Related papers: CM Stability And The Generalized Futaki Invariant …

Based on the Cayley, Grothendieck, Knudsen Mumford theory of determinants we extend the CM polarization to the Hilbert scheme. We identify the weight of this refined line bundle with the generalized Futaki invariant of Donaldson. We are…

We identify the difference between the CM polarisation and the Chow polarisation on the ``Hilbert scheme''. As a consequence, we give a numerical criterion for the CM stability as in Mumfords' G.I.T.. Also, we write down an explicit formula…

We study the Futaki invariant and the Mabuchi K-energy of a K\"ahler manifold $M$ using the Deligne pairing technique developed in earlier papers. We first prove a rather simple characterization of the Futaki character: The Futaki character…

In this paper, we discuss the relative $K$-stability and the modified $K$-energy associated to the Calabi's extremal metric on toric manifolds. We give a sufficient condition in the sense of convex polytopes associated to toric manifolds…

We consider the problem of existence of constant scalar curvature Kaehler metrics on complete intersections of sections of vector bundles. In particular we give general formulas relating the Futaki invariant of such a manifold to the weight…

Let ${\rm k}$ be an algebraically closed field of characteristic 0 and $G$ a connect, reductive group over it. Let $X$ be a projective $G$-variety of complexity 1. We classify $G$-equivariant normal test configurations of $X$ with integral…

We study the weighted constant scalar curvature, a modified scalar curvature introduced by Lahdili depending on weight functions $(v, w)$, on certain non-compact semisimple toric fibrations, a generalization of the Calabi Ansatz defined by…

We show that a compact weighted extremal Kahler manifold (as defined by the third named author) has coercive weighted Mabuchi energy with respect to a maximal complex torus in the reduced group of complex automorphisms. This provides a vast…

In this paper we compute the Futaki invariant of adiabatic Kaehler classes on resolutions of Kaehler orbifolds with isolated singularities. Combined with previous existence results of extremal metrics by Arezzo-Lena-Mazzieri, this gives a…

In this note we identify the leading terms of the (reduced) K-energy map with a universal linear combination of the principal and subdominant coefficients of the weight of the $mth$ Hilbert point. This shows that the weight…

This article is an expository introduction to our paper Convexity of the K-energy and Uniqueness of Extremal metrics. We present the main ideas behind the proof that Mabuchi's K-energy functional is convex along weak geodesics in the space…

The Futaki invariant is a fundamental tool in K\"ahler geometry representing an obstruction to the existence of K\"ahler-Einstein metrics. Recently, it was generalized to compact complex manifolds. In this paper, we prove that it vanishes…

The energy, the $p$-energy ($p\in\mathbb{R}$ with $p\geq 2$) and the extrinsic $k$-energy ($k\in\mathbb{N}$) for maps between Riemannian manifolds are central objects in the geometric calculus of variations. The equator map from the unit…

If $M$ is a projective manifold in $P^N$, then one can associate to each one parameter subgroup $H$ of $SL(N+1)$ the Mumford $\mu$ invariant. The manifold $M$ is Chow-Mumford stable if $\mu$ is positive for all $H$. Tian has defined the…

In this paper we extend the notion of Futaki invariant to big and nef classes in such a way that it defines a continuous function on the \K\ cone up to the boundary. We apply this concept to prove that reduced normal crossing singularities…

As recently pointed out by Li and Xu, the definition of K-stability, and the author's proof of K-stability for cscK manifolds without holomorphic vector fields, need to be altered slightly: the Donaldson-Futaki invariant is positive for all…

Consider a polarized complex manifold (X,L) and a ray of positive metrics on L defined by a positive metric on a test configuration for (X,L). For most of the common functionals in K\"ahler geometry, we prove that the slope at infinity…

We develop the connection between equivariant completions of algebraic homogeneous spaces of reductive groups and lower bounds for the Mabuchi energy of a polarized manifold over the space of Bergman metrics. We provide a new definition of…

In this paper, we study the limiting properties of the $K$ energy for smooth hypersurfaces in the projective spaces. Our result generalizes the result of Ding-Tian (W. Ding and G. Tian. K\"ahler-Einstein metrics and the generalized Futaki…

We give a formula of the Donaldson-Futaki invariants for certain type of semi test configurations, which essentially generalizes Ross-Thomas' slope theory. The positivity (resp. non-negativity) of those "a priori special" Donaldson-Futaki…