Let $X$ be a smooth, projective variety and $Y \subset X$ be a hyperplane section (possibly singular) of $X$. Suppose that the dimension of $X$ is $n$. Is it true that for any $k<n-1$, the induced morphism of intersection homologies:
$$IH\_k(Y) \to IH\_k(X)$$
is an isomorphism? 


I would think this is true as the version of the Lefschetz hyperplane section theorem I have it states this conclusion under the additional assumption that $Y$ is transverse to every strata of some Whitney stratification of $X$. As $X$ is smooth, in my case, I can take the only non-trivial strata to be $X$ and the other sub-strata to be empty sets. This should answer the above question, but I am not sure if I am making a mistake.



Lefschetz hyperplane section theorem for intersection homology
So, the answer is positive and follows from the "usual" Weak Lefschetz Theorem if you have transversality (and it suffices to assume that $Y$ is smooth). On the other hand, if $Y$ is singular then you surely don't have transversality and probably don't have the isomorphism in question (as far as I remember, the intersection homology of $X$ is isomorphic to its "usual" cohomology, whereas for a singular $Y$ the group $IH\_k(Y)$ is "usually smaller" than $H^k(Y)$).


