This paper is the written and expanded version of a talk at the twenty-fourth Gokova Geometry/Topology Conference (2017). We consider splittings of a symplectic rational homology \(\mathbb C\mathbb P^2\) by a contact rational homology sphere, which is embedded as a hypersurface of contact type inside the symplectic manifold. After going over some basic properties, we give evidence of subtle obstructions for such splittings in the case of \(\mathbb C\mathbb P^2\) and explain how these are related to interesting rigidity phenomena in symplectic and algebraic geometry. Based on the existence of such obstructions, we propose a method for constructing exotic \(\mathbb C\mathbb P^2\)'s and other related new small symplectic \(4\)-manifolds.