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?
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$\begingroup$ Gromov's Betti number theorem provides such examples. $\endgroup$– Anton PetruninCommented Sep 3, 2018 at 16:07
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$\begingroup$ Yes, but I had a hope for something more explicit. $\endgroup$– Alex GavrilovCommented Sep 3, 2018 at 16:16
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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.
answered Sep 3, 2018 at 18:39