On Pseudo-Effectivity of the Second Chern Classes for Terminal   Threefolds
  We give a reduction of the conjecture that for terminal projective threefolds whose anticanonical divisors are nef, the second Chern classes are pseudo-effective. On the other hand, some effective non-vanishing results are obtained as applications of the pseudo-effectivity of the second Chern classes. 
Qihong Xie