On a numerical criterion for Fano fourfolds

In this paper, we prove a special case of Campana--Peternell's conjecture in dimension 4. Specifically, we show that a projective smooth fourfold $X$ with $c^2_1(X)\cdot c_2(X)\neq 0$ and strictly nef anti-canonical divisor $-K_X$ is a Fano fourfold. To this aim, we completely solve the non-vanishing conjecture for strictly nef anti-canonical divisors in dimension 4.
Comment: 12 pages, comments are welcome. v2: we revise the introduction more clearly to show what we actually prove in this paper. v3: it is the published version, we revise the contents significantly under the suggestions of joural's referee