Main conjecture for Kähler–Einstein cubic surfaces

Let $\overline{\mathcal{M}_3}^{ALG}:=\mathbb{P}^{19}//SL(4,\mathbb{C})$ be the algebraic compactification of the moduli space of cubic surfaces, and let $\overline{\mathcal{M}_3}^{GH}$ be its Gromov–Hausdorff compactification. Cubic-surface degeneration conjecture. There exists a $2:1$ ramified covering map

$$D:\overline{\mathcal{M}_3}^{ALG}\longrightarrow\overline{\mathcal{M}_3}^{GH}$$

such that every Chow-polystable cubic admits a Kähler–Einstein orbifold metric and $D$ associates to each polystable Del Pezzo cubic its induced metric structure. In particular, Gromov–Hausdorff limits of cubics have at most $A_1$ singularities or exactly $3A_2$ singularities, and no other normal cubic surface admits a Kähler–Einstein metric. The conjecture gives a precise identification of algebraic and metric compactifications in degree three.