Non-algebraic hyperkahler manifolds

Frederic Campana Universit´e de Nancy Keiji Oguiso Osaka University Thomas Peternell Universit¨at Bayreuth

Differential Geometry mathscidoc:1609.10187

Journal of Differential Geometry, 85, (3), 397-424, 2010
We study the algebraic dimension a(X) of a compact hyperk ¨ahler manifold of dimension 2n. We show that a(X) is at most n unless X is projective. If a compact K¨ahler manifold with algebraic dimension 0 and Kodaira dimension 0 has a minimal model, then only the values 0, n and 2n are possible. In case of middle dimension, the algebraic reduction is holomorphic Lagrangian. If n = 2, then - without any assumptions - the algebraic dimension only takes the values 0, 2 and 4. The paper also gives structure results for ”generalised hyperk¨ahler” manifolds and studies nef lines bundles.
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  author={Frederic Campana, Keiji Oguiso, and Thomas Peternell},
  booktitle={Journal of Differential Geometry},
Frederic Campana, Keiji Oguiso, and Thomas Peternell. NON-ALGEBRAIC HYPERKAHLER MANIFOLDS. 2010. Vol. 85. In Journal of Differential Geometry. pp.397-424.
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