# Big Vector Bundles on Surfaces and Fourfolds

## TL;DR

This paper provides explicit criteria for the bigness of globally generated vector bundles on smooth projective varieties of dimension up to four, with applications to tangent bundles and Lazarsfeld-Mukai bundles.

## Contribution

It introduces new explicit criteria for the bigness of vector bundles on surfaces and fourfolds, connecting to recent research and specific geometric applications.

## Key findings

- Established sufficient conditions for vector bundle bigness
- Connected criteria to recent theoretical results
- Applied criteria to tangent bundles of Fano varieties

## Abstract

The aim of this note is to exhibit explicit sufficient criteria ensuring bigness of globally generated, rank-$r$ vector bundles, $r \geqslant 2$, on smooth, projective varieties of even dimension $d \leqslant 4$. We also discuss connections of our general criteria to some recent results of other authors, as well as applications to tangent bundles of Fano varieties, to suitable Lazarsfeld-Mukai bundles on four-folds, etcetera.