Let's work over the field $\mathbb{C}$ of complex numbers, and let $X\subset \mathbb{P}^n$ be a projective variety. Let $\tilde{X}\subset \mathbb{P}^n$ be any small open neighborhood of $X$, in the complex topology.



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> Question: Do there exist many algebraic varieties $\subset \tilde{X}$ of dimension equal to the dimension of $X$?
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Of course, $X$ itself lies in there, and one may apply some automorphism of $\mathbb{P}^n$ close to the identity to obtain some further examples. I would be interested in any construction of more subvarieties in $\tilde{X}$, or examples showing that in general these may not exist. Note that the degree of the subvarieties is allowed to be arbitrarily large, and that I do not want to consider only deformations of $X$ (which might be rigid).



Accumulation of algebraic subvarieties: Near one subvariety there are many others (?)
The answer is yes (as in the case of Sasha's answer we use ramified covers)


**Proof.** Let $X$ be any variety in $\mathbb CP^n$. Take a section $s\_m$ of $O(m)$ on $X$ such that $s\_m$ is not equal to $m$-th tensor power $s^{\otimes m}$ of any section $s$ of $O(1)$ restricted on $X$. Now, let $s\_m^{\frac{1}{m}}$ be the multi-section of $O(1)$ on $X$. This multi-section defines a subvarity $X\_m$ in the total space of $O(1)$ on $X$, that is the cover of $X$ of degree $m$.


Finally notice that there is a family of maps from the total space of $O(1)$ on $X$ to $\mathbb CP^n$ that sends the zero section of $O(1)$ on $X$ to $X$. The image of such a map in $\mathbb CP^n$ is just the union of all lines in $\mathbb CP^n$ that join a fixed point $p$ with all points of $X$, the point $p$ itself does not belong to the image. 
Then the image of $X\_m$ in $\mathbb CP^n$ is the desired variety. **END.**


We used here the fact that $O(1)$ on $\mathbb CP^n$ can be embedded in $T\mathbb CP^n$ as a subsheaf (in various ways). 


