A flatness criterion for pseudo-effective sheaves on compact K\"ahler spaces

Abstract

In this paper, we prove that if $E$ is a pseudo-effective sheaf with vanishing first Chern class on a klt compact K\"ahler space $X$, then, after passing to a finite quasi-\'etale cover, the reflexive pullback of $E$ is locally free and flat. This extends the flatness criterion of H\"oring--Peternell, originally established for projective varieties, to the K\"ahler setting. The proof relies on two main ingredients, both of which are new even in the projective case. The first is a flatness theorem for stable sheaves: we show that a slope-stable pseudo-effective sheaf with vanishing first Chern class is Hermitian flat. This is obtained by combining Hermitian--Einstein theory with the subharmonicity properties of direct image sheaves. The second is a singular K\"ahler analogue of Simpson's flatness theorem for extensions of locally free Hermitian flat sheaves.