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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?

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It seems natural to approach this by trying to compute the mixed Hodge structure on $Y \setminus X$. The cohomology carries a weight filtration $W_\bullet$ and the Gysin map will be surjective if and only if the lowest weight part $ W_{k+2r}H^{k+2r}(Y \setminus X, \mathbf Q) $ vanishes.

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I suppose that you are discussing Betti cohomology with coefficient in $\mathbb{Q}$.

Using the long exact sequence $$\cdots \to H^{k}(X,\mathbb{Q}) = H^{k+2r}(Y,Y \backslash X,\mathbb{Q}) \to H^{k+2r}(Y,\mathbb{Q}) \to H^{k+2r}(Y\backslash X,\mathbb{Q}) \to \cdots$$ we see that the surjectivity of the Gysin morphism $f_\*$ is equivalent to the vanishing of the pullback morphism $i^*:H^{k+2r}(Y,\mathbb{Q}) \to H^{k+2r}(Y\backslash X,\mathbb{Q})$, where $i$ is the open immersion $Y\backslash X \to Y$. (Since your $Y$ is smooth and projective, the Hodge structure on $H^{k+2r}(Y,\mathbb{Q})$ is pure, which explains the equivalence given by Dan)

We know that $Ker \ i^*$ is a sub-Hodge structure of (Hodge) coniveau $\ge r$ of $H^{k+2r}(Y,\mathbb{Q})$, it says in particular that if you want your Gysin morphism $f_{*}$ to be surjective, your $H^{k+2r}(Y, \mathbb{Q})$ should at least have coniveau $\ge r$! ($\leftarrow$ This should be seen as an exclamation mark, not a factorial notation...)

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