Semistability and Semisimplicity in representations of low height in positive characteristic

Ilangovan, Sheila; Mehta, V. B.; Parameswaran, A. J.

doi:10.1007/978-93-86279-11-8_18

Cited by 21 publications

(22 citation statements)

References 9 publications

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“…This is absolutely essential in the setting of Hitchin pairs since the reduction of structure group to the Kempf-Rousseau parabolic, which is key to proof of the main theorem, is realizable geometrically only if we employ Kirwan's stratification. Representation theoretic bounds such as low heights (see [11]) come up as expected when we work in char p (see Theorem 8.17). In characteristic zero we generalize Bogomolov's approach to the setting of Hitchin pairs and give a different proof of the main theorem; we do this for its aesthetic elegance.…”

Section: Introductionsupporting

confidence: 62%

Tensor product theorem for Hitchin pairs – An algebraic approach

Balaji¹,

Parameswaran²

2011

Ann. inst. Fourier

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Section: Introductionsupporting

confidence: 62%

Tensor product theorem for Hitchin pairs – An algebraic approach

Balaji¹,

Parameswaran²

2011

Ann. inst. Fourier

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“…There is also an analogue of Serre's theorem for semistable vector bundles (Balaji-Parameswaran [3], IlangovanMehta-Parameswaran [18]):…”

Section: Featurementioning

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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“…Since E has degree zero by assumption, all three vector bundles in this sequence have vanishing slope. The theorem of S. Ilangovan, V. B. Mehta, and A. J. Parameswaran [7] shows that the tensor product of semistable vector bundles whose ranks add up to less than p + 2 is again semistable. Therefore, E ⊗2 is semistable, so all quotients of E ⊗2 are of nonnegative degree.…”

Section: The Characteristic-3 Casementioning

confidence: 99%

Maximally Frobenius-destabilized vector bundles over smooth algebraic curves

Zhao

2017

Int. J. Math.

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Abstract. Vector bundles in positive characteristics have a tendency to be destabilized after pulling back by the Frobenius morphism. In this paper, we closely examine vector bundles over curves that are, in an appropriate sense, maximally destabilized by the Frobenius morphism. Then we prove that such bundles of rank 2 exist over any curve in characteristic 3, and are unique up to twisting by a line bundle. We also give an application of such bundles to the study of ample vector bundles, which is valid in all characteristics. IntroductionGiven a normal projective variety X over an algebraically closed field k of characteristic p = 0, with a fixed ample divisor, it can happen that pulling back by an inseparable morphism f : Y → X destroys semistability of vector bundles over X. In the simplest case, where X is a smooth curve and f = F rel is the (relative) Frobenius morphism, semistable vector bundles whose pullback under f fail to be semistable are called Frobenius-destabilized vector bundles. They are closely related to the study of the generalized Verschiebung.In this paper, we are primarily interested in rank-r vector bundles E over a curve X of genus g ≥ 2 with the following property: ( * ) the Harder-Narasimhan filtration of F * rel E admits line bundle quotientsIf themselves semistable, such bundles are the "most destabilized" ones possible. We will call a semistable vector bundle E with property ( * ) a maximally Frobenius-destabilized vector bundle (cf. Def. 2.11). These vector bundles appear in the works of many authors. Notably, over a genus-2 curve, all rank-2 Frobenius-destabilized vector bundles are maximally Prop. 3.3]); the work of S. Mochizuki [17] gave a precise formula counting the number of such bundles with trivial determinant over a general curve in arbitrary characteristic, and B. Osserman [18] counted them over an arbitrary curve in small characteristics. In another direction, K. Joshi et al. [9] gave a relation between certain Frobeniusdestabilized bundles and pre-opers, a concept originated from the geometric Langlands program; their observation was later used by X. Sun [20] to prove that stability is preserved under Frobenius-pushforward.We prove that vector bundles with property ( * ) exhibit an interesting trichotomy: Here L max and L min are measures of maximal slope of subbundles and minimal slope of quotient bundles respectively, taken over all finite pullbacks. We split Shepherd-Barron's inequality intoas well as giving an equality criterion for both. As we will see, the improved inequalities (0.1) have many independent applications as well. The result (ii) shows that maximally Frobenius-destabilized vector In characteristic p = 3, B. Osserman [18] has constructed rank-2 maximally Frobenius-destabilized vector bundles with trivial determinant over an arbitrary curve of genus 2, and showed that there are exactly #Pic(X)[2] = 16 of them. We derive from the previous theorem that, for arbitrary genus, such bundles are precisely the rank-2 vector bundles E satisfyingwhere S 2 (E) is ...

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Semistability and Semisimplicity in representations of low height in positive characteristic

Cited by 21 publications

References 9 publications

Tensor product theorem for Hitchin pairs – An algebraic approach

Tensor product theorem for Hitchin pairs – An algebraic approach

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

Maximally Frobenius-destabilized vector bundles over smooth algebraic curves

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