J-holomorphic curves in almost complex surfaces do not always minimize the genus

The adjunction formula computes the genus of an almost complex curve F embedded in an almost complex surface M in terms of the homology class of F. If M is Kahler (or at least symplectic) and the self-intersection of F is non-negative then the genus of any other surface embedded in M and homologous to F is not less then the genus of F (the proof of this statement (which is a generalization of the Them conjecture for CP2) was recently given by the Seiberg-Witten theory). This paper shows that the extra assumptions on M are essential for the genus-minimizing properties of embedded almost complex curves.