Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|cite|improve this question

2 Answers 2

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.

share|cite|improve this answer

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...)

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.