In this paper we prove there exists a Kähler-Ricci soliton, unique up to holomorphic automorphisms, on any toric Kähler manifold with positive first Chern class, and the Kähler-Ricci soliton is a Kähler-Einstein metric if and only if the Futaki invariant vanishes.

We study the existence of extremal Kähler metrics on Kähler manifolds. After introducing a notion of relative K-stability for Kähler manifolds, we prove that Kähler manifolds admitting extremal Kähler metrics are relatively K-stable. Along the way, we prove a general Lp lower bound on the Calabi functional involving test configurations and their associated numerical invariants, answering a ques...

The global holomorphic invariant αG(X) introduced by Tian [?], Tian and Yau [?] is closely related to the existence of Kähler-Einstein metrics. In his solution to the Calabi conjecture, Yau [?] proved the existence of a Kähler-Einstein metric on compact Kähler manifolds with nonpositive first Chern class. Kähler-Einstein metrics do not always exist in the case when the first Chern class is posi...

Sasakian manifolds provide rich source of constructing new Einstein manifolds in odd dimensions [2]. They play some important role in the superstring theory in mathematical physics [19, 20]. There is a renewed interest on Sasakian manifolds recently. The present paper is devoted to the regularity analysis of a geodesic equation in the space of Sasakian metrics H (definition in (1.2)) and some o...

We consider the situation of a compact irreducible subvariety of a smooth compact complex variety equipped with a Kähler form preserved by a torus action. We study the image of that subvariety under the moment map of the Kähler form. 1. Main result The moment map of a Hamiltonian T -symplectic/Kähler manifold has been one of the main interests of study for mathematical as well as physical reaso...

We discuss the fibre bundle of co-adjoint orbits compact Lie groups, and show how it admits a compatible Kähler structure. The case unitary group allows us to reformulate geometric framework quantum information theory. In particular, we that Fisher tensor gives rise structure is sufficiently close generalise some classical result on orbits.

There are three separate approaches to the challenge of constructing WCW Kähler geometry and spinor structure. The first one relies on a direct guess of Kähler function. Second approach relies on the construction of Kähler form and metric utilizing the huge symmetries of the geometry needed to guarantee the mathematical existence of Riemann connection. The third approach relies on the construct...

<abstract><p>We construct a toric generalised Kähler structure on $ \mathbb{C}P^2 and show that the various structures such as complex structure, metric etc are expressed in terms of certain elliptic functions. We also compute potential integrals functions.</p></abstract>

In this paper we first consider the Hamiltonian action of a compact connected Lie group on anH-twisted generalized complexmanifold M. Given such an action, we define generalized equivariant cohomology and generalized equivariant Dolbeault cohomology. If the generalized complex manifoldM satisfies the ∂̄∂-lemma, we prove that they are both canonically isomorphic to (Sg)⊗HH(M), where (Sg ) is the ...

We present a simple derivation of the Ricci-flat Kähler metric and its Kähler potential on the canonical line bundle over arbitrary Kähler coset space equipped with the Kähler-Einstein metric. ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected]