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Fix a positive integer $n$.

Let $S_{d, m}$ be the set of projective subvarieties $V\subset \mathbb{C}P^n$ that are the intersection of at most $m$ hypersurfaces of degree at most $d$.

Let $M$ be a smooth manifold. Consider the set $S_M$ of projective subvarieties $V\subset \mathbb{C}P^n$ diffeomorphic to $M$. Is it contained in $S_{d, m}$ for some $d, m$?

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    $\begingroup$ Welcome new contributor. That is not true. Consider the set of projective subvarieties of $\mathbb{C}P^3$ that are diffeomorphic to $\mathbb{C}P^1$. Since there are rational curves of arbitrarily positive degree embedded in $\mathbb{C}P^3$, there is no single $d$ and $m$ that work. $\endgroup$
    – Jason Starr
    Commented Aug 8, 2021 at 21:03
  • $\begingroup$ @JasonStarr What if we also fix the homology class of $M$ in $\mathbb{C}P^n$? $\endgroup$
    – cuo
    Commented Aug 8, 2021 at 21:10
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    $\begingroup$ Yes, if you fix the dimension and the degree of the subvariety, then since the Chow variety is finite type, there is also a finite type open parameterizing smooth subvarieties of that dimension and degree. Your parameter space is an open and closed subset of this finite type open. $\endgroup$
    – Jason Starr
    Commented Aug 8, 2021 at 21:22

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