Structure theorem for projective klt pairs with nef anti-canonical divisor

In this paper, we prove a Beauville–Bogomolov–Yau type decomposition theorem for projective klt pairs of log Calabi–Yau type: up to finite quasi-étale covers, such pairs are decomposed into products of building block varieties, namely, rationally connected varieties and Calabi–Yau varieties. To achieve this, we establish a structure theorem for maximal rationally connected fibrations applicable to a broader class, namely, projective klt pairs with nef anti-log canonical divisor. Our structure theorem reveals that, up to finite quasi-étale covers, these pairs admit locally trivial rationally connected fibrations onto Calabi–Yau varieties.