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If $X$ is a K3 surface over the complex (algebraic) then I wonder if the poincare pairing induces a polarization on the Hodge structure $H^2(X,\mathbb Z)$? The point is that I see that when constructing the Kuga-Satake abelian variety, people take the primitive cohomology in order to have a polarization on the induced Hodge structure, but I wonder if the reason they do not take the hole $H^2$ is because we do not have a polarization induced by the Poincare paring... I know that if this induces a polarization of the Hodge structure, then the Hodge decomposition is orthogonal but if this is not a polarization, is the Hodge decomposition stil orthogonal?

On the other hand, on etale cohomology, Faltings theorem give an analogous to the Hodge decomposition for (say) a $K3$-surface over a $p$-adic field $K$:

$H_{et}(X_{\bar H_{et}^2(X_{\bar K},Q_p)\otimes \mathbb C_p = H^2(X,\mathcal{O}_X)(0) \oplus H^1 (X, \Omega^1)(-1) \oplus H^0(X,\cal O_X)(-2)$ Is this decomposition orthogonal respect to the Poincare pairing on etale cohomology?

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# Poincare pairing and polarization of Hodge structure. Kuga-Satake construction.

If $X$ is a K3 surface over the complex (algebraic) then I wonder if the poincare pairing induces a polarization on the Hodge structure $H^2(X,\mathbb Z)$? The point is that I see that when constructing the Kuga-Satake abelian variety, people take the primitive cohomology in order to have a polarization on the induced Hodge structure, but I wonder if the reason they do not take the hole $H^2$ is because we do not have a polarization induced by the Poincare paring... I know that if this induces a polarization of the Hodge structure, then the Hodge decomposition is orthogonal but if this is not a polarization, is the Hodge decomposition stil orthogonal?

On the other hand, on etale cohomology, Faltings theorem give an analogous to the Hodge decomposition for (say) a $K3$-surface over a $p$-adic field $K$:

$H_{et}(X_{\bar K},Q_p)\otimes \mathbb C_p = H^2(X,\mathcal{O}_X)(0) \oplus H^1 (X, \Omega^1)(-1) \oplus H^0(X,\cal O_X)(-2)$ Is this decomposition orthogonal respect to the Poincare pairing on etale cohomology?