Rank 3 curve semistable Higgs bundles: the nilpotent case

Source abstract

A Higgs bundle on a smooth complex projective variety is curve semistable if its pullback to every smooth projective curve is semistable. A 2008 conjecture asserts that this is equivalent to slope semistability for some polarization plus vanishing discriminant. One direction is known, the converse holds in rank two, and Huang recently gave a rank-four counterexample. We prove the conjecture for rank three nilpotent Higgs bundles in any dimension. We treat separately the two possible cases: if the Higgs field $φ$ is nilpotent of order two, there is a divisorial component of a Higgs Grassmannian where nefness of the normalized tautological class forces the discriminant to vanish; otherwise, $φ$ is regular nilpotent, and then the canonical kernel flag, commutativity and Bogomolov's lemma on line subbundles of the cotangent bundle complete the proof.