Hitchin-Mochizuki morphism, opers and Frobenius-destabilized vector bundles over curves

Kirti N Joshi, Christian Pauly

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

Let X be a smooth projective curve of genus g ≥2 defined over an algebraically closed field k of characteristic p>0. For p>r(r-1)(r-2)(g-1) we construct an atlas for the locus of all Frobenius-destabilized bundles of rank r (i.e. we construct all Frobenius-destabilized bundles of rank r and degree zero up to isomorphism). This is done by exhibiting a surjective morphism from a certain Quot-scheme onto the locus of stable Frobenius-destabilized bundles. Further we show that there is a bijective correspondence between the set of stable vector bundles E over X such that the pull-back F*(E) under the Frobenius morphism of X has maximal Harder-Narasimhan polygon and the set of opers having zero p-curvature. We also show that, after fixing the determinant, these sets are finite, which enables us to derive the dimension of certain Quot-schemes and certain loci of stable Frobenius-destabilized vector bundles over X. The finiteness is proved by studying the properties of the Hitchin-Mochizuki morphism; an alternative approach to finiteness has been realized in [3]. In particular we prove a generalization of a result of Mochizuki to higher ranks.

Original languageEnglish (US)
Pages (from-to)39-75
Number of pages37
JournalAdvances in Mathematics
Volume274
DOIs
StatePublished - Apr 9 2015

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Morphism
Frobenius
Vector Bundle
Curve
Locus
Bundle
Finiteness
Stable Vector Bundles
Atlas
Pullback
Zero
Bijective
Algebraically closed
Polygon
Isomorphism
Genus
Determinant
Correspondence
Curvature
Alternatives

Keywords

  • Frobenius map
  • Local system
  • Moduli spaces
  • Primary
  • Secondary
  • Vector bundles

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Hitchin-Mochizuki morphism, opers and Frobenius-destabilized vector bundles over curves. / Joshi, Kirti N; Pauly, Christian.

In: Advances in Mathematics, Vol. 274, 09.04.2015, p. 39-75.

Research output: Contribution to journalArticle

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