Properness conjecture for non-archimedean Calabi energy

Assume $(X,L)$ is klt. For $1\le p<2$ , equip $\mathcal{E}_{\mathrm{NA}}^2(X,L)$ with the $d_p$ -topology. Then the normalized superlevel set

\{\varphi\in\mathcal{E}_{\mathrm{NA}}^2(X,L)\mid E(\varphi)=0,\ C_{\mathrm{NA}}(\varphi)\ge C\}

should be compact in the $d_p$ -topology. This is a compactness formulation of properness for the non-archimedean Calabi energy and is open; the source notes confirmation for $T$ -invariant metrics on toric varieties in a related boundedness result.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Properness conjecture for non-archimedean Calabi energy
Let $(X,L)$ be a polarized normal variety with only klt singularities. The normalized superlevel set
$\{\varphi\in\mathcal{E}_{\mathrm{NA}}^{\exp}(X,L)\mid \sup\varphi=0,\ \bm{\check{\mu}}_{\mathrm{NA}}(\varphi)\ge C\}$
should be compact in the $E_{\exp}$ -topology. This compactness is proposed as a properness principle for the non-archimedean entropy and is left open.
source: Eiji Inoue, “Entropies in μ-framework of canonical metrics and K-stability, II – Non-archimedean aspect: non-archimedean μ-entropy and μK-semistability”, arXiv:2202.12168 (2022).

Sources & referencesView supporting material

Primary source

Eiji Inoue, “Entropies in μ-framework of canonical metrics and K-stability, II – Non-archimedean aspect: non-archimedean μ-entropy and μK-semistability”, arXiv:2202.12168 (2022).

1 week ago

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

No solutions have been posted yet.