An odd Seiberg-Witten invariant imposes bounds on the signature of a closed, almost complex 4-manifold with vanishing first Chern class. This applies in particular to symplectic 4-manifolds of Kodaira dimension zero.
"Almost complex 4-manifolds with vanishing first Chern class." J. Differential Geom. 79 (1) 25 - 32, May 2008. https://doi.org/10.4310/jdg/1207834656