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Analytic Construction of Rational Curves on Fano Manifolds
Abstract
Inspired by constructions of entire curves in Oka geometry, we construct rational curves on complex Fano manifolds analytically, by alternately deforming a holomorphic disk to reduce its area and enlarging its source. Positive Ricci curvature yields an area-decreasing deformation, while an affine-lift perturbation enlarges the source at a controlled area cost. The two operations change the area and the weighted derivative by amounts we estimate explicitly, and analytic compactness passes from the resulting disks to a holomorphic sphere. The main result produces a sphere that meets a prescribed compact fiber without being contained in it, with an explicit area bound.
Related
- Transverse rational curves and the homogeneity of contact Fano manifolds
- Hilbert schemes of low-degree rational curves on a prime Fano threefold of degree $22$
- A duality principle for the neighbourhood of manifolds and applications to counting rational points near curves
- Fano orbifolds admit free Campana curves
- Extremal Kähler--Ricci solitons on Fano manifolds are Kähler--Einstein
BibTeX
@misc{du2026analytic,
title = {{Analytic Construction of Rational Curves on Fano Manifolds}},
author = {Yun-Heng Du and Bin Guo and Song-Yan Xie},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11612},
eprint = {2609.11612},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11612},
}
RIS
TY - UNPB
TI - Analytic Construction of Rational Curves on Fano Manifolds
AU - Yun-Heng Du
AU - Bin Guo
AU - Song-Yan Xie
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.11612
UR - https://arxiv.org/abs/2609.11612
AB - Inspired by constructions of entire curves in Oka geometry, we construct rational curves on complex Fano manifolds analytically, by alternately deforming a holomorphic disk to reduce its area and enlarging its source. Positive Ricci curvature yields an area-decreasing deformation, while an affine-lift perturbation enlarges the source at a controlled area cost. The two operations change the area and the weighted derivative by amounts we estimate explicitly, and analytic compactness passes from the resulting disks to a holomorphic sphere. The main result produces a sphere that meets a prescribed compact fiber without being contained in it, with an explicit area bound.
ER -