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**Nef reduction and anticanonical bundles.**
*(English)*
Zbl 1078.14055

Let \(X\) be a threedimensional projective manifold with nef anticanonical bundle \(-K_X\) and \(1\leq\nu(K_X):=\max\{k\in\mathbb N\,\,| \,\, K_X^k\not\equiv 0\,\}\leq 2\). The paper under review gives a detailed list of the possible structures of \(X\), using as main tools methods and results from Mori theory, the Albanese fibration, and the so-called nef reduction with respect to \(-K_X\), which is an almost holomorphic dominant meromorphic map \(f:X\rightharpoonup Y\), unique up to birational equivalence, such that \(-K_X\) is numerically trivial on all compact fibers \(F\) with dim\(\,X - \)dim \(F=\)dim \(Y=:n(-K_X)\), and \(-K_X\cdot C>0\) for every curve \(C\) in \(X\) which passes through a general point and is not contained in a fiber [see Th. Bauer et al., in: Complex geometry. Collection of papers dedicated to Hans Grauert on the occasion of his 70th birthday. 27–36 (2002; Zbl 1054.14019)]. The authors prove that the nef reduction of \(-K_X\) can be chosen to be holomorphic. If some non-trivial power \(-mK_X\) is spanned by global sections, then \(f\) can be taken, up to a finite covering, as the map defined by the sections of \(-mK_X\). If no positive multiple of \(-K_X\) is globally spanned, then \(X\) is either rationally connected with \(n(-K_X)=3\) or \(q(X)=1\), \(n(-K_X)=2\), or \(q(X)=2,\, n(-K_X)=3\) (Theorem 2.1, Proposition 5.1). The Albanese fibration \(X\rightarrow A(X)\) is a surjective submersion, therefore the pair \((q(X), n(-K_X))\) is a suitable parameter for a classification. Theorem 4.2 gives a detailed list for the possibilities of \(X\) if \(q(X)>0\) and \(n(-K_X)\leq 2\). The list shows in particular that a finite étale covering of \(X\) splits into a product \(A(X)\times\mathbb P_1\) or \(S\times C,\,C\) an elliptic curve. Using this list it is proved that the family of those \(X\) is bounded up to a finite étale cover (Theorem 4.5). If \(q(X)=0\) and if \(X\) is not rationally connected, then a finite étale covering of \(X\) splits biregularly into a product \(\mathbb P_1\times S\) with a \(K3\)-surface \(S\) (Theorem 3.1). Therefore the classification problem is more or less reduced to the case that \(X\) is rationally connected. The main part of the paper is devoted to this case and presents a detailed list of possible structures. Methods and results from Mori theory and from the minimal model program are essential tools for this part of the paper.

Reviewer: Eberhard Oeljeklaus (Bremen)

### MSC:

14J30 | \(3\)-folds |

14E30 | Minimal model program (Mori theory, extremal rays) |

32J17 | Compact complex \(3\)-folds |