Big Vector Bundles on Surfaces and Fourfolds

Abstract

The aim of this note is to exhibit explicit sufficient cohomological 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 fourfolds, etcetera.