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We introduce the trace operator for quasi-plurisubharmonic functions on compact K\"ahler manifolds, allowing to study the singularities of such functions along submanifolds where their generic Lelong numbers vanish. Using this construction we obtain novel Ohsawa--Takegoshi extension theorems and give applications to restricted volumes of big line bundles., Comment: v.1 comments welcome

We define the basic pluripotential-theoretic operations in terms of the transcendental theory of non-Archimedean metrics introduced in \cite{DXZ23}. In particular, we establish that the analogue of Boucksom--Jonsson's envelope conjecture holds in our theory., Comment: References updated

We show that the volume of transcendental big $(1,1)$-classes on compact K\"ahler manifolds can be realized by convex bodies, thus answering questions of Lazarsfeld-Musta\c{t}\u{a} and Deng. In our approach we use an approximation process by partial Okounkov bodies together with properties of the restricted volume, and we study the extension of K\"ahler currents, as well as the bimeromorphic behavior of currents with analytic singularities. We also establish a connection between transcendental Okounkov bodies and toric degenerations., Comment: comments are welcome!

We introduce the concept of non-Archimedean metrics attached to a transcendental pseudoeffective cohomology class on a compact K\"ahler manifold. This is obtained via extending the Ross-Witt Nystr\"om correspondence to the relative case, and we point out that our construction agrees with that of Boucksom-Jonsson when the class is induced by a pseudoeffective $\mathbb Q$-line bundle. We introduce the notion of a flag configuration attached to a transcendental big class, recovering the notion of a test configuration in the ample case. We show that non-Archimedean finite energy metrics are approximable by flag configurations, and very general versions of the radial Ding energy are continuous, a novel result even in the ample case. As applications, we characterize the delta invariant as the Ding semistability threshold of flag configurations and filtrations and prove a YTD type existence theorem in terms of flag configurations., Comment: v.2 references updated. Presentation improved v.3 accepted version

In this paper, we develop several pluripotential-theoretic techniques for singular metrics on vector bundles. 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 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., Comment: Various minor corrections. To appear on Mathematische Annalen

Let $X$ be a smooth projective variety. We construct partial Okounkov bodies associated to Hermitian pseudo-effective line bundles $(L,\phi)$ on $X$. We show that partial Okounkov bodies are universal invariants of the singularity of $\phi$. As an application, we generalize the theorem of Boucksom--Chen and construct Duistermaat--Heckman measures associated with finite energy metrics on the Berkovich analytification of an ample line bundle., Comment: Section 3 rewritten. Various minor corrections. To appear on Geometry & Topology

Let $(L,he^{-u})$ be a pseudoeffective line bundle on an $n$-dimensional compact K\"ahler manifold $X$. Let $h^0(X,L^k\otimes \mathcal I(ku))$ be the dimension of the space of sections $s$ of $L^k$ such that $h^k(s,s)e^{-ku}$ is integrable. We show that the limit of $k^{-n}h^0(X,L^k\otimes \mathcal I(ku))$ exists, and equals the non-pluripolar volume of $P[u]_\mathcal I$, the $\mathcal I$-model potential associated to $u$. We give applications of this result to K\"ahler quantization: fixing a Bernstein-Markov measure $\nu$, we show that the partial Bergman measures of $u$ converge weakly to the non-pluripolar Monge--Amp\`ere measure of $P[u]_\mathcal I$, the partial equilibrium., Comment: v3. addresses and references updated, typos fixed v.4 more typos fixed, to appear on Geometry & Topology

We prove an analytic Bertini theorem, generalizing a previous result of Fujino and Matsumura., Comment: Final version, to appear on Mathematische Zeitschrift

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: Final version, to appear in Israel Journal of Mathematics

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: Final version, to appear on IMRN

Given a K\"ahler manifold $X$ with an ample line bundle $L$, we consider the metric space of $L^1$ geodesic rays associated to the first Chern class $c_1(L)$. We characterize rays that can be approximated by ample test configurations. At the same time, we also characterize the closure of algebraic singularity types among all singularity types of quasi-plurisubharmonic functions, pointing out the very close relationship between these two seemingly unrelated problems. By Bonavero's holomorphic Morse inequalities, the arithmetic and non-pluripolar volumes of algebraic singularity types coincide. We show that in general the arithmetic volume dominates the non-pluripolar one, and equality holds exactly on the closure of algebraic singularity types. Analogously, we give an estimate for the Monge--Amp\`ere energy of a general $L^1$ ray in terms of the arithmetic volumes along its Legendre transform. Equality holds exactly for rays approximable by test configurations. Various other cohomological and potential theoretic characterizations are given in both settings. As applications, we give a concrete formula for the non-Archimedean Monge--Amp\`ere energy in terms of asymptotic expansion, and show the continuity of the projection map from $L^1$ rays to non-Archimedean rays. We also show that the closure of ample test configurations and filtrations gives the same set of rays., Comment: v.2

Let $X$ be a compact K\"ahler unibranch complex analytic space of pure dimension. Fix a big class $\alpha$ with smooth representative $\theta$ and a model potential $\phi$ with positive mass. We define and the study non-pluripolar products of quasi-plurisubharmonic functions on $X$. We study the spaces $\mathcal{E}^p(X,\theta;[\phi])$ of finite energy K\"ahler potentials with prescribed singularities 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: The contents of Section 4,5,6 will appear on Crelle's journal

In this paper, we prove the integration by parts formula for the non-pluripolar product on a compact K\"ahler manifold. Our result generalizes the special case of potentials with small unbounded loci proved in [BEGZ10]., Comment: 15 pages, comments are welcome

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 & PDE

This is the first of a series of papers. Our final goal is to establish Deligne-Riemann-Roch isomorphisms in various settings. In this paper, we establish a uniqueness theorem for Deligne pairings and prove the degree $1$ part of the Deligne-Riemann-Roch theorem., Comment: 38 pages, some typos are fixed