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Bigness of tangent bundles and dynamical rigidity of Fano manifolds of Picard number 1 (with an appendix by Jie Liu)

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Feng Shao at Institute for Basic Science
Feng Shao
Zhong Guolei at Institute for Basic Science
Zhong Guolei
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Abstract

Let f:X→Yf:X\rightarrow Y be a surjective morphism of Fano manifolds of Picard number 1 whose VMRTs at a general point are not dual defective. Suppose that the tangent bundle TXT_X is big. We show that f is an isomorphism unless Y is a projective space. As applications, we explore the bigness of the tangent bundles of complete intersections, del Pezzo manifolds, and Mukai manifolds, as well as their dynamical rigidity.
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Mathematische Annalen (2025) 391:1731–1752
https://doi.org/10.1007/s00208-024-02955-0
Mathematische Annalen
Bigness of tangent bundles and dynamical rigidity of Fano
manifolds of Picard number 1 (with an appendix by Jie Liu)
Feng Shao1·Guolei Zhong1
Received: 28 November 2023 / Revised: 9 April 2024 / Accepted: 23 July 2024 /
Published online: 12 August 2024
© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2024
Abstract
Let f:X→Ybe a surjective morphism of Fano manifolds of Picard number 1 whose
VMRTs at a general point are not dual defective. Suppose that the tangent bundle TXis
big. We show that fis an isomorphism unless Yis a projective space. As applications,
we explore the bigness of the tangent bundles of complete intersections, del Pezzo
manifolds, and Mukai manifolds, as well as their dynamical rigidity.
Mathematics Subject Classification 14J40 ·14J45
Contents
1 Introduction ............................................ 1732
2 Proofs of Theorems 1.2 and 1.4 .................................. 1735
3 Examples and Proof of Proposition 1.9 .............................. 1739
4 Bigness of tangent bundles of certain projective manifolds .................... 1741
4.1 Smooth complete intersections, Proof of Theorem 1.6 .................... 1741
4.2 Del Pezzo manifolds, Proof of Theorem 1.7 ......................... 1742
4.3 Mukai manifolds, Proof of Proposition 1.8 .......................... 1745
Appendix A. Finite covering of intersections of two quadrics .................... 1747
4.4 Intersection of two quadrics .................................. 1747
4.5 Proof of Theorem 4.5 ..................................... 1748
References ............................................... 1750
Jie Liu: Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Institute of
Mathematics, Beijing 100190, China. jliu@amss.ac.cn.
BFeng Shao
shaofeng@amss.ac.cn ; shaofeng@ibs.re.kr
Guolei Zhong
zhongguolei@u.nus.edu ; guolei@ibs.re.kr
1Center for Complex Geometry, Institute for Basic Science (IBS), 55 Expo-ro, Yuseong-gu, Daejeon
34126, Republic of Korea
123
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... The bigness of T X is a rather restrictive property for projective manifolds (see for instance [HLS22,KKL22,HL23,SZ23] and the references therein). The known interesting examples of projective manifolds with big tangent bundles include the quintic del Pezzo threefold [HLS22] and smooth projective horospherical varieties ([Liu23, § 3B2]), which contain toric varieties [Hsi15] and rational homogeneous spaces [Ric74]. ...
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