Generalized Bogomolov--Gieseker inequalities via Frobenius

Abstract

We establish a Bogomolov--Gieseker-type inequality for tilt-semistable objects on smooth projective varieties of dimension $\geq 3$, in the spirit of the Bayer--Macr\`i--Toda conjecture. A key ingredient is an analysis of the action of the Frobenius morphism on tilt-semistable objects. As an application, we prove the Castelnuovo Bound Conjecture, up to an explicit constant term. Combining our results with work of Feyzbakhsh and Thomas, we further show that on a Calabi--Yau threefold, arbitrary rank Donaldson--Thomas invariants of semistable sheaves are determined by rank one invariants, and hence are symplectic invariants.