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Abstract
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Given a complex manifold, a real structure is an anti-holomorphic involution. A central problem in real algebraic geometry is to classify real structures on complex manifolds. We address this problem for compact hyper-Kahler manifolds by showing that any such manifold admits only finitely many real structures up to equivalence. We actually prove more generally that the automorphism group and the Klein automorphism group of a compact hyper-kahler manifold contain only finitely many conjugacy classes of finite subgroups. If time permits, I also show the finite generation of automorphism group of a compact hyperkahler manifold, a conjecture of Oguiso. This is a joint work with Andrea Cattaneo, arXiv:1806.03864.
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