On the Kodaira problem for uniruled K\"ahler spaces

TL;DR

This paper presents a counterexample in uniruled K"ahler spaces showing that algebraic approximation may fail even when the minimal model program terminates with a Mori fiber space, challenging assumptions about their relationship.

Contribution

It constructs a specific uniruled K"ahler space where the MMP terminates but the space cannot be approximated algebraically, highlighting limitations in the current understanding.

Findings

Counterexample of a uniruled K"ahler space without algebraic approximation

Shows that approximability of the base does not imply approximability of the total space in Mori fibrations

Demonstrates limitations of the Kodaira problem in the context of uniruled K"ahler spaces

Abstract

We discuss the Kodaira problem for uniruled K\"ahler spaces. Building on a construction due to Voisin, we give an example of a uniruled K\"ahler 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.