Subvarieties with partially ample normal bundle

Halic, Mihai

doi:10.1007/s00209-018-2175-1

Cited by 3 publications

(4 citation statements)

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“…The author [11] proved that a local complete intersection subvariety with non-pseudoeffective co-normal bundle possesses the G2-property; this generalizes the classical result of Hartshorne [13], which assumes that the normal bundle is ample. Moreover, the author showed that the non-pseudo-effectiveness property holds for strongly movable subvarieties.…”

Section: Introductionmentioning

confidence: 58%

“…The author also proved [11,Proposition 3.5] that strongly movable subvarieties -this is a simple geometric condition-have non-pseudo-effective co-normal bundle, thus they are G2 in the ambient space. (The equivalence between the partial amplitude of a vector bundle and non-pseudo-effectiveness of the dual is explained in op.…”

Section: Movable and Generating Subvarietiesmentioning

confidence: 99%

“…Moreover, the strong mobility condition implies that the co-normal bundle of Y o is not pseudo-effective (cf. [11,Proposition 3.5]). Together, these two conditions mean that Y o Ă X is 1 ą0 .…”

Section: A Chow-type G3-criterion and Applicationsmentioning

confidence: 99%

“…Since Y is movable, after possibly replacing Y by a deformation, we may assume that F is a regular fibre of φ. Then N F {X ae Y (which is trivial, of rank at least one) is a quotient of N Y {X (which is pdim Y ´1q-ample, by [11]). This is impossible.…”

Section: 21mentioning

confidence: 99%

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Splitting Criteria for Vector Bundles Induced by Restrictions to Divisors

Halic¹

2018

Michigan Math. J.

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In this article we deduce criteria for the splitting and the triviality of vector bundles, by restricting them to partially ample divisors. This allows to study the problem of splitting on the total space of fibre bundles. The statements are illustrated with a number of examples. For products of minuscule homogeneous varieties, our results allow to test the splitting of vector bundles by restricting them to products of Schubert 2-planes.The triviality criteria obtained inhere are particularly suited to Frobenius split varieties, whose splitting is defined by a section in the anti-canonical line bundle. As an application, we prove that a vector bundle on a smooth toric variety X, whose anti-canonical bundle has stable base locus of co-dimension at least three, is trivial when its restrictions to the invariant divisors are trivial, with trivializations compatible along the various intersections.Theorem (splitting criteria). Let (X, O X (1)) be a smooth, complex projective variety, with dim X 3. Let V be a vector bundle on X, E := End(V ) the bundle of endomorphisms, L ∈ Pic(X) be q-ample, and D ∈ |dL|. The equivalenceholds in any of the following cases:

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Section: Introductionmentioning

confidence: 58%

Section: Movable and Generating Subvarietiesmentioning

confidence: 99%

Section: A Chow-type G3-criterion and Applicationsmentioning

confidence: 99%

Section: 21mentioning

confidence: 99%

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