Related papers: Test configurations and Geodesic rays

This article contains a detailed study, in the toric case, of the test configuration geodesic rays defined by Phong-Sturm. We show that the `Bergman approximations' of Phong-Sturm converge in C^1 to the geodesic ray and that the geodesic…

Geodesic rays of class C^{1,1} are constructed for any test configuration of a positive line bundle L on X using resolution of singularities. The construction reduces to finding a subsolution of the corresponding Monge-Ampere equation.…

Let $X$ be a compact complex manifold, $L\to X$ an ample line bundle over $X$, and ${\cal H}$ the space of all positively curved metrics on $L$. We show that a pair $(h_0,T)$ consisting of a point $h_0\in {\cal H}$ and a test configuration…

We establish the essentially optimal form of Donaldson's geodesic stability conjecture regarding existence of constant scalar curvature K\"ahler metrics. We carry this out by exploring in detail the metric geometry of Mabuchi geodesic rays,…

In this note, we consider a sequence of test configurations compatible with a Kaehler metric in $c_1(L)$ on a polarized algebraic manifold $(X,L)$. Then an explicit formula for the Donaldson-Futaki invariant $F_1$ for the sequence will be…

It is shown that the geodesic rays constructed as limits of Bergman geodesics from a test configuration are always of class $C^{1,\alpha}, 0<\alpha<1$. An essential step is to establish that the rays can be extended as solutions of a…

Let $(M, \omega, J)$ be a K\"ahler manifold, equipped with an effective Hamiltonian torus action $\rho: T \rightarrow \mathrm{Diff}(M, \omega, J)$ by isometries with moment map $\mu: M \rightarrow \mathfrak{t}^{*}$. We first construct a…

We study geodesics of the form $\gamma(t)=\pi(\exp(tX)\exp(tY))$, $X,Y\in \fr{g}=\operatorname{Lie}(G)$, in homogeneous spaces $G/K$, where $\pi:G\rightarrow G/K$ is the natural projection. These curves naturally generalise homogeneous…

We partially confirm an old conjecture of Donaldson that if there exists a cscK metrics in a given K\"ahler class, then there is no degenerated geodesic ray which is tamed by a bounded ambient geometry unless it parallels to a holomorphic…

We prove the Yau-Tian-Donaldson conjecture for cohomogeneity one manifolds, that is, for projective manifolds equipped with a holomorphic action of a compact Lie group with at least one real hypersurface orbit. Contrary to what seems to be…

We consider homogeneous spaces of Lie groups with compact stabilizer subgroups of two types: those with integrable invariant distributions and those with geodesic orbit invariant Riemannian metrics. The latter means that for an arbitrary…

This paper is concerned with a relative uniform Yau--Tian--Donaldson correspondence, in terms of test configurations, for the projectivization \( \mathbb{P}(E) \) of a holomorphic vector bundle \( E \) over a smooth curve. For any K\"ahler…

Let $(X, D)$ be a log variety with an effective holomorphic torus action, and $\Theta$ be a closed positive $(1,1)$-current. For any smooth positive function $g$ defined on the moment polytope of the torus action, we study the…

In this note we investigate the regularity of geodesics in the space of convex and plurisubharmonic functions. In the real setting we prove (optimal) local C^{1,1} regularity. We construct examples which prove that the global C^{1,1}…

Suppose $(X,\omega)$ is a compact K\"ahler manifold of dimension $n$, and $\theta$ is closed $(1,1)$-form representing a big cohomology class. We introduce a metric $d_1$ on the finite energy space $\mathcal{E}^1(X,\theta)$, making it a…

Let $Y$ be a compact K\"ahler normal space and $\alpha \in H^{1,1}(Y,\mathbb{R})$ a K\"ahler class. We study metric properties of the space $\mathcal{H}_\alpha$ of K\"ahler metrics in $\alpha$ using Mabuchi geodesics. We extend several…

Let $S$ be a compact, connected, oriented surface, possibly with boundary, of negative Euler characteristic. In this article we extend Lindenstrauss-Mirzakhani's and Hamenst\"adt's classification of locally finite mapping class group…

In this follow up work to [45, 33, 32, 46] we introduce and study a notion of geodesic stability restricted to rays with prescribed singularity types. A number of notions of interest fit into this framework, in particular algebraic- and…

We continue the research on the structure of complex geodesics in tube domains over (bounded) convex bases. In some special cases a more explicit form of the geodesics than the existing ones are provided. As one of the consequences of our…

The first goal of the article is to solve several fundamental problems in the theory of holomorphic bundles over non-algebraic manifolds: For instance we prove that stability and semi-stability are Zariski open properties in families when…