Hausdorff Distance Between Projective Varieties

Question

I wonder whether the following question was studied. Let $X,Y \subset \mathbb{P}^d(\mathbb{C})$ be two projective varieties, We can measure the Hausdorff distance $d_H(X,Y)$ between them in terms of the Fubini-Study metric. What happens if they are close to one another? For example, can one find an $\epsilon >0$ small enough, such that if $d_H(X,Y) < \epsilon$, then they have the same Hilbert polynomial? Are any results in this spirit known?