There is a wealth of research that I have found that characterize different classes of compact $4$-manifolds by the ability (or lack thereof, in the case of aspherical) to (symplectically or smoothly) embed spheres of certain self-intersections into them.
However, I am interested in what I can learn about those compact 6-manifolds which admit symplectic or smooth embeddings of spheres or tori into them. To be clear, I am only wishing to embed surfaces into compact six-manifolds... not spheres or tori of higher dimension. Does anybody here know a nice source of collected research on this topic? So far, I have not found it. I think the most energy over the last $35$ years has been spent on $4$-manifolds.