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Which finite groups are automorphism groups of polarized K3 surfaces (let's say over ℂ)?
...and does the answer change is I remove "polarized"?

(polarized = equipped with an ample line bundle)

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The answer does not change if you remove "polarized" provided your K3 surface is projective. Indeed take any positive integral (1,1) form corresponding to a polarization of K3 and sum it over the action of the group. This will give you an invariant positive integral (1,1) form and hence an invariant polarisation. I don't know if non-algebraic K3 surfaces can have an automorphism of finite order (if this is possible the automorphism will preserve the volume form so that the quotient is a singular K3 again) –  aglearner May 20 '13 at 14:36
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This is a comment for non-polarized ones. As you probably know, possible symplectic automorphis groups are classified by Mukai. Non-symplectic automorphism of prime order are studied in arxiv.org/abs/0903.3481 I think general case has not been explored. –  Atsushi Kanazawa May 20 '13 at 14:38
    
@Atsushi Kanazawa: I did not know that the distinction between symplectic and non-symplectic was important for this question. I ask because I am ignorant, and I would be interested to learn the answer. I would also be interested in a list of examples of Kahler K3 surfaces that realize the possible maximal automorphism groups (assuming the list is not too long). –  André Henriques May 20 '13 at 17:42
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@Andre Here is the famous paper by Mukai link.springer.com/article/10.1007%2FBF01394352#page-1 All symplectic automorphism groups are contained in Mathieu group M_{24}. He also gave examples of K3 surfaces that realize the possible maximal automorphism groups. There are 11 maximal groups. –  Atsushi Kanazawa May 22 '13 at 19:45

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