In a follow-up to our paper [Threefolds with big and nef anticanonical bundles I, Math. Ann., 2005, 333(3), 569–631], we classify smooth complex projective threefolds Xwith −K X big and nef but not ample, Picard number γ(X) = 2, and whose anticanonical map is small. We assume also that the Mori contraction of X and of its flop X + are not both birational.
IntroductionAs one of the first applications of Mori theory, Mori and Mukai classified (smooth) Fano threefolds with Picard (or second Betti) number at least 2. In differential geometric terms, this is the same as classifying smooth threefolds with positive Ricci curvature. It is clearly interesting to consider the situation when we "degenerate" the positivity condition, i.e. we consider threefolds whose anticanonical bundles are no longer ample but only big and nef. E.g. there exists a metric with semipositive Ricci curvature which is positive at some point. Recall that −K X nef is to say that (−K X ) · C ≥ 0 for all curves C ⊂ X. This automatically implies (−K X ) 3 ≥ 0 and bigness is just saying that (−K X ) 3 > 0. We will call a threefold X with −K X big and nef (but not ample) an almost Fano threefold.With this paper we begin the classification of smooth almost Fano threefolds X. Due to the complexity of the problem we first study almost Fano threefolds with Picard number two. The classification in the Fano case uses essentially the fact that there are two Mori contractions (= contractions of an extremal ray) which are transversal in some sense. In our case we only have one Mori contraction at our disposal. The second Mori contraction is substituted by the morphism associated with the base point free linear system |−mK X |, m ≫ 0. This morphism is clearly more difficult to handle than a "simple" Mori contraction. The present article is dealing with the case that |−mK X |, m ≫ 0, is divisorial, while the second part will treat the case where this morphism is small, i.e. contracts just finitely many curves.To be a little more precise, the setup of the paper is as follows. Call the extremal ray contraction φ : X −→ Y ; on the other hand, the base point free theorem guarantees that |−mK X | is spanned for m ≫ 0. After Stein factorization we get a second map ψ : X −→ X ′ and a diagramHere X ′ is a canonical Gorenstein Fano threefold, i.e., −K X ′ is ample, but X ′ is singular with mild singularities. Using be, φ is a either a del Pezzo fibration over P 1 , a conic bundle over P 2 or birational with a very precise structure. We shall treat all these cases separately.
IntroductionComplex projective space P n ,étale quotients of complex tori and compact complex manifolds whose universal cover is the unit ball B n ⊂ C n are standard examples of complex Kähler manifolds admitting a (flat) holomorphic normal projective connection. In particular, any compact complex curve admits a (flat) holomorphic normal projective connection. In Holomorphic projective structures on compact complex surfaces I and II, [KO], Kobayashi and Ochiai proved that the list of compact complex Kähler surfaces admitting a normal holomorphic projective connection is precisely this list of standard examples. Their result raised the question whether or not the list is complete even in higher dimensions.In this article we give a complete classification of projective threefolds admitting a holomorphic normal projective connection. The result shows in particular that the above list is not complete in general:Theorem 5.1 The class of 3-dimensional complex projective manifolds admitting a holomorphic normal projective connection consists precisely of 1.) P 3 , 2.)étale quotients of abelian threefolds, 3.)étale quotients of smooth modular families of false elliptic curves, 4.) manifolds, whose universal cover is the unit ball B 3 ⊂ C 3 . However, as in the case of curves and surfaces, this list coincides with the list of projective threefolds admitting a flat holomorphic normal projective connection.Recall that a false elliptic curve is an abelian surface, where the Q-endomorphism algebra End Q is a totally indefinite quaternion algebra. The moduli scheme of such a surface is known to be a Shimura curve; we briefly recall the * The authors were supported by a Forschungsstipendium of the Deutsche Forschungsgemeinschaft and the DFG-Schwerpunkt Globale Methoden in der komplexen Geometrie. 1 construction of the universal family in Example 1.2. Modular families of false elliptic curves are well-known, but seem not to have been considered as a source of examples of manifolds admitting a flat holomorphic normal projective connection. Note that their Kodaira dimension is one. The proof of Theorem 5.1 bases on Mori theory, which is so far only sufficiently settled in the projective case, and results from variation of Hodge structures.In On Fano manifolds with normal projective connections, [Y], Ye proved, that P n is the only Fano manifold with a holomorphic normal projective connection. Recall that a complex (projective) manifold X is called Fano if the dual of the canonical bundle K X is ample. It is called minimal if K X is nef, i.e., if K X has nonnegative intersection number with any irreducible curve in X. The following general structure Theorem 3.1 is one of the keys to the proof of Theorem 5.1:Theorem 3.1 Let X be a projective manifold of dimension n with a holomorphic normal projective connection. If X ≃ P n , then X is minimal and does not contain any rational curve. Furthermore: 1.) If K X ≡ 0, then X is covered by a torus. 2.)If K X is big, then K X is ample and X is covered by the unit ball.In general, if K X is ...
We classify all Gorenstein Fano threefolds with at worst canonical singularities for which the anticanonical system |−K| has a nonempty base locus.
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