Basis for hodge decomposition of Elliptic K3 Surfaces

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?

edited Dec 6 at 22:37

asked Dec 6 at 18:52

LMN
1,665●10

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 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 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 at 22:37

Yes, right. I should have been clearer in explaining my notation. – Donu Arapura Dec 6 at 23:57

Donu, thanks a lot! – LMN Dec 7 at 1:00

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