Corti–Prokhorov–Süß conjecture for index-two del Pezzo hypersurfaces

Let $S_d$ be a quasi-smooth and well-formed hypersurface of degree $d$ in $\mathbb{P}(a_0,a_1,a_2,a_3)$, where $a_0\leq a_1\leq a_2\leq a_3$ are positive integers. Define the index by

$$I=a_0+a_1+a_2+a_3-d$$

and assume that $I$ is positive. Then $S_d$ is a singular del Pezzo surface with at most quotient singularities. Corti–Prokhorov–Süß conjecture. If $I=2$, then $S_d$ admits an orbifold Kähler–Einstein metric.

The conjecture proposes that all quasi-smooth del Pezzo hypersurfaces of index two admit orbifold Kähler–Einstein metrics, extending the corresponding result for index one. The present paper is motivated by this conjecture and proves the stated metric existence for broad classes, while also identifying exceptional families without such metrics.