I am looking for some introductory reference concerning Automorphisms (of finite order) on K3 surfaces. Any suggestion?
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2$\begingroup$ Since you ask for introductory references I'll make this comment: if you don't already know it, it seems like a good idea to understand the Global Torelli Theorem of Piateckii--Shapiro and Shafarevich. One place where this is treated in detail is Barth--Hulek--Peters--van de Ven, Compact Complex Surfaces. $\endgroup$– user5117Oct 17, 2013 at 9:00
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2 Answers
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You may consult
Kondō, Shigeyuki Quadratic forms and K3. Enriques surfaces [translation of Sûgaku 42 (1990), no. 4, 346–360; MR1083944 (92b:14018)]. Sugaku Expositions. Sugaku Expositions 6 (1993), no. 1, 53–72.
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You can start with
Hans Sterk
Finiteness results for algebraic K3 surfaces
Mathematische Zeitschrift, 1985, Volume 189, Issue 4, pp 507-513
The aim of this paper is to prove the following theorem:
Theorem. Let $X$ be an algebraic K3 surface over the complex numbers. Then
a) the group $\mathrm{Aut}(X)$ of (biholomorphic) automorphisms is finitely generated, and
b) for every even integer $d\geq -2$, the number of $\mathrm{Aut}(X)$-orbits in the collection of complete linear systems which contain an irreducible curve of selfintersection $d$ is finite.
If this does not give you enough, look at the references that seem promising and especially at the papers that refer to this one. Very likely anything newer that's relevant will refer to this.
Also look at the many papers of Nikulin on K3's. He has made lots of computations in the K3 lattice about automorphisms.
I believe Oguiso also has a paper where he studies finite automorphisms of Calabi-Yau's, but this is only a vague memory, so I could be off. Someone might correct me on this....
answered Oct 15, 2013 at 3:41
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$\begingroup$ Dear Sándor: part b) of the theorem should read $d \geq -2$. (A small but crucial difference!) $\endgroup$– user5117Oct 15, 2013 at 8:45
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$\begingroup$ @Artie: Indeed. Thanks! I keep demonstrating my utmost talent for introducing typos at all places. $\endgroup$ Oct 15, 2013 at 15:40