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Finiteness results for algebraic K3 surfaces. (English) Zbl 0545.14032
Following suggestions of E. Looijenga we prove for an algebraic K3 surface X over the complex numbers: (a) The group Aut(X) of automorphisms is finitely generated. (b) For every even integer \(d\geq -2\), the number of Aut(X)-orbits in the collection of complete linear systems which contain an irreducible curve of selfintersection d is finite. - We prove this result by relating Aut(X) to certain arithmetic subgroups of the orthogonal group of \(NS\otimes_{{\mathbb{Z}}}{\mathbb{R}}\), where NS= Néron- Severi group of X.

MSC:
14J25 Special surfaces
32M05 Complex Lie groups, group actions on complex spaces
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
14L30 Group actions on varieties or schemes (quotients)
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