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We know the hodge numbers of K3 Surfaces. To work out some ideas, I'd like to know an explicit basis for the hodge decomposition $H^{p,q}$ of a smooth Elliptic K3 Surface over $\mathbb{C}$ (for all $p,q$, but no other restriction on the surface). Are there some such surfaces where particularly "nice" bases are known?

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Let $X$ be the Kummer surface associated to a product $A=E_1\times E_2$ of elliptic curves. Then $X$ fibers over $\mathbb{P}^1$. $H^{11}(X)$ is generated by $(1,1)$ classes $dz_1\wedge d\bar z_1$ on $A$ plus the 16 exceptional classes. $H^{20}$ by $dz_1\wedge dz_2$ etc. Does this help? –  Donu Arapura Dec 6 '12 at 19:48
    
(Can't edit) The first $2$-form is $dZ_1\wedge d\bar z_2$, this should be followed by an ellipsis "..." –  Donu Arapura Dec 6 '12 at 19:54
    
Thanks. Just to make sure, are the differential forms you mention pulled back from $(E_1\times E_2)/(involution)$ and the 16 exceptional classes the images under the cycle class map of the 16 rational curves that $X$ has because it is a blowup of $(E_1\times E_2)/(involution)$. –  LMN Dec 6 '12 at 22:37
    
Yes, right. I should have been clearer in explaining my notation. –  Donu Arapura Dec 6 '12 at 23:57
    
Donu, thanks a lot! –  LMN Dec 7 '12 at 1:00
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