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Can you tell me what the Hodge-Deligne polynomial of a point is, i.e. the polynomial $\sum_{p, q} (-1)^{p+q}h^{p, q}(X) x^py^q$ for some algebraic variety $X$ over the complex numbers? Can you also tell me what the Hodge-Deligne polynomial of $A_{\mathbb{C}}^1$ is, i.e. of the affine line?

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No, I can not. Thus I am asking. –  phil Sep 4 '11 at 22:15
    
Does $X$ have to be proper or projective or something? I assume $h^{p,q}(X)=\dim_\mathbb{C} H^q(X, \Omega^p)$ which doesn't have to be a finite number... –  Matt Sep 6 '11 at 17:38
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@Matt: Dear Matt, Presumably the $h^{p,q}$s are defined in terms of the mixed Hodge structure on the cohomology of $X$, which means that they are not the same as the dimension of $H^q(X,\Omega^p)$ when $X$ is not proper. Regards, –  Matt E Sep 6 '11 at 18:36

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Well $h^{p,q}(\mathbb{C}) = h^{p,q}(pt) = 0$ for $(p,q) \neq (0,0)$ and $h^{0,0}(\mathbb{C}) = h^{0,0}(pt) = 1$ (a generator of $H^0(\mathbb{C},A)$ is any non zero constant function $\mathbb{C} \to A$). So the Hodge polynomials are constant equal to 1.

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