We construct examples of geometrically decomposable aspherical 4-manifolds with non-zero signature. We show that all such 4-manifolds satisfy the inequality (of Bogomolov--Miyaoka--Yau type) χ >= 3|σ|. We also construct examples attaining the equality that are non-geometric and have non-zero signature. Finally, we prove that for higher graph 4-manifolds, with complex-hyperbolic vertices, the strict inequality always holds. Moreover, we construct infinitely many examples of higher graph 4-manifolds with non-zero signature and prove that the inequality is strict and sharp in this class.

LFDC thanks Grigori Avramidi, Igor Belegradek, Claude LeBrun, and Stefano Vidussi for interesting discussions around the signature of 4-manifolds. Part of this work was carried out at the University of Florida: MG thanks this institution for the hospitality and support. We would like to thank an anonymous referee for their comments, especially for catching a computational mistake in the proof of Proposition 4.3. This work is partially supported by the NSF grant DMS-2104662.