Year: 2026
Type: preprint
Abstract: In this paper, we prove that if $E$ is a pseudo-effective sheaf with vanishing first Chern class on a klt compact Kähler space $X$, then, after passing to a finite quasi-étale cover, the reflexive pullback of $E$ is locally free and flat. This extends the flatness criterion of Höring--Peternell, originally established for projective varieties, to the Kähler 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 Kähler analogue of Simpson's flatness theorem for extensions of locally free Hermitian flat sheaves.
Source: arXiv (Cornell University)
Authors: Cao, Junyan, Deng, Ya, Matsumura, Shin-ichi
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G776509466
OpenAlex ID: W7211885160