A flatness criterion for pseudo-effective sheaves on compact Kähler spaces
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.
Related
- Hard Lefschetz theorem for logarithmic sheaves twisted by pseudo-effective line bundles
- A note on compact Kähler varieties with pseudo-effective tangent sheaf
- Refined proper descent for left completed sheaves on locally compact Hausdorff spaces
- Parabolic Bogomolov--Gieseker Inequality and Pluriharmonic Metrics on Regular Parabolic Higgs Sheaves over Compact Kähler Manifolds
- Sharp effective equidistribution of closed geodesics in compact hyperbolic surfaces
BibTeX
@misc{cao2026flatness,
title = {{A flatness criterion for pseudo-effective sheaves on compact Kähler spaces}},
author = {Junyan Cao and Ya Deng and Shin-ichi Matsumura},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05154},
eprint = {2609.05154},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05154},
}
RIS
TY - UNPB
TI - A flatness criterion for pseudo-effective sheaves on compact Kähler spaces
AU - Junyan Cao
AU - Ya Deng
AU - Shin-ichi Matsumura
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05154
UR - https://arxiv.org/abs/2609.05154
AB - 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.
ER -