RCD structures and metric stability on klt Kähler spaces

Establish that every Kähler current in the class $K_{ heta,\Omega}(p,K;\lambda)$ on a compact normal Kähler space with klt singularities induces an RCD structure on the underlying space, and prove that sufficiently small $L^1$ differences between the volume measures of two such currents imply uniformly small differences between their induced distance functions.

Background

The paper extends its smooth-manifold framework to compact normal Kähler spaces with klt singularities by replacing the smooth reference volume form with an adapted volume measure Ω. The proposed class consists of positive currents with bounded local potentials, uniformly controlled Lp volume densities relative to Ω, and a plurisubharmonic logarithmic volume ratio.

The conjecture has two components: existence of an RCD metric-measure structure induced by each admissible Kähler current, and quantitative stability of the induced distances under L1L^1 convergence of volume measures. The authors state that the conjecture is known in complex dimension three using existing RCD structure and Hölder-continuity results.

References

\begin{conjecture} Let $(X, \theta)$ be a compact normal Kähler space with klt singularities equipped with a smooth Kähler metric and an adapted volume measure $\Omega$. Then \begin{enumerate} \item Each Kähler current $\omega\in K_\theta(p,K;\lambda)$ induces an RCD structure on $X$, \medskip \item for any $\epsilon>0$, there exists $\delta>0$ such that for any $\omega, \omega'\in K_{\theta, \Omega}(p,K)$ with $$| \omegan - (\omega')n|_{L1(X)}< \delta,$$ we have $$\sup_{x,y\in X} |d_{\omega'}(x, y) - d_{\omega}(x,y)| < \epsilon.$$

\end{enumerate} \end{conjecture}

— Geometric stability for complex Monge-Ampere equations  (2609.28978 - Guo et al., 24 Sep 2026) in Conjecture in the subsection following Corollary 1.2