PreprintarXiv licencearXiv
A note on compact Kähler varieties with pseudo-effective tangent sheaf
Abstract
In this paper, we discuss recent developments and open problems concerning the geometry of compact Kähler varieties whose tangent sheaves are pseudo-effective in a strong sense. As our main result, we prove that, after passing to a finite quasi-étale cover, any compact Kähler variety with quotient singularities and pseudo-effective tangent sheaf admits a flat fibration onto a complex torus, with rationally connected fibers. We also obtain an analogous structure theorem for klt compact Kähler varieties, assuming the existence of minimal models in numerical dimension zero; consequently, the result holds unconditionally in the projective setting and for low-dimensional compact Kähler varieties.
Related
- A flatness criterion for pseudo-effective sheaves on compact Kähler spaces
- Positivity of compact Kähler varieties admitting an int-amplified endomorphism
- Scalar curvature bounds and tangent cones for Kähler--Ricci shrinkers
- Non-null ruled surfaces and pseudo hyper-dual tangent sphere bundles
- Parametrization of ancient Ricci flows with smooth compact tangent shrinkers
BibTeX
@misc{matsumura2026note,
title = {{A note on compact Kähler varieties with pseudo-effective tangent sheaf}},
author = {Shin-ichi Matsumura and Guolei Zhong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07069},
eprint = {2609.07069},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07069},
}
RIS
TY - UNPB
TI - A note on compact Kähler varieties with pseudo-effective tangent sheaf
AU - Shin-ichi Matsumura
AU - Guolei Zhong
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07069
UR - https://arxiv.org/abs/2609.07069
AB - In this paper, we discuss recent developments and open problems concerning the geometry of compact Kähler varieties whose tangent sheaves are pseudo-effective in a strong sense. As our main result, we prove that, after passing to a finite quasi-étale cover, any compact Kähler variety with quotient singularities and pseudo-effective tangent sheaf admits a flat fibration onto a complex torus, with rationally connected fibers. We also obtain an analogous structure theorem for klt compact Kähler varieties, assuming the existence of minimal models in numerical dimension zero; consequently, the result holds unconditionally in the projective setting and for low-dimensional compact Kähler varieties.
ER -