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How to classify K3 surfaces over an arbitrary field k?

edited Feb 12 at 22:06

Gerry Myerson
15.6k●1●47●87

asked Oct 19 2009 at 19:28

Ilya Nikokoshev
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Although I completely agree with David Lehavi's answer, if you work over $\mathbb{Z}_p$, probably you can also make sense of rigid analytic K3 surfaces that are not necessarily algebraic schemes over $\mathbb{Z}_p$ (just as there are complex analytic K3 surfaces that are not algebraic schemes over $\mathbb{C}$). – Jason Starr Feb 13 at 22:12

David Lehavi · Accepted Answer · 2013-02-12 20:58:34Z UTC

The "standard" definition of a K3 surface is field independent (unless you are a physicist):

$p_g=1, q=0$, and trivial canonical class.

Some results:

Mumford and Bombieri showed that you get (just as in the complex case) a 19 dimensional family of K3 surfaces for any degree (the 19 dimensional thingy is a deformation theory argument which is completely algebraic).
Deligne showed that all the K3 surfaces in finite characteristics are reductions mod p.

What you obviously don't get is the fact that all these spaces sit together in a nice 20 dimensional complex ball. I also don't know if you can carry over any of the recent Kodaira dimension computation of these moduli (which are very analytic in nature).

Reference: Complex algebraic surfaces (Beauville): Chapter VIII and Appendix A.

K3 over fields other than C?

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