Five simply connected closed 4-manifolds are known to admit Riemannian metrics with nonnegative sectional curvature: 
$$\mathbb{S}^4,\,\mathbb{C}\mathbb{P}^2,\,\mathbb{S}^2\times\mathbb{S}^2,\,\mathbb{C}\mathbb{P}^2\#\mathbb{C}\mathbb{P}^2,\,\mathbb{C}\mathbb{P}^2\#\overline{\mathbb{C}\mathbb{P}^2}.$$
Hypothetically, this is all. I am interested in a special case of this problem: which (simply connected) projective complex surfaces are known to *not* belong to this list? 



Complex surfaces not admitting nonnegative sectional curvature metrics
Gromov proved that there is a constant $C(n)$ such that any complete $n$-manifold $M$ of non-negative curvature satisfies $dimH\_\*(M)\leq C(n)$. Where $C(n)\leq 10^{3n^4+9n^3+6n^2}$. For details reference look at Theorem 3.19 <https://www.math.upenn.edu/~wziller/math660/TopogonovTheorem-Myer.pdf>


And that will give the answer of your question now.


