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}^{n} \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}^{n} \mathrm{Hdg}^{ 2 k } ( \mathcal{H}\mathrm{ilb} ( \mathbb{P}^n ) , \mathbb{Q}) $ ?

Thanks in advance for your help.

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.