In this article we show that every closed oriented smooth 4–manifold can be decomposed into two codimension zero submanifolds (one with reversed orientation) so that both pieces are exact Kähler manifolds with strictly pseudoconvex boundaries and that induced contact structures on the common boundary are isotopic. Meanwhile, matching pairs of Lefschetz fibrations with bounded fibers are offered as the geometric counterpart of these structures. We also provide a simple topological proof of the existence of folded symplectic forms on 4–manifolds.
R Inanç Baykur. "Kähler decomposition of 4–manifolds." Algebr. Geom. Topol. 6 (3) 1239 - 1265, 2006. https://doi.org/10.2140/agt.2006.6.1239