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 $\Sigma_{g} \times \Sigma_{h}$, where $g \geq 1$ and $h \geq 0$, 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 $(2n-1) \mathbb{CP}^{2} \# (2n-1) \overline{\mathbb{CP}}^{2}$ for each integer $n \geq 9$, (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.