On the positivity of high-degree Schur classes of an ample vector bundle

Xiao, Jianjun

doi:10.1007/s11425-020-1868-7

Cited by 5 publications

(3 citation statements)

References 23 publications

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“…for the definition of this variant of hermitian positivity). Other related interesting results about finding positive representatives (not necessarily coming from the given positively curved metric) of the Schur polynomials in the Chern classes are obtained in [Pin18,Xia20].…”

Section: Question 43 ([Gri69]mentioning

confidence: 96%

Pointwise Universal Gysin formulae and Applications towards Griffiths' conjecture

Diverio¹,

Fagioli²

2020

Preprint

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Let X be a complex projective manifold and (E, h) → X be a rank r holomorphic hermitian vector bundle. Let Qj, j = 1, . . . , r, be the tautological quotient line bundles over the flag bundle F(E) → X, endowed with the natural metric induced by that of E, with Chern curvature Ξj. We show that the universal Gysin formula à la Darondeau-Pragacz for the push forward of a homogeneous polynomial in the Chern classes of the Qj's also hold pointwise at the level of Chern forms in this hermitianized situation.As an application, we show the positivity of several polynomials in the Chern forms of a Griffiths positive vector bundle not previously known, thus giving some new evidences towards a conjecture by Griffiths, which is in turn a pointwise hermitianized version of the Fulton-Lazarsfeld Theorem on numerically positive polynomials for ample vector bundles.

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Section: Question 43 ([Gri69]mentioning

confidence: 96%

Pointwise Universal Gysin formulae and Applications towards Griffiths' conjecture

Diverio¹,

Fagioli²

2020

Preprint

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“…Guler [13, Theorem 1.1] verified Question 1.1 for all signed Segre forms, and Diverio-Fagioli [6] showed the positivity of several other polynomials in the Chern forms of a Griffiths (semi)positive vector bundle by considering the pushforward of a flag bundle, including the later developments [7, 8]. See Xiao [20] and Ross-Toma [18] for other related results of ample vector bundles.…”

Section: Introductionmentioning

confidence: 99%

Positivity of Schur forms for strongly decomposably positive vector bundles

Wan

2024

Forum of Mathematics, Sigma

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In this paper, we define two types of strongly decomposable positivity, which serve as generalizations of (dual) Nakano positivity and are stronger than the decomposable positivity introduced by S. Finski. We provide the criteria for strongly decomposable positivity of type I and type II and prove that the Schur forms of a strongly decomposable positive vector bundle of type I are weakly positive, while the Schur forms of a strongly decomposable positive vector bundle of type II are positive. These answer a question of Griffiths affirmatively for strongly decomposably positive vector bundles. Consequently, we present an algebraic proof of the positivity of Schur forms for (dual) Nakano positive vector bundles, which was initially proven by S. Finski.

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“…Xiao in [55] proved that the cohomological class

P_a(c(E))

for

k = n - 1

, contains a positive form for any ample vector bundle

E

. The statement for any

k \in \mathbb {N}

k \leqslant n

, was conjectured in [55, Conjecture 1.4].…”

Section: Introductionmentioning

confidence: 99%

On characteristic forms of positive vector bundles, mixed discriminants, and pushforward identities

Finski

2022

Journal of London Math Soc

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We prove that Schur polynomials in Chern forms of Nakano and dual Nakano positive vector bundles are positive as differential forms. Moreover, modulo a statement about the positivity of a "double mixed discriminant" of linear operators on matrices, which preserve the cone of positive definite matrices, we establish that Schur polynomials in Chern forms of Griffiths positive vector bundles are weakly-positive as differential forms. This provides differential-geometric versions of Fulton-Lazarsfeld inequalities for ample vector bundles.An interpretation of positivity conditions for vector bundles through operator theory is in the core of our approach. Another important step in our proof is to establish a certain pushforward identity for characteristic forms, refining the determinantal formula of Kempf-Laksov for homolorphic vector bundles on the level of differential forms. In the same vein, we establish a local version of Jacobi-Trudi identity.Then we study the inverse problem and show that already for vector bundles over complex surfaces, one cannot characterize Griffiths positivity (and even ampleness) through the positivity of Schur polynomials, even if one takes into consideration all quotients of a vector bundle.

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On the positivity of high-degree Schur classes of an ample vector bundle

Cited by 5 publications

References 23 publications

Pointwise Universal Gysin formulae and Applications towards Griffiths' conjecture

Pointwise Universal Gysin formulae and Applications towards Griffiths' conjecture

Positivity of Schur forms for strongly decomposably positive vector bundles

On characteristic forms of positive vector bundles, mixed discriminants, and pushforward identities

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