In 1983, Futaki [2] introduced his invariants which generalize the obstruction of Kazdan-Warner to prescribe Gauss curvature on S. The Futaki invariants are defined for any compact Kähler manifold with positive first Chern class that has nontrivial holomorphic vector fields. Their vanishing are necessary conditions to the existence of Kähler-Einstein metric on the underlying manifold. Let M be ...

A symplectic cut of a manifold M with a Hamiltonian circle action is a symplectic quotient of M × C. If M is Kähler then, since C is Kähler, the cut space is Kähler as well. The symplectic structure on the cut is well understood. In this paper we describe the complex structure (and hence the metric) on the cut. We then generalize the construction to the case where M has a torus action and C is ...

Let (Mn, g) be a compact Kähler manifold with nonpositive bisectional curvature. We show that a finite cover is biholomorphic and isometric to a flat torus bundle over a compact Kähler manifold Nk with c1 < 0. This confirms a conjecture of Yau. As a corollary, for any compact Kähler manifold with nonpositive bisectional curvature, the Kodaira dimension is equal to the maximal rank of the Ricci ...

We prove that any Kähler manifold admitting a flat complex conformal connection is a Bochner-Kähler manifold with special scalar distribution and zero geometric constants. Applying the local structural theorem for such manifolds we obtain a complete description of the Kähler manifolds under consideration.

We prove the correspondence between the information geometry of a signal filter and a Kähler manifold. The information geometry of a minimum-phase linear system with a finite complex cepstrum norm is a Kähler manifold. The square of the complex cepstrum norm of the signal filter corresponds to the Kähler potential. The Hermitian structure of the Kähler manifold is explicitly emergent if and onl...

A Kähler-Nijenhuis manifold is a Kähler manifold M , with metric g, complex structure J and Kähler form Ω, endowed with a Nijenhuis tensor field A that is compatible with the Poisson structure defined by Ω in the sense of the theory of Poisson-Nijenhuis structures. If this happens, and if AJ = ±JA, M is foliated by im A into non degenerate Kähler-Nijenhuis submanifolds. If A is a non degenerate...

We prove a Liouville property of holomorphic maps from a complete Kähler manifold with nonnegative holomorphic bisectional curvature to a complete simply connected Kähler manifold with a certain assumption on the sectional curvature.

In this paper, we prove that Kähler-Ricci flow converges to a Kähler-Einstein metric (or a Kähler-Ricci soliton) in the sense of Cheeger-Gromov as long as an initial Kähler metric is very closed to gKE (or gKS) if a compact Kähler manifold with c1(M) > 0 admits a Kähler Einstein metric gKE (or a Kähler-Ricci soliton gKS). The result improves Main Theorem in [TZ3] in the sense of stability of Kä...

We consider hyper-Kähler manifolds of complex dimension 4 which are fibrations. It is known that the fibers are abelian varieties and the base is P. We assume that the general fiber is isomorphic to a product of two elliptic curves. We prove that such a hyper-Kähler manifold is deformation equivalent to a Hilbert scheme of two points on a K3 surface. 1. Preliminaries First we define our main ob...

where g(t) is a family of Kähler metrics and Ric(ωg) denotes the Ricci curvature of g. It is known that for any smooth Kähler metric g0, there is a unique solution g(t) of (1.1) for some maximal time T > 0 with g(0) = g0. In general, T will depend on the initial metric g0. However, in Kähler manifold, this only depends on the Kähler class and the first Chern class. This observation plays a impo...