On the kernels of Lefschetz operators
Abstract
We study the kernels of Lefschetz operators arising from complete intersections of nef classes. When the classes are semiample, we provide a complete characterization of mixed Lefschetz properties below a fixed degree -- all obstructions are detected by irreducible subvarieties of relevant codimensions. We further show that the kernels in the next degree range vanish off the diagonal, while on the diagonal they admit bases consisting of fundamental classes of irreducible subvarieties annihilated by the Lefschetz operator. We also establish a kernel decoupling theorem for nef classes on complex Lefschetz modules. As an application, we obtain a kernel characterization for supercritical collections of nef classes, thereby proving a conjecture of Shenfeld-van Handel and Hu-Xiao.
Related
- Weak type (1,1) boundedness of noncommutative singular integral operators with rough kernels
- Weak type (1,1) bound criterion for maximal truncated singular integral operators with rough kernels
- Coercivity and regularity for nonlocal divergence-form operators with measurable kernels
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- Weighted Hp−Lq boundedness of integral operators with rough kernels
BibTeX
@misc{hu2026kernels,
title = {{On the kernels of Lefschetz operators}},
author = {Jiajun Hu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12091},
eprint = {2610.12091},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12091},
}
RIS
TY - UNPB
TI - On the kernels of Lefschetz operators
AU - Jiajun Hu
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12091
UR - https://arxiv.org/abs/2610.12091
AB - We study the kernels of Lefschetz operators arising from complete intersections of nef classes. When the classes are semiample, we provide a complete characterization of mixed Lefschetz properties below a fixed degree -- all obstructions are detected by irreducible subvarieties of relevant codimensions. We further show that the kernels in the next degree range vanish off the diagonal, while on the diagonal they admit bases consisting of fundamental classes of irreducible subvarieties annihilated by the Lefschetz operator. We also establish a kernel decoupling theorem for nef classes on complex Lefschetz modules. As an application, we obtain a kernel characterization for supercritical collections of nef classes, thereby proving a conjecture of Shenfeld-van Handel and Hu-Xiao.
ER -