On the geography of simply connected nonspin symplectic 4-manifolds with nonnegative signature

In [8,5], the first author and his collaborators constructed the irreducible symplectic 4-manifolds that are homeomorphic but not diffeomorphic to (2n-1)CP²#(2n-1)CP² for each integer n≥ 25, and the families of simply connected irreducible nonspin symplectic 4-manifolds with positive signature that are interesting with respect to the symplectic geography problem. In this paper, we improve the main results in [8,5]. In particular, we construct (i) an infinitely many irreducible symplectic and nonsymplectic 4-manifolds that are homeomorphic but not diffeomorphic to (2n-1)CP²#(2n-1)CP² for each integer n≥ 12, and (ii) the families of simply connected irreducible nonspin symplectic 4-manifolds that have the smallest Euler characteristics among the all known simply connected 4-manifolds with positive signature and with more than one smooth structure. Our construction uses the complex surfaces of Hirzebruch and Bauer-Catanese on Bogomolov-Miyaoka-Yau line with c₁²=9χ_h=45, along with the exotic symplectic 4-manifolds constructed in [2,6,4,7,11].