Hermitian vector bundles and extension groups on arithmetic schemes. I. Geometry of numbers

Bost, Jean-Benôıt; Künnemann, Klaus

doi:10.1016/j.aim.2009.09.005

Cited by 27 publications

(39 citation statements)

References 55 publications

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“…[8] proposition 3.3.1), qui est une reformulation du premier théorème de Minkowski dans le cadre des fibrés vectoriels hermitiens. Si E = (E, ( · σ ) σ:K→C ) un fibré vecotiel hermitien sur Spec O K , on désigne par ε(E) le nombre réel Proposition 4.1.8.…”

Section: Théorème De Hilbertunclassified

Convergence des polygones de Harder-Narasimhan

Chen

2010

Mémoires Soc. Math. France

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Résumé. -On reformule la théorie des polygones de Harder-Narasimhan par le langage des R-filtrations. En utlisant une variante du lemme de Fekete et un argument combinatoire des monômes, onétablit la convergence uniforme des polygones associés a une algèbre graduée munie des filtrations. Cela conduità l'existence de plusieur invariants arithmétiques dont un cas très particulier est la capacité sectionnelle. Deux applications de ce résultat dans la géométrie d'Arakelov sont abordées : le théorème de Hilbert-Samuel arithmétique ainsi que l'existence et l'interprétation géométrique de la pente maximale asymptotique.

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Section: Théorème De Hilbertunclassified

Convergence des polygones de Harder-Narasimhan

Chen

2010

Mémoires Soc. Math. France

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“…4) Recently, Bost and Künnemann [BK07] have proved that, if K is a number field and if E and F are two non-zero Hermitian vector bundles on Spec O K , then…”

Section: Introductionmentioning

confidence: 99%

Maximal slope of tensor product of Hermitian vector bundles

Chen¹

2008

J. Algebraic Geom.

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We give an upper bound for the maximal slope of the tensor product of several non-zero Hermitian vector bundles on the spectrum of an algebraic integer ring. By Minkowski's First Theorem, we need to estimate the Arakelov degree of an arbitrary Hermitian line subbundle M of the tensor product. In the case where the generic fiber of M is semistable in the sense of geometric invariant theory, the estimation is established by constructing (through the classical invariant theory) a special polynomial which does not vanish on the generic fibre of M . Otherwise we use an explicte version of a result of Ramanan and Ramanathan to reduce the general case to the former one.

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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: 65%

“…It owes its name to Poincaré's upper-half plane h = {z ∈ C : Im z > 0} which can be seen as the "upper-half" of P 1 (C) P 1 (R). 6 Actually this definition is too restrictive, as one has to let the filtration F • V to be defined only on a finite extension of Q p .…”

Section: The Role Of Semi-stable Vector Bundlesmentioning

confidence: 99%

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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Hermitian vector bundles and extension groups on arithmetic schemes. I. Geometry of numbers

Cited by 27 publications

References 55 publications

Convergence des polygones de Harder-Narasimhan

Convergence des polygones de Harder-Narasimhan

Maximal slope of tensor product of Hermitian vector bundles

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

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