We construct a Petersson-Weil type Kähler form on the moduli spaces of Higgs bundles over a compact Kähler manifold. A fiber integral formula for this form is proved, from which it follows that the Petersson-Weil form is the curvature of a certain determinant line bundle, equipped with a Quillen metric, on the moduli space of Higgs bundles over a projective manifold. The curvature of the Peters...

In the same way that a contact manifold determines and is determined by a symplectic cone, a Sasaki manifold determines and is determined by a suitable Kähler cone. KählerSasaki geometry is the geometry of these cones. This paper presents a symplectic action-angle coordinates approach to toric Kähler geometry and how it was recently generalized, by Burns-Guillemin-Lerman and Martelli-Sparks-Yau...

Abstract In this paper, we prove a conjecture raised by Angella, Otal, Ugarte and Villacampa recently, which states that if the Strominger connection (also known as Bismut connection) of compact Hermitian manifold is Kähler-like, in sense its curvature tensor obeys all symmetries Kähler manifold, then metric must be pluriclosed. What actually showed bit more: for any given Kähler-like condition...

We prove the existence of Kähler-Einstein metric on a K-stable Fano manifold using the recent compactness result on Kähler-Ricci flows. The key ingredient is an algebro-geometric description of the asymptotic behavior of Kähler-Ricci flow on Fano manifolds. This is in turn based on a general finite dimensional discussion, which is interesting in its own and could potentially apply to other prob...

Theorem 1. Suppose (M; g) is a noncompact complete Riemannian manifold with Ric (n 1), we have 0 (n 1) =4. The estimate is sharp since the spectrum is the ray [(n 1) =4;+1) for the hyperbolic space H. For Kähler manifolds, this estimate can be improved. On a Kähler manifold (M; g) of complex dimension n, where g is the Riemannian metric, let ! = g (J ; ) be the Kähler form. In local holomorphic...

In the paper [SY80] it was shown that a Kähler manifold with strictly positive bisectional curvature was biholomorphic to CP. In this paper, we use the techniques developed by [SY80], to prove that a compact Kähler manifold with positive orthogonal bisectional curvature is biholomorphic to CP, a condition strictly weaker than positive bisectional curvature. This gives a direct elliptic proof of...

A para-Kähler manifold can be defined as a pseudoRiemannian manifold (M, g) with a parallel skew-symmetric paracomplex structures K, i.e. a parallel field of skew-symmetric endomorphisms with K = Id or, equivalently, as a symplectic manifold (M, ω) with a bi-Lagrangian structure L, i.e. two complementary integrable Lagrangian distributions. A homogeneous manifold M = G/H of a semisimple Lie gro...

Let M 1 and M 2 be two Kähler manifolds. We call M 1 and M 2 relatives if they share a non-trivial Kähler submanifold S, namely, if there exist two holomorphic and isometric immersions (Kähler immersions) h 1 : S → M 1 and h 2 : S → M 2. Moreover, two Kähler manifolds M 1 and M 2 are said to be weakly relatives if there exist two locally isometric (not necessarily holomorphic) Kähler manifolds ...

Let M1 and M2 be two Kähler manifolds. We call M1 and M2 relatives if they share a non-trivial Kähler submanifold S, namely, if there exist two holomorphic and isometric immersions (Kähler immersions) h1 : S → M1 and h2 : S → M2. Moreover, two Kähler manifolds M1 and M2 are said to be weakly relatives if there exist two locally isometric (not necessarily holomorphic) Kähler manifolds S1 and S2 ...

We know that the existence of Kähler structure on a compact complex manifold imposes certain homological or even homotopical restrictions on its underlining topological manifold. Hodge theory is of central importance in this line. There have been recently certain extensions and progresses in this area of research. Among them is the field of Kähler groups, in which the main subject to study is t...