Threefolds with big and nef anticanonical bundles I

Abstract

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.

Asymptotically cylindrical Calabi–Yau 3–folds from weak Fano 3–folds

“…Recall that in the classification of non-singular rank [40]) out of the 59 families does the AC morphism contract a divisor to a point. Hence we have 53(+2?)…”

“…Thanks to recent work of various authors -including Arap-Cutrone-Marshburn [2], Blanc-Lamy [8], Cutrone-Marshburn [18], Jahnke-Peternell-Radloff [40,41], Kaloghiros [45] and Takeuchi [93] -class (a) is known to consist of over 150 distinct deformation classes of semi-Fano 3-folds; many of these can be obtained by blowing up an appropriate smooth irreducible curve in an appropriate smooth rank one Fano 3-fold. This makes it relatively straightforward to determine many of the basic topological properties of such weak Fano 3-folds.…”

Examples of Calabi–Yau Threefolds as Covers of Almost-Fano Threefolds

“…2 and 3 in order compute the Euler-Poincaré characteristics of some concrete example of smooth Calabi-Yau threefold X obtained as cover : X → Y of degree d = 3, 4, 5, 6, 7, 8 of almost-Fano threefolds Y. Up to now there is no complete classification of almost-Fano threefolds, but only partial results (see [18]). …”

Threefolds with big and nef anticanonical bundles II

“…On the other hand, φ * (−K X ) = S 2 (F (1)) not globally generated implies ψ divisorial by [JPR05], Proposition 3.2. (compare the proof of 3.2., in particular p. 588.…”

“…By [JPR05] E is spanned, since ψ is not divisorial (compare the proof of Proposition 3.2 in [JPR05]). Thus we obtain an embedding…”