Gorenstein toric Fano varieties

Nill, Benjamin

doi:10.1007/s00229-004-0532-3

Cited by 60 publications

(78 citation statements)

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“…X 22 is a singular Fano 3-fold with Picard rank 1, AC degree 22 and 9 ODPs. Every Gorenstein toric Fano variety X has an associated combinatorial object called a reflexive polytope which determines X ; see Chapters 1 and 2 in the thesis of Nill [76] for a review of basic definitions and facts in toric Fano geometry. See also Section 8 for a brief overview of basic properties of toric weak Fano 3-folds in general.…”

Section: Example 72mentioning

confidence: 99%

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

et al. 2013

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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 1 s 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 [17] 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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Section: Example 72mentioning

confidence: 99%

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

et al. 2013

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“…As an application we finish in Theorem 6.3(3) the proof of [13,Thm. 6.4] saying that such a lattice polytope has at most 3 d lattice points, with equality if and only if it is isomorphic to [−1, 1] d .…”

Section: Theorem 14 Let X Be a Complete Toric Varietymentioning

confidence: 94%

“…There is the following fundamental result (see [1] or [13]): Theorem 2.3 Under the map P → X P reflexive polytopes correspond uniquely up to isomorphism to Gorenstein toric Fano varieties. There are only finitely many isomorphism types of d-dimensional reflexive polytopes.…”

Section: Notation and Basic Definitionsmentioning

confidence: 99%

Complete toric varieties with reductive automorphism group

Nill

2006

Math. Z.

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We give equivalent and sufficient criteria for the automorphism group of a complete toric variety, respectively a Gorenstein toric Fano variety, to be reductive. In particular we show that the automorphism group of a Gorenstein toric Fano variety is reductive, if the barycenter of the associated reflexive polytope is zero. Furthermore a sharp bound on the dimension of the reductive automorphism group of a complete toric variety is proven by studying the set of Demazure roots.Comment: AMS-LaTeX, 20 pages with 1 figur

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“…We refer the reader to [1], [13], [14], [15] and [17] for related works on toric Fano varieties or Gorenstein toric Fano varieties.…”

Section: Takayuki Hibi Akihiro Higashitani and Hidefumi Ohsugimentioning

confidence: 99%