Structure of projective varieties with nef anticanonical divisor: the case of log terminal singularities

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

“…The rationally connected fiber of ψ : X ′ → Y in the theorem is simply connected by [Tak03] (see Theorem 2.3 for details). Hence, our result reduces the uniformization problem predicted by [Wan20b, Conjecture 1] to the following conjecture. This conjecture is formulated only for the fundamental group of Calabi-Yau varieties (not of their smooth loci), and thus much simpler than [Wan20b, Conjecture 2].…”

“…Theorem 3.3 will be applied to Q-factorial log terminal models of given klt pairs to derive the holomorphicity and local constancy of MRC fibrations (up to a finite quasi-étale cover), which will be discussed in Section 4. The proof of Theorem 3.3 clarifies why we require a finite quasiétale cover of X in the statement of Theorem 1.1, which does not appear in the case where X reg is simply connected ( [Wan20b]) or where X is smooth ([CCM19]).…”

“…After Campana-Cao-Matsumura ( [CCM19]) began to study structure theorems for klt pairs, the second author ( [Wan20b]) revealed that difficulty in adopting the same strategy as above arises in Step (1) by settling the singular version of (2) and (3) for klt pairs with nef anti-log canonical divisor ( [Wan20b] establishes Step (3) for X with simply connected smooth locus). The desired structure theorem for klt pairs can be reduced to conjectures on the fundamental group of the regular locus X reg .…”

“…The key point in Step (3) (resp. in [Wan20b]) is to prove that ψ * (mL) is actually a trivial locally free sheaf when X (resp. X reg ) is simply connected, which leads to the desired structure theorem.…”

“…However, the new base Y ′ is not necessarily smooth, even we start from a smooth Y , in fact we can at most confirm that Y ′ has klt singularities. Therefore we need to extend most results established in [Wan20b] or [CCM19] to the case that Y is singular, which greatly adds to the complexity of the proof.…”