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.
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3$\begingroup$ I guess you want $X$ to be compact right? $\endgroup$– diveriettiCommented May 22, 2020 at 20:15
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2$\begingroup$ It should be true when $X$ is a projective manifold by considering the map from the Hilbert to Chow schemes. Perhaps the same thing works more generally using Douady and Barlet spaces (assuming compactness)? $\endgroup$– Donu ArapuraCommented May 22, 2020 at 20:44
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$\begingroup$ Yes, $X$ is compact and the projective case should follow from a Hilbert scheme argument. $\endgroup$– user110111Commented May 22, 2020 at 21:48
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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.
