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

Abstract

We present the various constructions of new symplectic 4-manifolds with non-negative signatures using the complex surfaces on the BMY line c 2 1 = 9χ h , the Cartwright-Steger surfaces, the quotients of Hirzebruch's certain line-arrangement surfaces, along with the exotic symplectic 4-manifolds constructed in [5,13]. In particular, our constructions yield to (i) an irreducible symplectic and infinitely many non-symplectic 4-manifolds that are homeomorphic but not diffeomorphic to (2n − 1)CP 2 #(2n − 1)CP 2 for each integer n ≥ 9, (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 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.