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. 



Curves on a Kahler manifold
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.


