On the Kodaira problem for uniruled Kähler spaces

We discuss the Kodaira problem for uniruled Kähler spaces. Building on a construction due to Voisin, we give an example of a uniruled Kähler space $X$ such that every run of the $K_X$-MMP immediately terminates with a Mori fibre space, yet $X$ does not admit an algebraic approximation. Our example also shows that for a Mori fibration, approximability of the base does not imply approximability of the total space.

Keywords

Kähler manifolds, uniruled Kähler spaces, Mori fibrations, algebraic approximation, small projective deformations, locally trivial deformations

2010 Mathematics Subject Classification

14E30, 32G05, 32J27

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The first author was partially supported by a DFG Research Fellowship. The second author was partially supported by the DFG Collaborative Research Centre SFB/TR 45.

Received 14 June 2019

Received revised 24 September 2019

Accepted 13 March 2020

Published 3 November 2020