Bogomolov’s inequality for Higgs sheaves in positive characteristic

Abstract

We prove Bogomolov's inequality for Higgs sheaves on varieties in positive characteristic p that can be lifted modulo p 2 . This implies the Miyaoka-Yau inequality on surfaces of non-negative Kodaira dimension liftable modulo p 2 . This result is a strong version of Shepherd-Barron's conjecture. Our inequality also gives the first algebraic proof of Bogomolov's inequality for Higgs sheaves in characteristic zero, solving the problem posed by Narasimhan.

Semistable Higgs bundles, periodic Higgs bundles and representations of algebraic fundamental groups

“…However, the theory developed in this paper turns out to be also useful in the study of Higgs bundles with nontrivial Chern classes. This has been beautifully demonstrated in the recent work [20] of A. Langer on a purely algebraic proof of the Bogomolov-Giesecker inequality for semistable Higgs bundles in the complex case ( [30,Proposition 3.4]) and the Miyaoka-Yau inequality for Chern numbers of complex algebraic surfaces of general type. In his work, the notion of (semistable) Higgs-de Rham flow in characteristic p has played as similar role as the Yang-Mills-Higgs flow over the field of complex numbers.…”

Rank 3 rigid representations of projective fundamental groups

“…of (E, θ ) toȲ is A-stable (one can also give a direct proof of this fact: see [La2,Theorem 12] and [La3]). Since the rational Chern classes of E|Ȳ vanish, (E|Ȳ , θ Y ) is stable with respect to every stable polarization and semistable with respect to every nef polarization.…”

Uniformisation of higher-dimensional minimal varieties

“…For Higgs bundles on manifolds with ample polarisation, the theorem appears in Simpson's work, [Sim92, Lemma 3.7]. Langer proves a similar theorem for sheaves on projective manifolds, polarised by tuples of divisors that need not be ample, [Lan15,Theorem 10]. He works in positive characteristic but says that mutatis mutandis, his arguments will also work in characteristic zero, cf.…”