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.

| cite | improve this question | | | | |
  • 2
    $\begingroup$ I guess you want $X$ to be compact right? $\endgroup$ – diverietti 2 days ago
  • 1
    $\begingroup$ 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 Arapura 2 days ago
  • $\begingroup$ Yes, $X$ is compact and the projective case should follow from a Hilbert scheme argument. $\endgroup$ – user110111 2 days ago

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.

| cite | improve this answer | | | | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.