Degeneration of Kähler-Einstein hypersurfaces in complex torus to generalized pair of pants decomposition

According to the conjecture of Calabi, on a complex manifold X with ample canonical bundle KX , there should exist a Kähler-Einstein metric g. Namely, a metric satisfying Ricg = −ωg, where ωg is the Kähler form of the Kähler metric g. The existence of such metric when X is compact was proved by Aubin and Yau ([23]) using complex Monge-Ampère equation. This important result has many applications in Kähler geometry. Starting with this important result, Yau initiated the program of applying Kähler-Einstein metrics to algebraic geometry ([20]). It was realized by him the need to study such metrics for quasi-projective manifolds ([21]) and their degenerations. The original proof of [23] was a purely existence result. Later the existence of C1(X) < 0 Kähler-Einstein metrics was generalized to complete complex manifolds by Cheng and Yau ([5]), where other than existence the proof also exhibits the asymptotic behavior of the KählerEinstein metric near the infinity boundary.