Numerically Flat Higgs Vector Bundles

Abstract

Abstract. After providing a suitable definition of numerical effectiveness for Higgs bundles, and a related notion of numerical flatness, in this paper we prove, together with some side results, that all Chern classes of a Higgs-numerically flat Higgs bundle vanish, and that a Higgs bundle is Higgs-numerically flat if and only if it is has a filtration whose quotients are flat stable Higgs bundles. We also study the relation between these numerical properties of Higgs bundles and (semi)stability.

Metrics on semistable and numerically effective Higgs bundles

“…We conclude this section by proving two results that hold when X is a smooth projective curve. The first Proposition generalizes results given in [14,5,4].…”

“…However, if E is 1-H-nef and d = 0 (i.e., it is 1-H-nflat) we know it is semistable (Theorem 3.11; this also follows from Corollary 3.6 of [4] since, as we shall see in the next Section, on a curve the notions of H-nefness and 1-H-nefness coincide). △…”

“…Several criteria of this type have appeared recently [5,2,3], all generalizing Miyaoka's result according to which a vector bundle E on a smooth projective curve X is semistable if and only if the numerical class λ = c 1 (O PE (1)) − 1 r π * (c 1 (E)) is nef, where π : PE → X is the projection. Also results by Gieseker [14], generalized in [5,4], relate the notions of nefness and semistability.…”

“…A notion of nefness for Higgs bundles was introduced [4] in the case of projective varieties, and the basic properties of such bundles were there studied. In particular, it was proved there that numerically flat Higgs bundles (i.e., Higgs bundles that are numerically flat together with their duals) have vanishing Chern classes.…”

“…This will also allow us to complete some of the results given in [4]. In particular, we provide new criteria for the characterization (in terms of the numerical effectiveness of certain associated Higgs bundles) of those Higgs bundles on projective manifolds that are semistable and satisfy the condition ∆(E) = c 2 (E) − r−1 2r c 1 (E) 2 = 0.…”