On sharp lower bounds for Calabi type functionals and destabilizing properties of gradient flows

Let $X$ be a compact K\"ahler manifold with a given ample line bundle $L$. In \cite{Don05}, Donaldson proved that the Calabi energy of a K\"ahler metric in $c_1(L)$ is bounded from below by the supremum of a normalized version of the minus Donaldson--Futaki invariants of test configurations of $(X,L)$. He also conjectured that the bound is sharp. In this paper, we prove a metric analogue of Donaldson's conjecture, we show that if we enlarge the space of test configurations to the space of geodesic rays in $\mathcal{E}^2$ and replace the Donaldson--Futaki invariant by the radial Mabuchi K-energy $\mathbf{M}$, then a similar bound holds and the bound is indeed sharp. Moreover, we construct explicitly a minimizer of $\mathbf{M}$. On a Fano manifold, a similar sharp bound for the Ricci--Calabi energy is also derived.Comment: Final version. Statement of Theorem 4.1 corrected. To appear on Analysis & PD

Analytic Bertini theorem

We prove an analytic Bertini theorem, generalizing a previous result of Fujino and Matsumura

Analytic Bertini theorem

We prove an analytic Bertini theorem, generalizing a previous result of Fujino and Matsumura

Non-pluripolar products on vector bundles and Chern--Weil formulae on mixed Shimura varieties

In this paper, we develop the pluripotential-theoretic techniques for constructing the arithmetic intersection theory on mixed Shimura varieties. We first introduce the theory of non-pluripolar products on holomorphic vector bundles on complex manifolds. Then we define and study a special class of singularities of Hermitian metrics on vector bundles, called $\mathcal{I}$-good singularities, partially extending Mumford's notion of good singularities. Next, we derive a Chern--Weil type formula expressing the Chern numbers of Hermitian vector bundles with $\mathcal{I}$-good singularities on mixed Shimura varieties in terms of the associated b-divisors. We also define an intersection theory on the Riemann--Zariski space and apply it to reformulate our Chern--Weil formula. Finally, we define and study the Okounkov bodies of b-divisors.Comment: 67 pages, comments are welcome

Mabuchi geometry of big cohomology classes with prescribed singularities

Let $X$ be a compact K\"ahler manifold. Fix a big class $\alpha$ with smooth representative $\theta$ and a model potential $\phi$ with positive mass. We study the space $\mathcal{E}^p(X,\theta;[\phi])$ of finite energy K\"ahler potentials with prescribed singularity for each $p\geq 1$. We define a metric $d_p$ and show that $(\mathcal{E}^p(X,\theta;[\phi]),d_p)$ is a complete metric space. This construction generalizes the usual $d_p$-metric defined for an ample class.Comment: 38 pages, comments are welcom

On Liu morphisms in non-Archimedean geometry

We define Liu morphisms and quasi-Liu morphisms between Berkovich analytic spaces. We show that Liu morphisms and quasi-Liu morphisms behave exactly as affine morphisms and quasi-affine morphisms of schemes.Comment: V2: Original Section 6 removed. Details added. Typos fixe

Pluripotential-theoretic methods in K-stability and the space of K\ue4hler metrics

It is a natural problem, dating back to Calabi, to find canonical metrics on complex manifolds. In the case of polarized compact K\ue4hler manifolds, a natural candidate is a metric with constant scalar curvature (cscK metric).Since the 80s, Yau, Tian, Donaldson among others proposed that the existence of these special metrics are equivalent to an algebrico-geometric notion of K-stability. There are several known approaches to the study of K-stability and canonical metrics, using various tools from the theory of PDEs, algebraic geometry and non-Archimedean geometry for example. In this thesis, we study a different approach, based on pluripotential theory. In geometric terms, pluripotential theory is the study of positively curved metrics on vector bundles. For the purpose of K-stability, we only need pluripotential theory on an ample line bundle. In this case, pluripotential theory can be identified with the study of quasi-plurisubharmonic functions on the manifold. The application of pluripotential theory in K-stability is not completely new, but previously, people are principally interested in the regular (or mildly singular) quasi-plurisubharmonic functions. In this thesis, we put more emphasis on the role of singular\ua0quasi-plurisubharmonic functions and their singularities. In Paper 1 and Paper 2, we prove a criterion for the existence of canonical metrics on Fano manifolds in terms of quasi-plurisubharmonic functions. In Paper 3, we are concerned with the case when there are no canonical metrics, we prove that there is always an optimal destabilizer to the K-stability

Pluripotential-theoretic stability thresholds

Given a compact polarized manifold $(X,L)$, we introduce two new stability thresholds in terms of singularity types of global quasi-plurisubharmonic functions on $X$. We prove that in the Fano setting, the new invariants can effectively detect the K-stability of $X$. We study some functionals of geodesic rays in the space of K\"ahler potentials by means of the corresponding test curves. In particular, we introduce a new entropy functional of quasi-plurisubharmonic functions and relate the radial entropy functional to this new entropy functional.Comment: 38 pages. Comments are welcome