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

Abstract

In this paper, we establish a structure theorem for projective Kawamata log terminal (klt) pairs (X, ∆) with nef anti-log canonical divisor; specifically, we prove that up to replacing X with a finite quasi-étale cover, X admits a locally trivial rationally connected fibration onto a projective klt variety with numerically trivial canonical divisor. Our structure theorem generalizes previous works for smooth projective varieties and reduces the structure problem to the singular Beauville-Bogomolov decomposition for Calabi-Yau varieties. As an application, the projective varieties of klt Calabi-Yau type, which naturally appear as an outcome of the Log Minimal Model Program, are decomposed into building block varieties, namely, rationally connected varieties and Calabi-Yau varieties. SHIN-ICHI MATSUMURA AND JUANYONG WANG 4.2. Case of Q-factorial terminal pairs with splitting tangent sheaf 40 4.3. Case of klt pairs with nef anti-log canonical divisor 46 References 51

The Nonvanishing problem for varieties with nef anticanonical bundle

“…This part of the proof is similar in spirit to the proof from the 1980s, which relied on the solution of particular cases of Iitaka's conjecture C n,m , see for instance [MP97,]. An analogue of the subadditivity of the Iitaka dimension does not exist in the context of this paper; instead, we use in Sections 6 and 7 the recently developed structure theory for MRC fibrations of varieties with nef anticanonical class [MW21].…”

“…We now come to the following very recent structure theorem from [MW21,EIM20] for varieties with nef anticanonical class. It is fundamental for our work, and many of our proofs use it as a starting point.…”

“…We treat next the case where X is uniruled and −K X is nef. The structure theory of such varieties was recently completed in [MW21], building on [Cao19, CH19, Wan20] and other works. We now know that X admits an analytically locally trivial MRC fibration to a K-trivial variety.…”

On the minimal model program for projective varieties with pseudo-effective tangent sheaf

“…In this subsection, following [MW21], we review singular Hermitian metrics on torsion-free sheaves, taking them on vector bundles as known (see [Rau15,HPS18,PT18]).…”

The Nonvanishing Problem for varieties with nef anticanonical bundle

“…The key concept is that of locally constant fibrations, see Section 3. By [64], the assumption on the existence of a locally constant fibration in Theorem F is always satisfied after passing to a quasi-étale cover.…”