Necessity of the delta-invariant threshold for twisted Kähler–Einstein metrics

Determine whether the condition \(\delta_\psi(X,\Delta,\{\theta\})>1\) is necessary for the existence of twisted Kähler–Einstein metrics on compact normal Kähler klt pairs when the twisting current \(\eta_\psi\) is semipositive.

Background

The paper proves that the valuative and analytic twisted delta invariants coincide and derives an existence criterion: if δψ(X,Δ,{θ})>1\delta_\psi(X,\Delta,\{\theta\})>1, then the corresponding twisted Kähler–Einstein equation admits a solution with minimal singularities. The authors note that this threshold is unlikely to be necessary in complete generality, but expect necessity under the additional assumption that the twisting current ηψ\eta_\psi is semipositive. Establishing or refuting this expectation would clarify whether the divisorial delta-invariant threshold gives a full existence characterization in the semipositive-twist setting.

References

As pointed out in , the condition $\delta_\psi(X,\Delta,{\theta})>1$ is unlikely to be necessary for the existence of twisted KE metrics in full generality. We nevertheless expect it to be necessary when $\eta_\psi\geq 0$, a compelling open problem.

— Demailly-Kollár continuity on klt pairs, and applications to alpha and delta invariants  (2608.23505 - Darvas et al., 24 Aug 2026) in Section 1, Introduction, paragraph “Applications to (twisted) delta invariants”