# Curves on a Kahler manifold

Let $$(X,\omega )$$ be a compact Kahler manifold. For any $$d>0$$ are there only finitely many families of curves $$C_i$$ such that $$C_i\cdot \omega ? (More precisely, if $$C$$ is any curve such that $$C\cdot \omega , 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.

• I guess you want $X$ to be compact right? – diverietti 2 days ago
• 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)? – Donu Arapura 2 days ago
• Yes, $X$ is compact and the projective case should follow from a Hilbert scheme argument. – user110111 2 days ago