https://maths-people.anu.edu.au/~wang/publications/2-AdvMath2004.pdf

Kahler-Ricci solitons on toric manifolds with positive first Chern class

In this paper we study compact Sasaki manifolds in view of transverse Kahler geometry and extend some results in Kahler geometry to Sasaki manifolds. In particular we define integral invariants which obstruct the existence of transverse Kahler metric with harmonic Chern forms. The integral invariant $f1$ for the first Chern class case becomes an obstruction to the existence of transverse Kahler metric of constant scalar curvature. We prove the existence of transverse Kahler-Ricci solitons (or Sasaki-Ricci soliton) on compact toric Sasaki manifolds whose basic first Chern form of the normal bundle of the Reeb foliation is positive and the first Chern class of the contact bundle is trivial. We will further show that if $S$ is a compact toric Sasaki manifold with the above assumption then by deforming the Reeb field we get a Sasaki-Einstein structure on S. As an application we obtain Sasaki-Einstein metrics on the $U(1)$-bundles associated with the canonical line bundles of toric Fano manifolds, including as a special case an irregular toric Sasaki-Einstein metrics on the unit circle bundle associated with the canonical bundle of the two-point blow-up of the complex projective plane.

/pdf/transverse-kahler-geometry-of-sasaki-manifolds-and-toric-4r49cmr84z.pdf

Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds

In this paper, we prove a theorem on convergence of Kahler-Ricci flow on a compact Kahler manifold which admits a Kahler-Ricci soliton

/pdf/convergence-of-kahler-ricci-flow-1k04vjx5ci.pdf

Convergence of Kähler-Ricci flow

For $\phi $ a metric on the anticanonical bundle, $-K_X$, of a Fano manifold $X$ we consider the volume of $X$

$$\begin{aligned} \int _X e^{-\phi }. \end{aligned}$$

In earlier papers we have proved that the logarithm of the volume is concave along geodesics in the space of positively curved metrics on $-K_X$. Our main result here is that the concavity is strict unless the geodesic comes from the flow of a holomorphic vector field on $X$, even with very low regularity assumptions on the geodesic. As a consequence we get a simplified proof of the Bando–Mabuchi uniqueness theorem for Kahler–Einstein metrics. A generalization of this theorem to ‘twisted’ Kahler–Einstein metrics and some classes of manifolds that satisfy weaker hypotheses than being Fano is also given. We moreover discuss a generalization of the main result to other bundles than $-K_X$, and finally use the same method to give a new proof of the theorem of Tian and Zhu on uniqueness of Kahler–Ricci solitons.

A Brunn-Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry

Well-known conjectures of Tian predict that existence of canonical Kahler metrics should be equivalent to various notions of properness of Mabuchi's K-energy functional. In some instances this has been verified, especially under restrictive assumptions on the automorphism group. We provide counterexamples to the original conjecture in the presence of continuous automorphisms. The construction hinges upon an alternative approach to properness that uses in an essential way the metric completion with respect to a Finsler metric and its quotients with respect to group actions. This approach also allows us to formulate and prove new optimal versions of Tian's conjecture in the setting of smooth and singular Kahler-Einstein metrics, with or without automorphisms, as well as for Kahler-Ricci solitons. Moreover, we reduce both Tian's original conjecture (in the absence of automorphisms) and our modification of it (in the presence of automorphisms) in the general case of constant scalar curvature metrics to a conjecture on regularity of minimizers of the K-energy in the Finsler metric completion. Finally, our results also resolve Tian's conjecture on the Moser-Trudinger inequality for Fano manifolds with Kahler-Einstein metrics.

Tian’s properness conjectures and Finsler geometry of the space of Kähler metrics

In this paper, we discuss the relation between the existence of Kahler–Ricci solitons and a certain functional associated to some complex Monge–Ampere equation on compact complex manifolds with positive first Chern class. In particular, we obtain a strong inequality of Moser–Trudinger type on a compact complex manifold admitting a Kahler–Ricci soliton.

/pdf/kahler-ricci-solitons-on-compact-complex-manifolds-with-c-1-1uz1zelai6.pdf

Xiaohua Zhu

Papers

Kähler-Ricci solitons on compact complex manifolds with C 1(M) > 0