Maximal slope of tensor product of Hermitian vector bundles

Chen, Huayi

doi:10.1090/s1056-3911-08-00513-4

Cited by 13 publications

(16 citation statements)

References 20 publications

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“…25 this is the usual convention in the context of p-adic Frobenius slopes. 26 this is the usual convention in the context of ramification theory and asymptotic analysis of differential equations. 27 that is nothing but the flag on M with respect to the dual slope filtrationF ≥.…”

Section: Definitionmentioning

confidence: 99%

Slope filtrations

André

2009

Confluentes Mathematici

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Many slope filtrations occur in algebraic geometry, asymptotic analysis, ramification theory, p-adic theories, geometry of numbers... These functorial filtrations, which are indexed by rational (or sometimes real) numbers, have a lot of common properties.We propose a unified abstract treatment of slope filtrations, and survey how new ties between different domains have been woven by dint of deep correspondences between different concrete slope filtrations.

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Section: Definitionmentioning

confidence: 99%

Slope filtrations

André

2009

Confluentes Mathematici

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“…Results towards a positive answer to this question have been proved by André [1], Bost, de Shalit-Parzanovski, Chen [7] and Bost-Künneman [6]. The best available result is the following:…”

Section: Semi-stable Latticesmentioning

confidence: 98%

Tensor Product and Semi-stability: Four Variations on a Theme

Maculan

2018

EMS Newsl.

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Chevalley proved that in characteristic 0 the tensor product of semi-simple representations is semi-simple. This result has analogues in rather diverse contexts: three of them are presented here in independent sections, focusing on the differences of the frameworks and the similarities of the proofs. Algebraic groups will play a crucial role, sometimes in unexpected ways. RepresentationsLet G be a group and k a field. In this note, a representation of G is a finite-dimensional k-vector space V together with a group homomorphism ρ : G → GL(V). Semi-simple representationsA representation is said to be:• irreducible if there are exactly two sub-vector spaces of V stable under the action of G: the zero subspace 0 and the whole vector space V. In particular the zero representation is not considered to be irreducible.• semi-simple if it can be decomposed into irreducible ones:there are irreducible sub-representationsThe proof of Chevalley's theorem is a beautiful application of the theory of linear algebraic groups (that is, groups of matrices defined by polynomial equations), even though the group G may not at all be of this form.Indeed, in order to prove theorem 1, one may suppose that the field k is algebraically closed and look at GL(V 1 ), GL(V 2 ) as algebraic groups.Then one can suppose G to be itself a linear algebraic group. For, it suffices to take the Zariski-closureḠ of the image of G in GL(V 1 ) × GL(V 2 ), namely the set of pointssuch that f (x) = 0 for all polynomial functions f vanishing identically on the image of G. The semi-simplicity of the representation V 1 ⊗ k V 2 is equivalent for G andḠ, because it is a condition that can be expressed as the vanishing of some polynomials. Now the theory of linear algebraic groups applies: there is an algebraic subgroup rad u (G) of G called the unipotent radical which is connected, unipotent (meaning that all the eigenvalues of its elements are 1), normal and contains any other subgroup of G with these three properties. The unipotent radical controls the semi-simplicity of the representations of G: Applying the preceding fact, the proof of Chevalley's theorem is easily achieved: since the representations V 1 and V 2 are supposed to be semi-simple, an element g of the unipotent radical of G acts as the identity on V 1 and V 2 . Therefore g operates trivially on V 1 ⊗ k V 2 too.

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“…has been proved in [17] for any family hermitian vector bundles (E i ) n i=1 of positive rank over an arbitrary arithmetic curve Spec O K .…”

Section: Satisfiesmentioning

confidence: 99%

“…An extension of Theorem 1.1, concerning semistable G-bundles for G an arbitrary reductive group, has been established by Ramanan and Ramanathan [44]. Their technique of proof, which like Bogomolov's one relies on geometric invariant theory, turned out to be crucial in the study of the preservation of semistability by tensor products in various contexts (see notably [51]), in particular in Arakelov geometry ( [17]).…”

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confidence: 98%

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Concerning the semistability of tensor products in Arakelov geometry

Bost

Chen

2013

Journal de Mathématiques Pures et Appliquées

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We study the semistability of the tensor product of hermitian vector bundles by using the ε-tensor product and the geometric (semi)stability of vector subspaces in the tensor product of two vector spaces.Notably, for any number field K and any hermitian vector bundles E and F over Spec O K , we show that the maximal slopes of E, F , and E ⊗ F satisfy the following inequality :We also prove that, for any. This shows that, if E and F are semistable and if rk E. rk F 9, then E ⊗ F also is semistable.Résumé. -Nous étudions la semi-stabilité du produit tensoriel de fibrés vectoriels hermitiens en utlisant le produit ε-tensoriel et la (semi-)stabilité géométrique des sous-espaces vectoriels dans le produit tensoriel de deux espaces vectoriels.En particulier, si K désigne un corps de nombres et si E et F sont deux fibrés vectoriels hermitiens sur Spec O K , nous montrons que les pentes maximales de E, F et E ⊗ F satisfont à l'inégalité :Nous prouvons aussi que, pour tout sous O K -module V de E ⊗ F de rang 4, la pente de V vérifie :μ (V ) μmax(E) +μmax(F ). Cela entraîne que, si rk E. rk F 9 et que E et F sont semistables, le produit tensoriel E ⊗ F l'est aussi.

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Maximal slope of tensor product of Hermitian vector bundles

Cited by 13 publications

References 20 publications

Slope filtrations

Slope filtrations

Tensor Product and Semi-stability: Four Variations on a Theme

Concerning the semistability of tensor products in Arakelov geometry

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