Complex Ball Quotients and New Symplectic 4-manifolds with Nonnegative Signatures

We construct new symplectic 4-manifolds with non-negative signatures and with the smallest Euler characteristics, using fake projective planes, Cartwright–Steger surfaces and their normal covers and product symplectic 4-manifolds Σ_g×Σh, where g≥1 and h≥0, along with exotic symplectic 4-manifolds constructed in [7, 12]. In particular, our constructions yield to (1) infinitely many irreducible symplectic and infinitely many non-symplectic 4-manifolds that are homeomorphic but not diffeomorphic to (2n−1)CP²#(2n−1)CP² for each integer n≥9, (2) infinite families of simply connected irreducible nonspin symplectic and such infinite families of non-symplectic 4-manifolds that have the smallest Euler characteristics among the all known simply connected 4-manifolds with positive signatures and with more than one smooth structure. We also construct a complex surface with positive signature from the Hirzebruch’s line-arrangement surfaces, which is a ball quotient.