Generalized Bogomolov Inequalities

Abstract

We introduce the notion of a Hodge-Riemann pair of cohomology classes that generalizes the classical Hodge-Riemann bilinear relations, and the notion of a Bogomolov pair of cohomology classes that generalizes the Bogomolov inequality for semistable sheaves. We conjecture that every Hodge-Riemann pair is a Bogomolov pair, and prove various cases of this conjecture. As an application we get new results concerning boundedness of semistable sheaves.

32 pages. v2: Corrected Proposition 6.1 and its corollaries; we thank Ziquan Zhuang for pointing out an error in the previous proof of the proposition