Arkiv för Matematik

Volume 58 (2020)

Number 2

On the Kodaira problem for uniruled Kähler spaces

Pages: 267 – 284



Patrick Graf (Department of Mathematics, University of Utah, Salt Lake City, Ut., U.S.A.)

Martin Schwald (Fakultät für Mathematik, Universität Duisburg–Essen, Essen, Germany)


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.


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

2010 Mathematics Subject Classification

14E30, 32G05, 32J27

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