Transverse rational curves and the homogeneity of contact Fano manifolds
Abstract
We prove that every smooth complex projective contact Fano manifold is homogeneous under its connected contact automorphism group. The argument extends and simplifies the incidence method of our earlier proof of the LeBrun--Salamon conjecture. After the classical reductions, balanced transverse rational maps through every point yield tangent sections extending across the nodal boundary. These establish irreducibility and nonlinearity of the general variety of minimal rational tangents associated with contact lines, enabling Hwang's descent theorem to produce global vector fields generating the tangent bundle. Consequences include very ampleness of the contact line bundle, embeddedness of every balanced rational map, Kähler--Einstein existence, K-polystability, and the LeBrun--Salamon conjecture.
Related
- Transverse rational curves and the LeBrun--Salamon conjecture
- Analytic Construction of Rational Curves on Fano Manifolds
- A duality principle for the neighbourhood of manifolds and applications to counting rational points near curves
- Contact 3-manifolds with infinitely many rational homology ball Stein fillings
- Rational Points on Drinfeld Modular Curves of Prime Level
BibTeX
@misc{hu2026transverse,
title = {{Transverse rational curves and the homogeneity of contact Fano manifolds}},
author = {Zhengyu Hu and Kefeng Liu and Xueyuan Wan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38736},
eprint = {2610.38736},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38736},
}
RIS
TY - UNPB
TI - Transverse rational curves and the homogeneity of contact Fano manifolds
AU - Zhengyu Hu
AU - Kefeng Liu
AU - Xueyuan Wan
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38736
UR - https://arxiv.org/abs/2610.38736
AB - We prove that every smooth complex projective contact Fano manifold is homogeneous under its connected contact automorphism group. The argument extends and simplifies the incidence method of our earlier proof of the LeBrun--Salamon conjecture. After the classical reductions, balanced transverse rational maps through every point yield tangent sections extending across the nodal boundary. These establish irreducibility and nonlinearity of the general variety of minimal rational tangents associated with contact lines, enabling Hwang's descent theorem to produce global vector fields generating the tangent bundle. Consequences include very ampleness of the contact line bundle, embeddedness of every balanced rational map, Kähler--Einstein existence, K-polystability, and the LeBrun--Salamon conjecture.
ER -