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I try to calculate the rational cohomology algebra $ \mathrm{Hdg}^{ 2 \bullet } ( \mathcal{H}\mathrm{ilb} ( \mathbb{P}^n ),\mathbb{Q} ) = \displaystyle \bigoplus_{k=0}^{+ \infty} \mathrm{Hdg}^{ 2 k } ( \mathcal{H}\mathrm{ilb} ( \mathbb{P}^n ) , \mathbb{Q}) $, such that : $ \mathrm{Hdg}^{ 2 k } ( \mathcal{H}\mathrm{ilb} ( \mathbb{P}^n ) , \mathbb{Q} ) = H^{2k} ( \mathcal{H}\mathrm{ilb} ( \mathbb{P}^n ), \mathbb{Q} ) \cap H^{k,k} ( \mathcal{H}\mathrm{ilb} ( \mathbb{P}^n ) ) $, but it's hard for me to do it alone.

I'm remembring, i was able to calculate for instance objects like $ H^{2k} ( \mathbb{P}^{n}, \mathbb{Q} ) $ for $ k = 0 , ... , n $ when i was young, but now, i think i unfortunatly forgot a lot of things about cohomology theory.

All thing that i know for this moment is that : $ \mathcal{H}\mathrm{ilb} ( \mathbb{P}^n ) = \displaystyle \coprod_P \mathcal{H}\mathrm{ilb}_P ( \mathbb{P}^n ) $ such that, $ \mathcal{H}\mathrm{ilb}_P ( \mathbb{P}^n ) = \{ \text{ subvarieties } X \subset \mathbb{P}^{n} \text{ with Hilbert polynomial } P = P(m)\}$ is the Hilbert variety ( i.e : parameter space ) with Hilbert polynomial $P=P(m)$, defined by : $$ \Psi : \mathcal{H}\mathrm{ilb} ( \mathbb{P}^n ) \to G ( q(m) , N(m) ) $$ $$ X \ \ \to I(X)_m $$

$ \Psi $ is an injection; the image, called the open Hilbert variety, is a quasi projective variety. $$ N(m) = \begin{pmatrix} m+n \\ n \end{pmatrix} \ \ \mathrm{and} \ \ q(m) = N(m) - P(m) $$ $ G ( q(m) , N(m) ) $ is the Grassmannian.

$ I(X)_m $ is the $m$-th graded piece of the ideal $ I(X) $ such that : $ S(X) = \mathbb{C} [X_1 , \dots , X_n ] / I(X) $ is the homogeneous coordinate ring.

Could you please explain to me how to calculate : $ \mathrm{Hdg}^{ 2 \bullet } ( \mathcal{H}\mathrm{ilb} ( \mathbb{P}^n ),\mathbb{Q} ) = \displaystyle \bigoplus_{k=0}^{ + \infty } \mathrm{Hdg}^{ 2 k } ( \mathcal{H}\mathrm{ilb} ( \mathbb{P}^n ) , \mathbb{Q}) $ ?

Thanks in advance for your help.

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  • $\begingroup$ There is an algebraic cell decomposition of the Hilbert schemes of finite length closed subschemes of $\mathbb{P}^n$, cf. the work of Ellingsrud and Stromme. Thus, all of the rational cohomology is Tate type. $\endgroup$
    – Jason Starr
    Commented Aug 23, 2018 at 1:17
  • $\begingroup$ Thank you very much @Jason Starr for your answer. :-) Can you give me please, some titles of books or articles of Ellingsrud and Stromme where i can find about the algebraic cell decomposition of $ \mathcal{H} \mathrm{ilb} ( \mathbb{P}^n ) $ that you told me in your comment above ? Thank you. :-) $\endgroup$
    – YoYo
    Commented Aug 23, 2018 at 2:42
  • $\begingroup$ Are you aware that $\mathcal{H}\mathrm{ilb}(\mathbb{P}^n)$ is highly singular for $n\geq 3$? Given that, what is $H^{k,k}(\mathcal{H}\mathrm{ilb}(\mathbb{P}^n))$? $\endgroup$
    – abx
    Commented Aug 23, 2018 at 7:54
  • $\begingroup$ @abx. Perhaps the OP just wanted the direct sum of the $(k,k)$-components of the associated graded pure Hodge structures of the weight filtration ("diagonal" coefficients of the Hodge-Deligne polynomial). $\endgroup$
    – Jason Starr
    Commented Aug 23, 2018 at 9:19
  • $\begingroup$ I can't follow you in these explanations since I do not have yet the level for that, nevertheless, can you tell me which are the books that treat of these subjects there : the (k,k)-components of the associated graded pure Hodge structures of the weight filtration, "diagonal" coefficients of the Hodge-Deligne polynomial. @Jason Starr : Can you please tell me where exactly can i find about the algebraic cell decomposition of $ \mathcal{H} \mathrm{ilb} ( \mathbb{P}^n ) $ in the work of Ellingsrud and Stromme ?. In which book ? What is its title please ? Thank you very much. $\endgroup$
    – YoYo
    Commented Aug 23, 2018 at 9:45

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