Fano manifolds of index \(n-1\) and the cone conjecture (Q2929681)

Let \(Z\) be an \(n\)-dimensional Fano manifold of index \(n - 1\) (i.e., \(-K_Z = (n-1)H\) for some ample divisor \(H\)) over an algebraically closed field \(k\subset\mathbb{C}\). Let \(V\subset |H|\) be a general \((n-1)\)-dimensional linear system with base locus \(\Gamma\). Consider the blowup of \(Z\) at \(\Gamma\) and a smooth divisor \(D\in |-mK_X|\) for some \(m > 1\). (According to the classification of all such \(Z\) (called \textit{del Pezzo manifolds}) in [\textit{T. Fujita}, Proc. Japan Acad., Ser. A 58, 113--116 (1982; Zbl 0568.14018), Tohoku Math. J. (2) 34, 319--341 (1982; Zbl 0489.14002)], all the stated properties of \(|H|,V\), etc. are actually satisfied.) Then the pair \((X,\frac{1}{m}D)\) is klt Calabi-Yau and the main result of the paper under review confirms the so-called Kawamata-Morrison conjecture (= Conjecture 1.1 in the text) for it.NEWLINENEWLINENamely, projecting away from \(V\) one obtains a morphism \(f: X \longrightarrow \mathbb{P}^{n-1}\), for which the authors prove (Proposition 2.2) that degenerate fibers vary over a codimension \(>1\) locus on \(\mathbb{P}^{n-1}\). Next, intersecting with proper transforms of lines on \(Z\), the authors show that (the closure of) the nef cone \(\overline{A(X)}\) is rational polyhedral and spanned by (finitely many) effective classes (see Theorem 3.1). It is then straightforward that the group \(\text{Aut}(X)\) acts with rational polyhedral fundamental domain on (effective) \(\overline{A(X)}\) and the first part of K-M conjecture follows.NEWLINENEWLINEFurther, since the general fiber of \(f\) is an elliptic curve, all flopping (\(K\)-trivial) curves on \(X\) are contained in reducible fibres of \(f\). Using their Lemma 4.1 the authors show how the numerical classes of these curve change under flops. Also, applying the fact that the cone \(\overline{\text{Eff}(X)}\) of pseudo-effective divisors on \(X\) is dual to the cone of moving curves on \(X\) (see \textit{S. Boucksom}, \textit{J.-P. Demailly}, \textit{M. Pǎun}, and \textit{Th. Peternell} [\textit{S. Boucksom} et al., J. Algebr. Geom. 22, No. 2, 201--248 (2013; Zbl 1267.32017)]), the authors prove that any isolated \(K\)-trivial extremal ray on \(X\) corresponds to a flop (see Lemma 4.2). The same holds for any SQM model of \(X\). Let us stress that the fact the degenerate fibers of \(f\) vary in codimension \( > 1\) is crucially used here.NEWLINENEWLINEUsing the preceding data plus the results of \textit{C. Birkar}, \textit{P. Cascini}, \textit{C. Hacon}, and \textit{J. McKernan} [J. Am. Math. Soc. 23, No. 2, 405--468 (2010; Zbl 1210.14019)] the authors prove (Corollary 5.2) that the effective movable cone \(\overline{M(X)}\) of \(X\) equals the union of the cones \(\overline{A(X')}\) coming from various SQMs \(X' \dashrightarrow X\). Furthermore, they show that if \(X'\) was obtained by a sequence of flops, then \(\overline{A(X')}\) is rational polyhedral, spanned by semi-ample divisors, and every \(K\)-trivial extremal ray on \(X'\) can be flopped (see Proposition 5.6 and Corollary 5.7). This is used to show first that finite union of these \(\overline{A(X')}\) gives a locus \(\subset N^1(X/\mathbb{P}^{n-1})\) whose orbit under the (elliptic) pseudo-automorphisms from \(\text{Aut}(X/\mathbb{P}^{n-1})\) coincides with the relative cone \(\overline{M(X/\mathbb{P}^{n-1})}\) (see Propositions 5.5, 5.8). Finally, the authors also show (Claim 5.10) that every codimension \(1\) face of \(\overline{A(X')}\), intersecting the movable cone, is dual to the class of a fiber component. It is now not difficult to conclude K-M conjecture. (The proof of Claim 5.10 uses the previous results about extremal rays on \(X'\) and the smoothness of \(X'\)s (starting with \(\dim = 3\)) under flops.)