Symplectic symmetries of a hypothetical exotic CP2: a potential construction or a nonexistence proof
Abstract
This paper is primarily concerned with the existence of a symplectic $4$-manifold, homeomorphic but not diffeomorphic to CP2, which admits a nontrivial finite order symplectomorphism. We reduced the existence of such an exotic CP2 to a question in contact geometry. More concretely, we showed that there are precisely $12$ plumbed manifolds, each equipped with a canonical contact structure associated to the plumbing, such that an exotic CP2 with a nontrivial finite order symplectomorphism exists if and only if one of the $12$ contact plumbed manifolds admits a certain specific symplectic filling. As the main technical result, we completely determined the order and local representations of a pseudofree symplectic Zp-action for an odd prime p on a rational homology CP2 with positive canonical class. Our result suggests that potentially, there are symplectic Zp-actions which are not a finite automorphism of a fake projective plane. Moreover, we introduced a construction, which, under the minimal assumptions on the symplectic fillings of the $12$ contact plumbed manifolds, produces a symplectic $4$-manifold on the Bogomolov-Miyaoka-Yau line via a reverse-engineering process.
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BibTeX
@misc{chen2026symplectic,
title = {{Symplectic symmetries of a hypothetical exotic $CP^2$: a potential construction or a nonexistence proof}},
author = {Weimin Chen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06746},
eprint = {2610.06746},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06746},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Symplectic symmetries of a hypothetical exotic CP^2: a potential construction or a nonexistence proof
AU - Weimin Chen
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.06746
UR - https://arxiv.org/abs/2610.06746
AB - This paper is primarily concerned with the existence of a symplectic $4$-manifold, homeomorphic but not diffeomorphic to CP^2, which admits a nontrivial finite order symplectomorphism. We reduced the existence of such an exotic CP^2 to a question in contact geometry. More concretely, we showed that there are precisely $12$ plumbed manifolds, each equipped with a canonical contact structure associated to the plumbing, such that an exotic CP^2 with a nontrivial finite order symplectomorphism exists if and only if one of the $12$ contact plumbed manifolds admits a certain specific symplectic filling. As the main technical result, we completely determined the order and local representations of a pseudofree symplectic Z_p-action for an odd prime p on a rational homology CP^2 with positive canonical class. Our result suggests that potentially, there are symplectic Z_p-actions which are not a finite automorphism of a fake projective plane. Moreover, we introduced a construction, which, under the minimal assumptions on the symplectic fillings of the $12$ contact plumbed manifolds, produces a symplectic $4$-manifold on the Bogomolov-Miyaoka-Yau line via a reverse-engineering process.
ER -