Q:

Small deformations of smooth projective varieties

Let $X\subseteq \mathbf P^n(\mathbb C)$ be a non-singular projective variety over the complex numbers.  Suppose that $X$ is given by the vanishing of homogeneous polynomials $f_1, \dots, f_r$. Is it true that for sufficiently small variations of the coefficients of the polynomials $f_1, \dots, f_r$, the resulting variety $X'$ is isomorphic as a topological manifold to $X$? 
Thanks! 

A:

If $r=\operatorname{codim} X$, I think this is likely to be true, maybe by continuity.  But if $X$ isn't a complete intersection, the question can't possibly be true, as perturbing the defining equations should (generically) give rise to something of codimension $r$.
For example, take the twisted cubic in $\mathbb{P}^3$. It requires three defining quadratic equations, but three general quadratic equations will just give 8 points. 
So the answer is "no", at least to the question as stated.