Let $(X,\omega )$ be a compact Kahler manifold. For any $d>0$ are there only finitely many families of curves $C_i$ such that $C_i\cdot \omega <d$? (More precisely, if $C$ is any curve such that $C\cdot \omega <d$, then $C$ belongs to one of the families $C_i$.) I believe the analogous statement for projective varieties follows from well known results for Hilbert or Chow schemes.
From "Bounded sets of sheaves on Kähler manifolds" By Matei Toma, J. reine angew. Math. 710 (2016), 77–93
Lemma 4.4. Let X be a Kähler manifold, r be an integer and F be a set of compact reduced subspaces of X of bounded degree and all of whose components are of dimension r and contained in a ﬁxed compact subset of X. Then F is bounded.