From complex to non-Archimedean geometry: an approach to the YTD conjecture

TL;DR

This work develops and integrates complex, analytic, and non-Archimedean viewpoints to address when a polarized smooth projective variety (X,L) admits a constant scalar curvature Kähler metric in c_1(L). It builds a variational framework using the Mabuchi K-energy on the space of finite-energy metrics E^1, and links geodesic stability to non-Archimedean K-stability via the Mabuchi slope along maximal geodesic rays, connecting to Berkovich and test-configuration data. A central achievement is proving that, under the assumption Aut^0(X,L) is trivial, the existence of a cscK metric is equivalent to widehat{K}-stability (and uniformly so), through a precise asymptotic analysis of energy functionals and a deep non-Archimedean-calibrated picture of the energy landscape. The results illuminate a robust algebro-geometric criterion for canonical metrics, unify Archimedean and non-Archimedean pluripotential theories, and set the stage for extensions via K-stability, beta-stabilized variants, and divisorial criteria with entropy regularization.

Abstract

These notes expand on talks given by the authors at the 2025 Summer Research Institute in Algebraic Geometry in Fort Collins, Colorado. We discuss the relation between algebraic, analytic, and non-Archimedean geometry over the complex numbers, and sketch a proof of a version of the Yau--Tian--Donaldson conjecture for constant scalar curvature Kähler metrics.

Figures (2)

• Figure 1: The left figure shows the Berkovich space $\mathbb{P}^1_\mathrm{na}=\mathbb{P}^1_\beth$. The branch point is the trivial valuation on $\mathbb{C}(\mathbb{P}^1)$, and there is one branch for each closed point $\xi\in\mathbb{P}^1$. The interior of each branch is parametrized by $c\mathop{\mathrm{ord}}\nolimits_\xi$, $0<c<\infty$, and the endpoint is the point $\xi$ itself ( i.e. the trivial valuation on $\kappa(\xi)\simeq\mathbb{C})$. Any open neighborhood of the branch point contains all but finitely many full branches. The right figure shows $\mathbb{G}_{m,\mathrm{na}}=\mathbb{P}^1_\mathrm{na}\setminus\{0,\infty\}$. The space $\mathbb{G}_{m,\beth}$ is the compact subset of $\mathbb{G}_{m,\mathrm{na}}$ obtained by removing the open branches leading to $0$ and $\infty$. Finally, $\mathbb{P}^1_\mathrm{div}=\mathbb{G}_{m,\mathrm{div}}$ is the dense, totally disconnected subset of $\mathbb{P}^1_\mathrm{na}$ consisting of the branch point and all rational points $c\mathop{\mathrm{ord}}\nolimits_\xi$, $c\in\mathbb{Q}_{>0}$.
• Figure 2: The hybrid projective line $\mathbb{P}^1_\mathrm{hyb}$, fibering over the interval $[0,1]$. The round sphere on the right is the fiber over $t=1$ and equals the Riemann sphere. The spiky sphere in the middle illustrates the fiber over $t\in(0,1)$, which is homeomorphic to the Riemann sphere. The left-hand picture shows the fiber over $t=0$, the Berkovich projective line (see Figure \ref{['fig:Berkcurves']}).

Theorems & Definitions (3)

• Definition 6.1
• Theorem 6.2
• Theorem 6.3