A Bochner flat Kähler manifold is a Kähler manifold with vanishing Bochner curvature tensor. We shall give a uniformization of Bochner flat Kähler manifolds. One of the aims of this paper is to give a correction to the proof of our previous paper [9] concerning uniformization of Bochner flat Kähler manifolds. A Bochner flat locally conformal Kähler manifold is a locally conformal Kähler manifol...

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 ...

We simplify the classification of supersymmetric solutions with compact holonomy of the Killing spinor equations of heterotic supergravity using the field equations and the additional assumption that the 3-form flux is closed. We determine all the fractions of supersymmetry that the solutions preserve and find that there is a restriction on the number of supersymmetries which depends on the iso...

For certain compact complex Fano manifolds M with reductive Lie algebras of holomorphic vector fields, we determine the analytic subvariety of the second cohomology group of M consisting of Kähler classes whose Bando-Calabi-Futaki character vanishes. Then a Kähler class contains a Kähler metric of constant scalar curvature if and only if the Kähler class is contained in the analytic subvariety....

A geometry with parallel skew-symmetric torsion is a Riemannian manifold carrying metric connection torsion. Besides the trivial case of Levi-Civita connection, geometries non-vanishing arise naturally in several geometric contexts, e.g. on reductive homogeneous spaces, nearly Kähler or G2-manifolds, Sasakian and 3-Sasakian manifolds, twistor spaces over quaternion-Kähler manifolds positive sca...

We calculate the Chern classes and Chern numbers for the natural almost Hermitian structures of the partial flag manifolds Fn = SU(n + 2)/S(U(n) × U(1) × U(1)). For all n > 1 there are two invariant complex algebraic structures, which arise from the projectivizations of the holomorphic tangent and cotangent bundles of CP. The projectivization of the cotangent bundle is the twistor space of a Gr...

1 Riemann Poisson manifolds were introduced by the author in [1] and studied in more details in [2]. Kähler-Riemann foliations form an interesting subset of the Riemannian foliations with remarkable properties ( see [3]). In this paper we will show that to give a regular Riemann Poisson structure on a manifold M is equivalent to to give a Kähler-Riemann foliation on M such that the leafwise sym...

One of the methods to obtain Frobenius manifold structures is via DGBV (differential Gerstenhaber-Batalin-Vilkovisky) algebra construction. An important problem is how to identify Frobenius manifold structures constructed from two different DGBV algebras. For DGBV algebras with suitable conditions, we show the functorial property of a construction of deformations of the multiplicative structure...

We give an exposition of a theorem of Hirzebruch, Kodaira and Yau which proves the uniqueness of the Kähler structure of complex projective space, and of Yau’s resolution of the Severi Conjecture.

In this note we show that Tolman’s example (of a six dimensional Hamiltonian T -space with isolated fixed points and no compatible Kähler structure) can be constructed from the flag variety U(3)/U(1) by U(2)-equivariant symplectic surgery. This implies that Tolman’s space has a “transversal multiplicity-free” action of U(2) and that Delzant’s theorem “every compact multiplicity-free torus actio...