manifold with a polarized initial Kähler metric. We prove that the Kähler-Ricci flow becomes extinct in finite time if and only if the manifold has positive first Chern class and the initial Kähler class is proportional to the first Chern class of the manifold. This proves a conjecture of Tian for the smooth ...

In this note, we prove that on an n-dimensional compact toric manifold with positive first Chern class, the Kähler-Ricci flow with any initial (S)-invariant Kähler metric converges to a Kähler-Ricci soliton. In particular, we give another proof for the existence of Kähler-Ricci solitons on a compact toric manifold with positive first Chern class by using the Kähler-Ricci flow. 0. Introduction. ...

This is a survey of recent contributions to the area of special Kähler geometry. It is based on lectures given at the 21st Winter School on Geometry and Physics held in Srni in January 2001. 1 Remarkable features of special Kähler manifolds A (pseudo-) Kähler manifold (M,J, g) is a differentiable manifold endowed with a complex structure J and a (pseudo-) Riemannian metric g such that (i) J is ...

In this paper we study curvature properties of semi - symmetric type of totally umbilical radical transversal lightlike hypersurfaces $(M,g)$ and $(M,widetilde g)$ of a K"ahler-Norden manifold $(overline M,overline J,overline g,overline { widetilde g})$ of constant totally real sectional curvatures $overline nu$ and $overline {widetilde nu}$ ($g$ and $widetilde g$ are the induced metrics on $M$...

We construct a Kähler structure (which we call a generalised Kähler cone) on an open subset of the cone of a strongly pseudo-convex CR manifold endowed with a 1-parameter family of compatible Sasaki structures. We determine those generalised Kähler cones which are Bochner-flat and we study their local geometry. We prove that any Bochner-flat Kähler manifold of complex dimension bigger than two ...

This paper discusses the Norden golden manifold having a constant sectional curvature. First, it is shown that if has real curvature, flat. For this reason, notions of holomorphic-like curvature and bisectional on are investigated, but seen these do not work manifold. shows need for new concept In direction, notion (Norden curvature) proposed, an example given, constant, tensor field expressed ...

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...

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 a theorem of Tian [5]. By [8], we may assume that each X ′ ∈ M is a Kähler-Einstein manifold. i.e. the associated Kähler metric (g′ αβ ) is Ricci flat. The

A Poincaré type Kähler metric on the complement X\D of a simple normal crossing divisor D, in a compact Kähler manifold X, is a Kähler metric onX\D with cusp singularity alongD. We relate the Futaki character for holomorphic vector fields parallel to the divisor, defined for any fixed Poincaré type Kähler class, to the classical Futaki character for the relative smooth class. As an application ...