We study the Kähler-Ricci flow on Fano manifolds. We show that if the curvature is bounded along the flow and if the manifold is K-polystable and asymptotically Chow semistable, then the flow converges exponentially fast to a Kähler-Einstein metric.

It has been shown recently by Kapustin and Tomasiello that the mathematical notion of Hamiltonian actions on twisted generalized Kähler manifolds is in perfect agreement with the physical notion of general (2, 2) gauged sigma models with three-form fluxes. In this article, we study the twisted equivariant cohomology theory of Hamiltonian actions on H-twisted generalized complex manifolds. If th...

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 review the information geometry of linear systems and its application to Bayesian inference, and the simplification available in the Kähler manifold case. We find conditions for the information geometry of linear systems to be Kähler, and the relation of the Kähler potential to information geometric quantities such as α-divergence, information distance and the dual αconnection structure. The...

Let X be a compact Kähler manifold of dimension n. For 1 ⩽ j m , let T closed positive (1,1)-current on whose cohomology class is Kähler. current bi-degree ( p ) with + n . We prove that if … are T-relative full mass intersection, then the non-pluripolar product equal to Dinh–Sibony

According to the conjecture of Calabi, on a complex manifold X with ample canonical bundle KX , there should exist a Kähler-Einstein metric g. Namely, a metric satisfying Ricg = −ωg, where ωg is the Kähler form of the Kähler metric g. The existence of such metric when X is compact was proved by Aubin and Yau ([23]) using complex Monge-Ampère equation. This important result has many applications...

We prove that a Kähler manifold with positive bisectional curvature and maximal diameter is isometric to complex projective space the Fubini-Study metric.

An important question with a rich history is the extent to which the symplectic category is larger than the Kähler category. Many interesting examples of non-Kähler symplectic manifolds have been constructed [T] [M] [G]. However, sufficiently large symmetries can force a symplectic manifolds to be Kähler [D] [Kn]. In this paper, we solve several outstanding problems by constructing the first sy...

It is proved that a compact Kähler manifold whose Ricci tensor has two distinct constant non-negative eigenvalues is locally the product of two Kähler–Einstein manifolds. A stronger result is established for the case of Kähler surfaces. Without the compactness assumption, irreducible Kähler manifolds with Ricci tensor having two distinct constant eigenvalues are shown to exist in various situat...

The celebrated Kodaira theorem [6] says that a compact complex manifold is projective if and only if it admits a Kähler form whose cohomology class is integral. This suggests that Kähler geometry is an extension of projective geometry, obtained by relaxing the integrality condition on a Kähler class. This point of view, together with the many restrictive conditions on the topology of Kähler man...