  We provide notions of numerical effectiveness and numerical flatness for
Higgs vector bundles on compact K\"ahler manifolds in terms of fibre metrics.
We prove several properties of bundles satisfying such conditions and in
particular we show that numerically flat Higgs bundles have vanishing Chern
classes, and that they admit filtrations whose quotients are stable flat Higgs
bundles. We compare these definitions with those previously given in the case
of projective varieties. Finally we study the relations between numerically
effectiveness and semistability, establishing semistability criteria for Higgs
bundles on projective manifolds of any dimension.
