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

TL;DR

This paper introduces new symplectic 4-manifolds with non-negative signatures via complex surface constructions, including BMY line surfaces, Cartwright-Steger surfaces, and line-arrangement quotients, expanding the known landscape of such manifolds.

Contribution

It provides novel constructions of symplectic and non-symplectic 4-manifolds with specific topological properties, including examples homeomorphic but not diffeomorphic to certain connected sums and minimal Euler characteristics.

Findings

Constructed infinitely many non-symplectic 4-manifolds homeomorphic but not diffeomorphic to (2n-1)CP^2 #(2n-1)bar{CP}^2 for n≥9.

Produced simply connected irreducible nonspin symplectic 4-manifolds with minimal Euler characteristics among known examples.

Developed a complex ball quotient surface from Hirzebruch's line arrangements with positive signature.

Abstract

We present the various constructions of new symplectic $4$-manifolds with non-negative signatures using the complex surfaces on the BMY line $c_1^2 = 9\chi_h$, the Cartwright-Steger surfaces, the quotients of Hirzebruch's certain line-arrangement surfaces, along with the exotic symplectic $4$-manifolds constructed in \cite{AP2, AS}. In particular, our constructions yield to (i) an irreducible symplectic and infinitely many non-symplectic $4$-manifolds that are homeomorphic but not diffeomorphic to $(2n-1)CP^{2}\#(2n-1)\bar{CP}^{2}$ for each integer $n \geq 9$, (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.