Some quadratic inequalities on product varieties

TL;DR

This paper establishes new quadratic inequalities for semi-stable objects on product varieties, extending classical stability results and providing a weaker form of Bogomolov's inequality applicable in arbitrary characteristics.

Contribution

It introduces novel quadratic inequalities for semi-stable objects on product varieties, generalizing classical stability conditions and Bogomolov's inequality.

Findings

Derived positivity results for coefficients in the complexified Hilbert polynomial.

Established a sequence of quadratic inequalities for semi-stable objects.

Presented a weak version of Bogomolov's inequality valid in arbitrary characteristics.

Abstract

We prove some positivity results on the coefficients in the complexified Hilbert polynomial of a semi-stable object. After applying these results on the classical slope stability conditions, we get sequences of quadratic inequalities for semi-stable objects on product varieties. The leading quadratic inequality can be viewed as a weak version of Bogomolov's inequality holding in arbitrary characteristics.