Let $f:X \to Y$ be a closed immersion between smooth projective complex varieties. Suppose that the codimension of (the image of) $Y$ in $X$ is equal to $r \ge 1$. This induces the Gysin morphism $f_{\*}:H^{k}(X) \to H^{k+2r}(Y)$. Is there any known example/criterion for which the Gysin morphism $f_*$ is surjective at least when $k$ is even?

Let $f:X \to Y$ be a closed immersion between smooth projective complex varieties. Suppose that the codimension of (the image of) $Y$ in $X$ is equal to $r \ge 1$. This induces the Gysin morphism $ f_{*} : H^{k} (X) \to H^{k+2r} (Y) $. 

Is there any known example/criterion for which the Gysin morphism $f_*$ is surjective, at least when $k$ is even?

Let $f:X \to Y$ be a closed immersion between smooth projective complex varieties. Suppose that the codimension of (the image of) $Y$ in $X$ is equal to $r \ge 1$. This induces the Gysin morphism $f_{\*}:H^{k}(X) \to H^{k+2r}(Y)$. Is there any known example/criterion for which the Gysin morphism $f_*$ is surjective?

Let $f:X \to Y$ be a closed immersion between smooth projective complex varieties. Suppose that the codimension of (the image of) $Y$ in $X$ is equal to $r \ge 1$. This induces the Gysin morphism $f_{\*}:H^{k}(X) \to H^{k+2r}(Y)$. Is there any known example/criterion for which the Gysin morphism $f_*$ is surjective at least when $k$ is even?