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Asymptotically cylindrical Calabi-Yau 3-folds from weak Fano 3-folds
Algebraic Geometry · Submitted 11 Jun 2012
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We prove the existence of asymptotically cylindrical (ACyl) Calabi-Yau 3-folds starting with (almost) any deformation family of smooth weak Fano 3-folds.
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We prove the existence of asymptotically cylindrical (ACyl) Calabi-Yau 3-folds starting with (almost) any deformation family of smooth weak Fano 3-folds. This allow us to exhibit hundreds of thousands of new ACyl Calabi-Yau 3-folds; previously only a few hundred ACyl Calabi-Yau 3-folds were known. We pay particular attention to a subclass of weak Fano 3-folds that we call semi-Fano 3-folds. Semi-Fano 3-folds satisfy stronger cohomology vanishing theorems and enjoy certain topological properties not satisfied by general weak Fano 3-folds, but are far more numerous than genuine Fano 3-folds. Also, unlike Fanos they often contain P^1s with normal bundle O(-1) + O(-1), giving rise to compact rigid holomorphic curves in the associated ACyl Calabi-Yau 3-folds. We introduce some general methods to compute the basic topological invariants of ACyl Calabi-Yau 3-folds constructed from semi-Fano 3-folds, and study a small number of representative examples in detail. Similar methods allow the computation of the topology in many other examples. All the features of the ACyl Calabi-Yau 3-folds studied here find application in arXiv:1207.4470 where we construct many new compact G_2-manifolds using Kovalev's twisted connected sum construction. ACyl Calabi-Yau 3-folds constructed from semi-Fano 3-folds are particularly well-adapted for this purpose.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We prove the existence of asymptotically cylindrical (ACyl) Calabi-Yau 3-folds starting with (almost) any deformation family of smooth weak Fano 3-folds. | result 0.72 | 0.97 | 0.93 | asserted 1.00 |
| s1 | This allow us to exhibit hundreds of thousands of new ACyl Calabi-Yau 3-folds; previously only a few hundred ACyl Calabi-Yau 3-folds were known. | result 0.74 | 0.89 | 0.15 | asserted 1.00 |
| s2 | We pay particular attention to a subclass of weak Fano 3-folds that we call semi-Fano 3-folds. | contribution 0.98 | 0.82 | 0.03 | asserted 0.96 |
| s3 | Semi-Fano 3-folds satisfy stronger cohomology vanishing theorems and enjoy certain topological properties not satisfied by general weak Fano 3-folds, but are far more numerous than genuine Fano 3-folds. | contribution 0.51 | 0.60 | 0.04 | asserted 1.00 |
| s4 | Also, unlike Fanos they often contain P^1s with normal bundle O(-1) + O(-1), giving rise to compact rigid holomorphic curves in the associated ACyl Calabi-Yau 3-folds. | result 0.80 | 0.54 | 0.03 | asserted 0.92 |
| s5 | We introduce some general methods to compute the basic topological invariants of ACyl Calabi-Yau 3-folds constructed from semi-Fano 3-folds, and study a small number of representative examples in detail. | contribution 0.64 | 0.95 | 0.05 | asserted 1.00 |
| s6 | Similar methods allow the computation of the topology in many other examples. | method 0.42 | 0.69 | 0.02 | asserted 0.78 |
| s7 | All the features of the ACyl Calabi-Yau 3-folds studied here find application in arXiv:1207.4470 where we construct many new compact G_2-manifolds using Kovalev's twisted connected sum construction. | interpretation 0.75 | 0.68 | 0.05 | asserted 1.00 |
| s8 | ACyl Calabi-Yau 3-folds constructed from semi-Fano 3-folds are particularly well-adapted for this purpose. | interpretation 0.87 | 0.49 | 0.06 | asserted 0.98 |
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The paper
Original abstract
We prove the existence of asymptotically cylindrical (ACyl) Calabi-Yau 3-folds starting with (almost) any deformation family of smooth weak Fano 3-folds. This allow us to exhibit hundreds of thousands of new ACyl Calabi-Yau 3-folds; previously only a few hundred ACyl Calabi-Yau 3-folds were known. We pay particular attention to a subclass of weak Fano 3-folds that we call semi-Fano 3-folds. Semi-Fano 3-folds satisfy stronger cohomology vanishing theorems and enjoy certain topological properties not satisfied by general weak Fano 3-folds, but are far more numerous than genuine Fano 3-folds. Also, unlike Fanos they often contain P^1s with normal bundle O(-1) + O(-1), giving rise to compact rigid holomorphic curves in the associated ACyl Calabi-Yau 3-folds. We introduce some general methods to compute the basic topological invariants of ACyl Calabi-Yau 3-folds constructed from semi-Fano 3-folds, and study a small number of representative examples in detail. Similar methods allow the computation of the topology in many other examples. All the features of the ACyl Calabi-Yau 3-folds studied here find application in arXiv:1207.4470 where we construct many new compact G_2-manifolds using Kovalev's twisted connected sum construction. ACyl Calabi-Yau 3-folds constructed from semi-Fano 3-folds are particularly well-adapted for this purpose.
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References
How the literature uses it
6 arXiv papers cite this in 45 sentences
Papers citing it · 29 sentences without a confident label
Builds on it 1 paper
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It is also possible to do a similar two-stage (or multiple-stage) resolution process for examples of $$-manifolds coming from the `twisted connected sum' constructions of [ref] .
A new construction of compact torsion-free $G_2$-manifolds by gluing families of Eguchi-Hanson spaces · Resolving orbifolds T⁷/Γ in several stages · 0.90
Uses its method or tool 1 paper
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Gluing $^1Y_1$ and $^1Y_2$ together using [ref] would give a family of compact $$-orbifolds $(M',_s,g_s)$, with singular set $(^1_1)⨿(^1_2)$ locally modelled on $^3(^4/{±1})$.
A new construction of compact torsion-free $G_2$-manifolds by gluing families of Eguchi-Hanson spaces · Resolving orbifolds T⁷/Γ in several stages · 0.94
Uses its data or values 1 paper
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Note that there are $899$ three-dimensional reflexive polytopes of the semi-Fano type [ref] .
Effective action from M-theory on twisted connected sum $G_2$-manifolds · Asymptotically cylindrical Calabi–Yau examples · 0.96
Cites it as support 1 paper
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In this case the forgetful morphism [equation] is generically surjective [ref] .
Tops as Building Blocks for G2 Manifolds · Relation to Construction via Semi-Fano Threefolds · 0.98
Cites it as background 4 papers
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The construction of $G_2$ manifolds as twisted connected sums [ref] starts from $K3$-fibred threefolds $Z$ which are called building blocks.
Tops as Building Blocks for G2 Manifolds · Building Blocks · 0.85
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In this section we focus on the construction of $G_2$-manifolds as put forward by Kovalev [ref] and further developed by Corti et al. [ref] .
Effective action from M-theory on twisted connected sum $G_2$-manifolds · Kovalev's construction of $G_2$-manifolds · 0.84
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These include Asymptotically Cylindrical Riemannian manifolds or submanifolds [ref] , and Asymptotically Conical (sub)manifolds [ref] .
Manifolds with analytic corners · Introduction · 0.84
What it cites · 43 sentences without a confident label
Contests 1 paper
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For this reason, in both this paper and in [ref] we chose not to rely on arguments from [ref] , and to give proofs or alternative references for the main results we need.
G_2-manifolds and associative submanifolds via semi-Fano 3-folds · Introduction · contest 0.83
Uses data or values from 3 papers
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So geometric realisability currently remains open for approximately $50$ of the $111$ numerical links of type E1–E1 listed in [ref] .
Towards the Classification of Weak Fano Threefolds with \rho = 2 · Rank $2$ weak Fanos with small AC morphism · 0.95
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Takeuchi [ref] gives a list of $33$ families of del Pezzo fibred non-singular rank two Fano 3–folds with small AC morphism and shows that none of them are deformation equivalent [ref] .
Weak Fano threefolds with del Pezzo fibration · Rough enumeration of rank two weak Fano 3–folds with small AC morphism · 0.92
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In addition we have $26$ cases from [ref] where both Mori contractions $ψ$ and $ψ^+$ are birational and over $40$ further examples of link type E1–E1 from Arap–Cutrone–Marshburn [ref] and Blanc–Lamy [ref] .
On the existence of certain weak Fano threefolds of Picard number two · Rough enumeration of rank two weak Fano 3–folds with small AC morphism · 0.95
Cites as background 6 papers
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The simplest class of such ACyl Einstein metrics are the asymptotically cylindrical Calabi–Yau metrics studied in the present paper; see also Haskins–Hein–Nordström [ref] .
Asymptotically cylindrical Calabi-Yau manifolds · Introduction · 0.82
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The classification of non-singular weak Fano 3–folds with $ρ=2$ was initiated recently by Jahnke–Peternell–Radloff [ref] with subsequent contributions by Takeuchi [ref] , Cutrone–Marshburn [ref] , Arap–Cutrone–Marshburn [ref] and Blanc–Lamy [ref] ; see also related work by Kaloghiros [ref] .
The topology of terminal quartic 3-folds · The classification scheme for rank two weak Fano 3–folds · 0.98
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The classification of non-singular weak Fano 3–folds with $ρ=2$ was initiated recently by Jahnke–Peternell–Radloff [ref] with subsequent contributions by Takeuchi [ref] , Cutrone–Marshburn [ref] , Arap–Cutrone–Marshburn [ref] and Blanc–Lamy [ref] ; see also related work by Kaloghiros [ref] .
Weak Fano threefolds with del Pezzo fibration · The classification scheme for rank two weak Fano 3–folds · 0.98
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