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.
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.
"On the Kodaira problem for uniruled Kähler spaces." Ark. Mat. 58 (2) 267 - 284, October 2020. https://doi.org/10.4310/ARKIV.2020.v58.n2.a3