Bogomolov type inequalities and Frobenius semipositivity
Hao Max Sun
TL;DR
The paper addresses extending Bogomolov-type inequalities to higher Chern characters under Frobenius-semipositivity constraints. It develops a high-rank asymptotic Riemann–Roch framework together with Langer’s global-section estimates to bound $H^{d-t}\mathrm{ch}_t(\mathcal{E})$ for torsion-free semistable sheaves with $\phi_{sp}(\mathcal{E})\le 1$, and obtains abelian-variety analogues via GV$_{-1}$. The main contributions include a suite of Bogomolov-type inequalities in characteristic 0 and positive characteristic, a Fujita-type vanishing theory for F-semipositivity, and an asymptotic Riemann–Roch formula, all of which yield concrete bounds on Chern characters of threefolds and varieties of small codimension. These results connect stability theory, Frobenius techniques, and Chern-class inequalities, with implications for Calabi–Yau, Fano, and abelian settings. The findings enhance understanding of positivity phenomena in algebraic geometry and provide new tools for bounding Chern characters in geometric applications.
Abstract
We prove Bogomolov type inequalities for high Chern characters of semistable sheaves satisfying certain Frobenius semipositivity. The key ingredients in the proof are a high rank generalization of the asymptotic Riemann-Roch theorem and Langer's estimation theorem of the global sections of torsion free sheaves. These results give some Bogomolov type inequalities for semistable sheaves with vanishing low Chern characters. Our results are also applied to obtain inequalities of Chern characters of threefolds and varieties of small codimension in projective spaces and abelian varieties.