Rigid and Nonrigid Isotropic Nef Classes on K3 Surfaces
Abstract
Let X be a fixed complex K3 surface and let 0≠α∈KahX̅ be a (1,1)-class on the boundary of the Kähler cone with α2=0. We build a novel connection between two seemingly unrelated, intensively studied problems: the rigidity of a closed positive current in the class α and the collapse geometry of a sequence of Ricci-flat Kähler metrics in the Kähler class α+tκ as t→ 0+. More precisely, we prove that α contains a unique closed positive (1,1)-current if and only if the Ricci-flat metrics in class α+tκ collapse to a point in the Gromov--Hausdorff sense. Then we exhibit the first example of a nonrigid irrational nef class α on a Kummer surface, which answers a question of Filip--Tosatti and Sibony--Soldatenkov--Verbitsky. We also construct many new examples of rigid nef classes on hyperkähler manifolds, which extend the work of Filip--Tosatti and Sibony--Soldatenkov--Verbitsky.
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BibTeX
@misc{fu2026rigid,
title = {{Rigid and Nonrigid Isotropic Nef Classes on K3 Surfaces}},
author = {Xin Fu and Zexuan Ouyang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33632},
eprint = {2609.33632},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33632},
}
RIS
TY - UNPB
TI - Rigid and Nonrigid Isotropic Nef Classes on K3 Surfaces
AU - Xin Fu
AU - Zexuan Ouyang
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33632
UR - https://arxiv.org/abs/2609.33632
AB - Let X be a fixed complex K3 surface and let 0≠α∈Kah_X̅ be a (1,1)-class on the boundary of the Kähler cone with α^2=0. We build a novel connection between two seemingly unrelated, intensively studied problems: the rigidity of a closed positive current in the class α and the collapse geometry of a sequence of Ricci-flat Kähler metrics in the Kähler class α+tκ as t→ 0^+. More precisely, we prove that α contains a unique closed positive (1,1)-current if and only if the Ricci-flat metrics in class α+tκ collapse to a point in the Gromov--Hausdorff sense. Then we exhibit the first example of a nonrigid irrational nef class α on a Kummer surface, which answers a question of Filip--Tosatti and Sibony--Soldatenkov--Verbitsky. We also construct many new examples of rigid nef classes on hyperkähler manifolds, which extend the work of Filip--Tosatti and Sibony--Soldatenkov--Verbitsky.
ER -