Note on new symplectic 4-manifolds with nonnegative signature

TL;DR

This paper constructs new symplectic 4-manifolds with non-negative signature by utilizing complex surfaces on the Bogomolov-Miyaoka-Yau line, including fake projective planes and Cartwright-Steger surfaces, resulting in an infinite family of manifolds with specific homology properties.

Contribution

It introduces a novel construction method for symplectic 4-manifolds with non-negative signature using complex surfaces on the BMY line, expanding known examples.

Findings

Infinite family of fake rational homology manifolds for 3 ≤ n ≤ 22

Construction of new symplectic 4-manifolds with non-negative signature

Utilization of complex surfaces on the BMY line, including fake projective planes

Abstract

In this short note, we present a construction of new symplectic 4-manifolds with non-negative signature using the complex surfaces on Bogomolov-Miyaoka-Yau line $c_1^2 = 9\chi_h$, the fake projective planes and Cartwright-Steger surfaces. Our construction yields an infinite family of fake rational homology $(2n-1)\CP#(2n-1)\CPb$ for any integer $3 \leq n \leq 22$.