  According to a conjecture attributed to Hartshorne and Lichtenbaum and proven
by Ellingsrud and Peskine, the smooth rational surfaces in $\mathbb{P}^4$
belong to only finitely many families. We formulate and study a collection of
analogous problems in which $\mathbb{P}^4$ 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.
