PreprintarXiv licencearXiv
Instability of the Fundamental Group for Ricci-Flat Kähler 4-Manifolds
Abstract
We show that the fundamental group is not locally stable under non-collapsed Gromov--Hausdorff convergence, even among closed Ricci-flat Kähler four-manifolds. More precisely, we construct families of Ricci-flat Kähler metrics on K3 surfaces and Enriques surfaces that converge to the same compact flat orbifold. Using the same mechanism, we further construct two families of closed locally hyperkähler Ricci-flat four-manifolds, with fundamental groups ℤ2 and ℤ2×ℤ2, respectively, which converge to the same compact flat orbifold.
Related
- Extremal Kähler--Ricci solitons on Fano manifolds are Kähler--Einstein
- Finite-Time Singularities of the Kähler--Ricci Flow on a ℂPm-Bundle over a Product of Kähler--Einstein Manifolds
- Complete Kähler Manifolds with pinched Ricci curvature
- An L1 bound for Ricci curvature on Kähler manifolds
- On complete Ricci--flat Kähler manifolds
BibTeX
@misc{bellati2026instability,
title = {{Instability of the Fundamental Group for Ricci-Flat Kähler 4-Manifolds}},
author = {Alejandro Bellati and Ignacio Bustamante},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05795},
eprint = {2610.05795},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05795},
}
RIS
TY - UNPB
TI - Instability of the Fundamental Group for Ricci-Flat Kähler 4-Manifolds
AU - Alejandro Bellati
AU - Ignacio Bustamante
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.05795
UR - https://arxiv.org/abs/2610.05795
AB - We show that the fundamental group is not locally stable under non-collapsed Gromov--Hausdorff convergence, even among closed Ricci-flat Kähler four-manifolds. More precisely, we construct families of Ricci-flat Kähler metrics on K3 surfaces and Enriques surfaces that converge to the same compact flat orbifold. Using the same mechanism, we further construct two families of closed locally hyperkähler Ricci-flat four-manifolds, with fundamental groups ℤ_2 and ℤ_2×ℤ_2, respectively, which converge to the same compact flat orbifold.
ER -