Smooth surfaces in smooth fourfolds with vanishing first Chern class

According to a conjecture attributed to Hartshorne and Lichtenbaum and proven by Ellingsrud and Peskine [18], the smooth rational surfaces in P4 belong to only finitely many families. We formulate and study a collection of analogous problems in which P4 is replaced by a smooth fourfold X with vanishing first integral Chern class. We embed such X into a smooth ambient variety and count families of smooth surfaces which arise in X from the ambient variety. We obtain various finiteness results in such settings. The central technique is the introduction of a new numerical invariant for smooth surfaces in smooth fourfolds with vanishing first Chern class.