A classical theorem of Frankel for compact Kähler manifolds states that a Kähler S-action is Hamiltonian if and only if it has fixed points. We prove a metatheorem which says that when Hodge theory holds on non-compact manifolds, then Frankel’s theorem still holds. Finally, we present several concrete situations in which the assumptions of the metatheorem hold. 1. The Classical Frankel Theorem ...

A simple geometric procedure is proposed for constructing Wick symbols on cotangent bundles to Riemannian manifolds. The main ingredient of the construction is a method of endowing the cotangent bundle with a formal Kähler structure. The formality means that the metric is lifted from the Riemannian manifold Q to its phase space T ∗Q in the form of formal power series in momenta with the coeffic...

Statements analogous to the Hard Lefschetz Theorem (HLT) and the Hodge-Riemann bilinear relations (HRR) hold in a variety of contexts: they impose severe restrictions on the cohomology algebra of a smooth compact Kähler manifold or on the intersection cohomology of a projective toric variety; they restrict the local monodromy of a polarized variation of Hodge structure; they impose conditions o...

We use hyperbolic geometry to construct simply-connected symplectic or complex manifolds with trivial canonical bundle and with no compatible Kähler structure. We start with the desingularisations of the quadric cone in C: the smoothing is a natural S-bundle over H, its holomorphic geometry is determined by the hyperbolic metric; the small-resolution is a natural S-bundle over H with symplectic...

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 ...

In this paper we present a two-field inflation model, which distinguishes itself with a non-canonical kinetic lagrangian and comes from the large volume approach to the moduli stabilization in flux compactification of type IIB superstring on a Calabi-Yau orientifold of h(1,2) > h(1,1) ≥ 4. The Kähler moduli are classified as volume modulus, heavy moduli and two light moduli. The axion-dilaton, ...

We study complete noncompact long time solutions (M, g(t)) to the Kähler-Ricci flow with uniformly bounded nonnegative holomorphic bisectional curvature. We will show that when the Ricci curvature is positive and uniformly pinched, i.e. Ri̄ ≥ cRgi̄ at (p, t) for all t for some c > 0, then there always exists a local gradient Kähler-Ricci soliton limit around p after possibly rescaling g(t) alon...

The study of compact Kähler manifolds made by Hodge and others shows that a Kähler structure imposes very strong conditions on the homotopy type of a compact complex manifold X . In particular, unlike in the case of compact differentiable or closed complex manifolds, not every finitely presented group G is the fundamental group of a compact Kähler manifold. Such groups are called Kähler groups....

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. ...

The main purpose of the present work is to investigate statistical manifolds endowed with almost product structures. We prove that structure a para-Kähler-like manifold constant curvature in Kurose's sense Hessian structure. also derive properties submersions which are compatible results illustrated by several nontrivial examples.