Smooth projective surfaces with pseudo-effective tangent bundles

Let S be a non-uniruled (i.e., non-birationally ruled) smooth projective surface. We show that the tangent bundle TS is pseudo-effective if and only if the canonical divisor KS is nef and the second Chern class vanishes, i.e., c₂(S) = 0. Moreover, we study the blow-up of a non-rational ruled surface with pseudo-effective tangent bundle.