Real Monge-Ampere equations and Kahler-Ricci solitons on toric log Fano varieties

We show, using a direct variational approach, that the second boundary value problem for the Monge-Amp\`ere equation in R^n with exponential non-linearity and target a convex body P is solvable iff 0 is the barycenter of P. Combined with some toric geometry this confirms, in particular, the (generalized) Yau-Tian-Donaldson conjecture for toric log Fano varieties (X,D), saying that (X,D) admits a (singular) K\"ahler-Einstein metric iff it is K-stable in the algebro-geometric sense. We thus obtain a new proof and extend to the log Fano setting the seminal result of Zhou-Wang concerning the case when X is smooth and D is trivial. Li's toric formula for the greatest lower bound on the Ricci curvature is also generalized. More generally, we obtain K\"ahler-Ricci solitons on any log Fano variety and show that they appear as the large time limit of the K\"ahler-Ricci flow. Furthermore, using duality, we also confirm a conjecture of Donaldson concerning solutions to Abreu's boundary value problem on the convex body P. in the case of a given canonical measure on the boundary of P.Comment: 53 page

A Brunn-Minkowski type inequality for Fano manifolds and some uniqueness theorems in K\"ahler geometry

For $\phi$ a metric on the anticanonical bundle, $-K_X$, of a Fano manifold $X$ we consider the volume of $X$ $$\int_X e^{-\phi}.$$ We prove that the logarithm of the volume is concave along bounded geodesics in the space of positively curved metrics on $-K_X$ and that the concavity is strict unless the geodesic comes from the flow of a holomorphic vector field on $X$. As a consequence we get a simplified proof of the Bando-Mabuchi uniqueness theorem for K\"ahler - Einstein metrics. A generalization of this theorem to 'twisted' K\"ahler-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 of uniqueness of K\"ahler-Ricci solitons. This is an expanded version of an earlier preprint, "A Brunn-Minkowski type inequality for Fano manifolds and the Bando-Mabuchi uniqueness theorem", arXiv:1103.0923Comment: This is a revised and expanded version of ArXiv 1103.092

On Sasaki-Ricci solitons and their deformations

We extend to the Sasakian setting a result of Tian and Zhu about the decomposition of the Lie algebra of holomorphic vector fields on a Kähler manifold in the presence of a Kähler-Ricci soliton. Furthermore we apply known deformations of Sasakian structures to a Sasaki-Ricci soliton to obtain a stability result concerning generalized Sasaki-Ricci solitons, generalizing results of Li in the Kähler setting and of He and Sun by relaxing some of their assumptions. © 2016 by Walter de Gruyter

K\"ahler immersions of K\"ahler-Ricci solitons into definite or indefinite complex space forms

Let $(g, X)$ be a K\"ahler-Ricci soliton on a complex manifold $M$. We prove that if the K\"ahler manifold $(M, g)$ can be K\"ahler immersed into a definite or indefinite complex space form of constant holomorphic sectional curvature $2c$, then $g$ is Einstein. Moreover, its Einstein constant is a rational multiple of $c$