On large families of bundles over algebraic surfaces

TL;DR

This paper constructs sequences of vector bundles with unbounded rank and discriminant on any algebraic surface, demonstrating the failure of the Strong Bogomolov Inequality for all l>4.

Contribution

It introduces a method to generate large families of vector bundles on algebraic surfaces and disproves the Strong Bogomolov Inequality for all l>4.

Findings

Sequences of vector bundles with unbounded rank and discriminant constructed

Strong Bogomolov Inequality is false for all l>4 on any surface

Addresses conjectures related to algebraic geometry and vector bundles

Abstract

The aim of this note is to construct sequences of vector bundles with unbounded rank and discriminant on an arbitrary algebraic surface. This problem, on principally polarized abelian varieties with cyclic Neron-Severi group generated by the polarization, was considered by Nakashima in connection with the Douglas-Reinbacher-Yau conjecture on the Strong Bogomolov Inequality. In particular we show that on any surface, the Strong Bogomolov Inequality $SBI_l$ is false for all $l>4$.