Symplectic Bogomolov–Miyaoka–Yau inequality

Prove that the Bogomolov–Miyaoka–Yau inequality c_1^2(M)\leq 3e(M) holds for every simply connected symplectic 4-manifold, thereby determining whether any such manifold violates the inequality.

Background

The paper considers the symplectic analogue of the Bogomolov–Miyaoka–Yau inequality, which is known for complex surfaces of general type. Fintushel and Stern conjectured that the inequality should hold for all simply connected symplectic 4-manifolds.

The authors note that their constructed symplectic 4-manifolds approach the line c_1^2=3e from below. Establishing the conjecture would yield the lower bound \Lambda_s\geq 8s-1 and, together with existing upper bounds, determine the asymptotic behavior of \Lambda_s/s.