Big Vector Bundles on Surfaces and Fourfolds

The aim of this note is to exhibit explicit sufficient criteria ensuring bigness of globally generated, rank-$r$ vector bundles, $r⩾2$, on smooth, projective varieties of even dimension $d⩽4$. 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.