Plurisubharmonic geodesics in spaces of non-Archimedean metrics of finite energy

平衡多面体空间上的Monge-Ampère测度

Monge-Ampère measures on balanced polyhedral spaces

本文研究了平衡多面体空间上的凸函数和Monge-Ampère测度，通过变分方法探讨了多面体Monge-Ampère方程的解的存在性，并与非阿基米德 pluripotential 理论建立了联系。

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 Kähler 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.

K-稳定性的非阿基米德方法，I：测试构型与赋值的度量几何

A non-Archimedean approach to K-stability, I: Metric geometry of spaces of test configurations and valuations

本文用非阿基米德方法研究K-稳定性，通过Berkovich解析化上的Fubini-Study函数刻画测试构型空间，并建立除子范数与除子测度的一一对应。

Geodesic rays and stability in the cscK problem

We prove that any finite energy geodesic ray with a finite Mabuchi slope is maximal in the sense of Berman-Boucksom-Jonsson, and reduce the proof of the uniform Yau-Tian-Donaldson conjecture for constant scalar curvature Kähler metrics to Boucksom-Jonsson's regularization conjecture about the convergence of non-Archimedean entropy functional. As further applications, we show that a uniform K-stability condition for model filtrations and the $\mathcal{J}^{K_X}$-stability are both sufficient conditions for the existence of cscK metrics. The first condition is also conjectured to be necessary. Ou…

Singular semipositive metrics in non-Archimedean geometry

Let X be a smooth projective Berkovich space over a complete discrete valuation field K of residue characteristic zero, endowed with an ample line bundle L. We introduce a general notion of (possibly singular) semipositive (or plurisubharmonic) metrics on L, and prove the analogue of the following two basic results in the complex case: the set of semipositive metrics is compact modulo constants, and each semipositive metric is a decreasing limit of smooth semipositive ones. In particular, for continuous metrics our definition agrees with the one by S.-W. Zhang. The proofs use multiplier ideals…

关于混合特征下的非阿基米德Monge-Ampère方程

On the non-archimedean Monge-Ampère equation in mixed characteristic

本文在混合特征下，通过引入扰动友好的测试理想，解决了非阿基米德Monge-Ampère方程的问题，扩展了变分方法的应用范围。

正度量空间无穷远处的几何：测试配置、测地射线与弦距离

Geometry at the infinity of the space of positive metrics: test configurations, geodesic rays and chordal distances

本文证明了正度量空间中测地射线间的Mabuchi弦距离等于截面环上滤过的谱距离，从而为无穷远边界提供了代数描述。

Differentiability of non-archimedean volumes and non-archimedean Monge-Ampère equations (with an appendix by Robert Lazarsfeld)

Let $X$ be a normal projective variety over a complete discretely valued field and $L$ a line bundle on $X$. We denote by $X^\textrm{an}$ the analytification of $X$ in the sense of Berkovich and equip the analytification $L^\textrm{an}$ of $L$ with a continuous metric $\| \ \|$. We study non-archimedean volumes, a tool which allows us to control the asymptotic growth of small sections of big powers of $L$. We prove that the non-archimedean volume is differentiable at a continuous semipositive metric and that the derivative is given by integration with respect to a Monge-Ampère measure. Such a …

The space of finite-energy metrics over a degeneration of complex manifolds

Given a degeneration of compact projective complex manifolds $X$ over the punctured disc, with meromorphic singularities, and a relatively ample line bundle $L$ on $X$, we study spaces of plurisubharmonic metrics on $L$, with particular focus on (relative) finite-energy conditions. We endow the space $\hat \cE^1(L)$ of relatively maximal, relative finite-energy metrics with a $d_1$-type distance given by the Lelong number at zero of the collection of fibrewise Darvas $d_1$-distances. We show that this metric structure is complete and geodesic. Seeing $X$ and $L$ as schemes $X_\K$, $L_\K$ over …

滤子与非阿基米德范数在截面环上的渐近几何

Filtrations and asymptotic geometry of non-Archimedean norms on section rings

本文研究了非阿基米德范数在截面环上的滤子作用及其渐近几何特性，推广了之前的成果并提供了新的测度收敛结果。

Entropies in $μ$-framework of canonical metrics and K-stability, I -- Archimedean aspect: Perelman's W-entropy and $μ$-cscK metrics

This is the first in a series of two papers studying mu-cscK metrics and muK-stability, from a new perspective evoked from observations in arXiv:2004.06393 and in this first article. The first paper is about a characterization of mu-cscK metrics in terms of Perelman's W-entropy $\check{W}^λ$. We regard Perelman's W-entropy as a functional on the tangent bundle $T \mathcal{H} (X, L)$ of the space $\mathcal{H} (X, L)$ of K"ahler metrics in a given K"ahler class $L$. The critical points of $\check{W}^λ$ turn out to be $μ^λ$-cscK metrics. When $λ\le 0$, the supremum along the fibres gives a smooth…

仿射簇的极化紧化空间

The space of polarized compactifications of an affine variety

本文通过非阿基米德几何，将仿射簇的极化紧化空间与富比尼-施图迪函数空间等对象建立联系，赋予其自然度量并确定完备化，拓展了相关几何空间的研究。

Fano流形Kähler度量空间上的反向Hölder不等式与有效开性

Reverse Hölder inequalities on the space of Kähler metrics of a Fano variety and effective openness

本文在Fano流形Kähler度量空间上建立反向Hölder不等式，利用隐藏对数凹性证明并推广至对数终端奇点Fano簇，进而得到测地线速度不等式、有效开性结果及与Igusa zeta函数的联系。