Projective manifolds whose tangent bundle contains a strictly nef subsheaf
Authors:
Jie Liu, Wenhao Ou and Xiaokui Yang
Journal:
J. Algebraic Geom. 33 (2024), 1-53
DOI:
https://doi.org/10.1090/jag/807
Published electronically:
October 11, 2022
Full-text PDF
Abstract |
References |
Additional Information
Abstract: Suppose that $X$ is a projective manifold whose tangent bundle $T_X$ contains a locally free strictly nef subsheaf. We prove that $X$ is isomorphic to either a projective space or a projective bundle over a hyperbolic manifold of general type. Moreover, if the fundamental group $\pi _1(X)$ is virtually solvable, then $X$ is isomorphic to a projective space.
References
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- Daniel Greb, Stefan Kebekus, and Thomas Peternell, Projective flatness over klt spaces and uniformisation of varieties with nef anti-canonical divisor, J. Algebraic Geom., 31 (2022), no. 3, 467–496.
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- Duo Li, Wenhao Ou, and Xiaokui Yang, On projective varieties with strictly nef tangent bundles, J. Math. Pures Appl. (9) 128 (2019), 140–151 (English, with English and French summaries). MR 3980847, DOI 10.1016/j.matpur.2019.04.007
- Frank Loray, Jorge Vitório Pereira, and Frédéric Touzet, Singular foliations with trivial canonical class, Invent. Math. 213 (2018), no. 3, 1327–1380. MR 3842065, DOI 10.1007/s00222-018-0806-0
- Ngaiming Mok, The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature, J. Differential Geom. 27 (1988), no. 2, 179–214. MR 925119
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- Shigefumi Mori, Projective manifolds with ample tangent bundles, Ann. of Math. (2) 110 (1979), no. 3, 593–606. MR 554387, DOI 10.2307/1971241
- Roberto Muñoz, Gianluca Occhetta, Luis E. Solá-Conde, Kiwamu Watanabe, and Jarosław A. Wiśniewski, A survey on the Campana-Peternell conjecture, Rend. Istit. Mat. Univ. Trieste 47 (2015), 127–185. MR 3456582, DOI 10.13137/0049-4704/11223
- V. B. Mehta and A. Ramanathan, Restriction of stable sheaves and representations of the fundamental group, Invent. Math. 77 (1984), no. 1, 163–172. MR 751136, DOI 10.1007/BF01389140
- V. B. Mehta and A. Ramanathan, Semistable sheaves on projective varieties and their restriction to curves, Math. Ann. 258 (1981/82), no. 3, 213–224. MR 649194, DOI 10.1007/BF01450677
- Noboru Nakayama, Zariski-decomposition and abundance, MSJ Memoirs, vol. 14, Mathematical Society of Japan, Tokyo, 2004. MR 2104208
- M. S. Narasimhan and C. S. Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. (2) 82 (1965), 540–567. MR 184252, DOI 10.2307/1970710
- R. Pandharipande, Convex rationally connected varieties, Proc. Amer. Math. Soc. 141 (2013), no. 5, 1539–1543. MR 3020841, DOI 10.1090/S0002-9939-2012-11429-7
- Th. Peternell, Minimal varieties with trivial canonical classes. I, Math. Z. 217 (1994), no. 3, 377–405. MR 1306667, DOI 10.1007/BF02571950
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Additional Information
Jie Liu
Affiliation:
Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
ORCID:
0000-0001-8131-5909
Email:
jliu@amss.ac.cn
Wenhao Ou
Affiliation:
Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
MR Author ID:
1089322
Email:
wenhaoou@amss.ac.cn
Xiaokui Yang
Affiliation:
Department of Mathematics and Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, People’s Republic of China
Email:
xkyang@mail.tsinghua.edu.cn
Received by editor(s):
June 24, 2021
Published electronically:
October 11, 2022
Additional Notes:
The first author was supported by the National Key R&D Program of China (No. 2021YFA1002300), the China Postdoctoral Science Foundation (2019M650873) and the NSFC grants (No. 11688101 and No. 12001521). The second author was supported by the National Key R&D Program of China (No. 2021YFA1002300). The third author was supported by NSFC grants (No. 12171262 and No. 12141101).
Article copyright:
© Copyright 2022
University Press, Inc.
