Guedj–Trusiani conjecture on local alpha invariants

Let $(X,x)$ be an $n$ -dimensional isolated log terminal singularity. If $\alpha_x^{\mathrm{an}}(X)$ and $\alpha_x^{\mathrm{alg}}(X)$ denote the two local alpha invariants introduced by Guedj and Trusiani, then $\alpha_x^{\mathrm{an}}(X)=\alpha_x^{\mathrm{alg}}(X)=\widehat{\operatorname{vol}}(x,X)^{1/n}$ , where $\widehat{\operatorname{vol}}(x,X)$ is Li's normalized volume of the singularity.

References

Primary source

arXiv

Additional references

Demailly-Kollár continuity on klt pairs, and applications to alpha and delta invariants — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

An unrefereed preprint claims to settle the conjecture by proving the predicted link between analytic singularities and an algebraic volume.

The Guedj–Trusiani conjecture predicts an equality connecting local alpha invariants with normalized volumes. No proposer or original date is identified in the retrieved sources.

August 24, 2026 claimed solution

A preprint claims that a Demailly–Kollár continuity theorem for klt pairs proves the conjectured equality and its normalized-volume formula, with applications to Kähler–Einstein geometry. This is an unrefereed claim and has not been independently verified.

Current status (as of August 2026): The conjecture has a claimed proof via Demailly–Kollár continuity, but that claim remains unverified.

Sources

No solutions have been posted yet.