Miyaoka-type inequalities for terminal threefolds with nef anti-canonical divisors

In this paper, we study Miyaoka-type inequalities on Chern classes of terminal projective 3-folds with nef anti-canonical divisors. Let X be a terminal projective 3-fold such that −K_X is nef. We show that if c₁(X) · c₂(X) ≠ 0, then \(c_{1}(X)\cdot c_{2}(X)\geqslant {1\over 252}\) $$c_1(X)\cdot c_2(X)⩾\frac{1}{252}$$ ; if further X is not rationally connected, then \(c_{1}(X)\cdot c_{2}(X)\geqslant {4\over 5}\) $$c_1(X)\cdot c_2(X)⩾\frac{4}{5}$$ and this inequality is sharp. In order to prove this, we give a partial classification of such varieties along with many examples. We also study the nonvanishing of c₁(X)^{dim X − 2} ·c₂(X) for terminal weak Fano varieties and prove a Miyaoka-Kawamata-type inequality.