Kähler thresholds
Abstract
The topology of Kähler manifolds is largely determined by the geometry due to its rigidity. In particular, the cohomology and Hodge theory of compact Kähler manifolds is quite restricted. We prove that within a certain threshold, the odd Betti numbers of any compact almost-hermitian manifold satisfying a degenerate Kähler condition are even, and the even Betti numbers are strictly positive. We call this new type of degenerated Kähler manifold acK. Our approach to proving these results makes use of a compactness theorem for acK manifolds, and a new version of Hodge theory for compact manifolds endowed with a Sobolev regular Kähler strucutre. In addition, we lay out a program to pursue the study of acK geometry that accommodates not only the classical viewpoint, but also constructive and finitary proofs, as well as formalization with proof assistants.
Related
- Extremal Kähler--Ricci solitons on Fano manifolds are Kähler--Einstein
- Kähler-Finsler rigidity of the Kobayashi metric and the Kähler base of weakly Kähler-Finsler metrics
- Finite-Time Singularities of the Kähler--Ricci Flow on a ℂPm-Bundle over a Product of Kähler--Einstein Manifolds
- Psh variation of singular Kähler-Einstein metrics and Invariance of plurigenera for smooth Kähler families
- Finite Time Singularities of the Kähler Ricci Flow on Compact Kähler Surfaces are of Type I
BibTeX
@misc{clemente2026kähler,
title = {{Kähler thresholds}},
author = {Gabriella Clemente and Carlos Simpson},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2606.08224},
eprint = {2606.08224},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.08224},
}
RIS
TY - UNPB
TI - Kähler thresholds
AU - Gabriella Clemente
AU - Carlos Simpson
PY - 2026
DA - 2026/08/27
PB - arXiv
DO - 10.48550/arxiv.2606.08224
UR - https://arxiv.org/abs/2606.08224
AB - The topology of Kähler manifolds is largely determined by the geometry due to its rigidity. In particular, the cohomology and Hodge theory of compact Kähler manifolds is quite restricted. We prove that within a certain threshold, the odd Betti numbers of any compact almost-hermitian manifold satisfying a degenerate Kähler condition are even, and the even Betti numbers are strictly positive. We call this new type of degenerated Kähler manifold acK. Our approach to proving these results makes use of a compactness theorem for acK manifolds, and a new version of Hodge theory for compact manifolds endowed with a Sobolev regular Kähler strucutre. In addition, we lay out a program to pursue the study of acK geometry that accommodates not only the classical viewpoint, but also constructive and finitary proofs, as well as formalization with proof assistants.
ER -