Degeneration of Kähler-Einstein metrics on complete Kähler manifolds (Q1293497)

A degeneration of a projective variety is a smooth holomorphic family \(\pi : \mathcal X \to \Delta \subset {\mathbb{C}}\) such that any fiber \(X_t = \pi ^{-1}(t)\) is smooth except for \(t =0\). Assume that the central fiber \(X_0\) is the union of smooth hypersurfaces \( X_{01}, \dots , X_{0m}\) with normal crossings and each line bundle \(K_{X_{0i}} + \sum_{j \neq i}X_{0j}\) is ample on \(X_{0i}, 1 \leq i \leq m \). The authors prove that a family \(g_t, t \in \Delta \setminus 0 \) of complete Kähler-Einstein metrics on \(X_t\) with \(\text{Ric}(g_t) = -g_t \) converges to a complete Kähler-Einstein metric on \(X_0 \setminus \text{Sing}(X_0)\) in the sense of Cheeger-Gromov. Under the additional assumption that no three of divisors \(X_{0i}\) have non-empty intersection, this theorem was proved by G.Tian.