Kähler–Ricci flow on singular klt varieties

Prove that, for a compact normal Kähler variety with klt singularities and an initial Kähler current inducing an RCD structure, the unique weak Kähler–Ricci flow induces a metric-measure space homeomorphic to the variety at every positive time and converges to the initial metric space in Gromov–Hausdorff distance as time tends to zero.

Background

The paper studies weak Kähler–Ricci flow starting from Kähler currents with bounded potentials and controlled Monge–Ampère densities. On smooth compact Kähler manifolds, the authors prove a distance non-inflation result: as positive time tends to zero, the evolving distance functions do not exceed the initial canonical distance in the limit.

They then propose an unresolved extension to compact normal Kähler varieties with klt singularities. The conjecture asks for preservation of the topological type of the underlying variety by the evolving metric-measure spaces and convergence back to the initial RCD structure. The authors expect this to hold in complex dimension three with canonical singularities.

References

\begin{conjecture} Let $X$ be a compact normal Kähler variety with klt singularities. Suppose $\omega_0\in \overline{K_\theta(p, K;\lambda)}$ induces an RCD structure on $(X, \omega_0)$. Then the unique solution $(X, \omega(t))$ induces a metric measure space homeomorphic to $X$ for each $t\in (0, T)$ with \begin{equation}\lim_{t\rightarrow 0+}((X, \omega(t)), (X, \omega_0)) =0. \end{equation}

\end{conjecture}

— Geometric stability for complex Monge-Ampere equations  (2609.28978 - Guo et al., 24 Sep 2026) in Conjecture in the subsection “Almost non-inflation under the Kähler-Ricci flow”