Slope Inequalities for the Geography Problem of Spin Symplectic 4-Manifolds

We construct infinitely many new pairwise nondiffeomorphic smooth structures on infinitely many closed simply connected spin 4-manifolds with positive signature. Our construction builds on an infinite family of simply connected complex surfaces of general type due to Roulleau and Urzúa that populate points arbitrarily near the Bogomolov-Miyaoka-Yau line in the complex geography plane. We also discuss how the conjectural symplectic Bogomolov-Miyaoka-Yau inequality implies that our results are close to being optimal.