Fano $5$-folds with nef tangent bundles

Kanemitsu, Akihiro

doi:10.4310/mrl.2017.v24.n5.a6

Cited by 20 publications

(15 citation statements)

References 20 publications

(50 reference statements)

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“…To prove this result, it suffices by a result of [DPS94] to show that a Fano variety with nef tangent bundle is rational homogeneous. This conjecture has been verified up to dimension 5 and in several related circumstances (see [Mok88], [Hwa01], [Mok02], [Wat14], [Kan15], [Wat15]).…”

Section: Higher Dimensional Examplesmentioning

confidence: 78%

Positivity of the diagonal

Lehmann

Ottem

2018

Advances in Mathematics

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We study how the geometry of a projective variety X is reflected in the positivity properties of the diagonal ∆X considered as a cycle on X × X. We analyze when the diagonal is big, when it is nef, and when it is rigid. In each case, we give several implications for the geometric properties of X. For example, when the cohomology class of ∆X is big, we prove that the Hodge groups H k,0 (X) vanish for k > 0. We also classify varieties of low dimension where the diagonal is nef and big.

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Section: Higher Dimensional Examplesmentioning

confidence: 78%

Positivity of the diagonal

Lehmann

Ottem

2018

Advances in Mathematics

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“…Proof. This follows from arguments as in [11,14,38,20,22]. We sketch the proof for the reader's convenience.…”

Section: Special Varietiesmentioning

confidence: 94%

“…Since the case where d = 4 is more complicated, so we omit the details. We refer the reader to [22] and [49,Theorem 2.4].…”

Section: Special Varietiesmentioning

confidence: 99%

Positivity of the second exterior power of the tangent bundles

Watanabe

2020

Preprint

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Let X be a smooth complex projective variety with nef 2 T X and dim X ≥ 3. We prove that, up to a finite étale cover X → X, the Albanese map X → Alb( X) is a locally trivial fibration whose fibers are isomorphic to a smooth Fano variety F with nef 2 T F . As a bi-product, we see that either T X is nef or X is a Fano variety. Moreover we study a contraction of a K X -negative extremal ray ϕ : X → Y . In particular, we prove that X is isomorphic to the blow-up of a projective space at a point if ϕ is of birational type. We also prove that ϕ is a smooth morphism if ϕ is of fiber type. As a consequence, we give a structure theorem of varieties with nef 2 T X .

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“…The main result of this article is Theorem 2.6, we prove that a birational map of simple type is a standard flop or a twisted Mukai flop if and only if P and P + are projective bundles over a common variety. As an application of our main result, we classify simple type birational maps when dim X ≤ 5, here recent results of [6] and [17] play important roles in our classification.…”

Section: Twisted Mukai Flop (Over a Base C)mentioning

confidence: 97%