A note on co-Higgs bundles

We show that for any ample line bundle on a smooth complex projective variety with nonnegative Kodaira dimension, the semistability of co-Higgs bundles of implies the semistability of bundles. Then we investigate the criterion for surface $X$ to have $H^0(T_X)=H^0(S^2T_X)=0$, which implies that any co-Higgs structure of rank two is nilpotent.