Density of vector bundles periodic under the action of Frobenius

Ducrohet, Laurent; Mehta, V. B.

doi:10.1016/j.bulsci.2009.11.001

Cited by 5 publications

(1 citation statement)

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“…Hence, M ef GL 1 is dense inside M ss,0 GL 1 = Jac 0 (X) since torsion points are dense in any abelian variety. In positive characteristic Ducrohet and Mehta have shown that M ef,0 GLn ⊂ M ss,0 GLn is dense for all n when g ≥ 2, and similarly for vector bundles with trivial determinant (they show in fact that a smaller set of objects, called Frobenius periodic vector bundles, are dense; see [DM10]). However, in characteristic zero much less seems to be known about the density of essentially finite bundles when the rank is greater than 1.…”

Section: Let Now M Ssmentioning

confidence: 98%

Essentially Finite $G$-torsors

Ghiasabadi¹,

Stefan²

2022

Preprint

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Let X be a smooth projective curve of genus g, defined over an algebraically closed field k, and let G be a connected reductive group over k. We say that a G-torsor is essentially finite if it admits a reduction to a finite group, generalising the notion of essentially finite vector bundles to arbitrary groups G. We prove that all such G-torsors are (strongly) semistable of torsion degree, and we then study the density of the set of k-points of essentially finite G-torsors of degree 0, denoted M ef,0 G , inside M ss,0 G , the k-points of all semistable degree 0 G-torsors. We show that when g = 1, M ef,0 G ⊂ M ss,0 G is dense. When g > 1 and when char(k) = 0, we show that for any reductive group of semisimple rank 1, M ef,0 G ⊂ M ss,0 G is not dense.

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Section: Let Now M Ssmentioning

confidence: 98%