The arithmetic version of the frequency transition conjecture for the almost Mathieu operators was recently proved by Jitomirskaya and Liu [
We show the validity of the relative dlt MMP over ${\mathbb{Q}}$-factorial threefolds in all characteristics $p>0$. As a corollary, we generalise many recent results to low characteristics including: $W{\mathcal{O}}$-rationality of klt singularities, inversion of adjunction, and normality of divisorial centres up to a universal homeomorphism.
We prove the following result: if a $\,\,\,\,\,{\mathbb {Q}}\,\,\,\,\,$-Fano variety is uniformly K-stable, then it admits a Kähler–Einstein metric. This proves the uniform version of Yau–Tian–Donaldson conjecture for all (singular) Fano varieties with discrete automorphism groups. We achieve this by modifying Berman–Boucksom–Jonsson’s strategy in the smooth case with appropriate perturbative arguments. This perturbation approach depends on the valuative criterion and non-Archimedean estimates, and is motivated by our previous paper.
In this paper, we show that for a connected compact Lie group to be acceptable, it is necessary and sufficient that its derived subgroup is isomorphic to a direct product of the groups ${\text {SU}}(n)$, ${\text {Sp}}(n)$, ${\text {SO}}(2n+1)$, ${\text {G}}_2$, ${\text {SO}}(4)$. We show that there are invariant functions on ${\text {SO}}_{4}({\mathbb {C}})^{2}$ which are not generated by 1-argument invariants, though the group ${\text {SO}}_{4}({\mathbb {C}})$ is acceptable.