An overview on curve semistable and numerically flat Higgs bundles
Armando Capasso
TL;DR
This paper reviews the concepts of Higgs-Grassmannian schemes and positivity conditions like numerically flatness for Higgs bundles, establishing properties and relations to semistability over smooth projective varieties.
Contribution
It introduces generalized positivity notions for Higgs bundles using Higgs-Grassmannian schemes and explores their properties and connection to curve semistability.
Findings
Properties of Higgs bundles satisfying positivity conditions
Relations between positivity and semistability of Higgs bundles
Framework for analyzing Higgs bundles over smooth projective varieties
Abstract
After recalling the basic notions concerning Higgs-Grassmannian schemes, I review how these latter can be used to define generalisations of the notion of positivity conditions, such as numerically flatness, which "feel" the Higgs field. Then I prove several properties of Higgs bundles, over smooth projective varieties defined over an algebraically closed field of characteristic , satisfying these conditions. Finally, I discuss how one can relate them to semistability of the so-called "curve semistable" Higgs bundles.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Geometry and complex manifolds · Algebraic structures and combinatorial models