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Algebraic approximation of Kähler threefolds of Kodaira dimension zero

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Abstract

We prove that for a compact Kähler threefold with canonical singularities and vanishing first Chern class, the projective fibres are dense in the semiuniversal deformation space. This implies that every Kähler threefold of Kodaira dimension zero admits small projective deformations after a suitable bimeromorphic modification. As a corollary, we see that the fundamental group of any Kähler threefold is a quotient of an extension of fundamental groups of projective manifolds, up to subgroups of finite index. In the course of the proof, we show that for a canonical threefold with \(c_1 = 0\), the Albanese map decomposes as a product after a finite étale base change. This generalizes a result of Kawamata, valid in all dimensions, to the Kähler case. Furthermore we generalize a Hodge-theoretic criterion for algebraic approximability, due to Green and Voisin, to quotients of a manifold by a finite group.

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Acknowledgements

I would like to thank Thomas Peternell for suggesting the topic of algebraic approximation to me, and for many fruitful discussions. Furthermore I would like to thank Zsolt Patakfalvi, Junyan Cao and in particular Christian Lehn for interesting conversations and for answering my questions. Last but not least, the anonymous referee’s input has been very helpful both in simplifying some of the arguments and in improving the statement of Theorem 1.10 significantly.

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  1. Lehrstuhl für Mathematik I, Universität Bayreuth, 95440, Bayreuth, Germany

    Patrick Graf

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Correspondence to Patrick Graf.

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Communicated by Ngaiming Mok.

To sing, to brag, to blaze, to grumble...

The author was partially supported by the DFG Grant “Zur Positivität in der komplexen Geometrie”.

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Graf, P. Algebraic approximation of Kähler threefolds of Kodaira dimension zero. Math. Ann. 371, 487–516 (2018). https://doi.org/10.1007/s00208-017-1577-4

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