Rigidity properties of holomorphic isometries into homogeneous Kähler manifolds
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- by Andrea Loi and Roberto Mossa;
- Proc. Amer. Math. Soc. 152 (2024), 3051-3062
- DOI: https://doi.org/10.1090/proc/16754
- Published electronically: May 21, 2024
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Abstract:
We prove two rigidity results on holomorphic isometries into homogeneous Kähler manifolds. The first shows that a Kähler-Ricci soliton induced by the homogeneous metric of the Kähler product of a special generalized flag manifold (i.e. a flag of classical type or integral type) with a bounded homogeneous domain is trivial, i.e. Kähler-Einstein. In the second one we prove that: (i) a flat space is not relative to the Kähler product of a special generalized flag manifold with a homogeneous bounded domain, (ii) a special generalized flag manifold is not relative to the Kähler product of a flat space with a homogeneous bounded domain and (iii) a homogeneous bounded domain is not relative to the Kähler product of a flat space with a special generalized flag manifold. Our theorems strongly extend the results of Cheng and Hao [Ann. Global Anal. Geom. 60 (2021), pp. 167–180], Cheng, Di Scala, and Yuan [Internat. J. Math. 28 (2017), p. 1750027], Loi and Mossa [Proc. Amer. Math. Soc. 149 (2021), pp. 4931–4941], Loi and Mossa [Proc. Amer. Math. Soc. 151 (2023), pp. 3975–3984] and Umehara [Tokyo J. Math. 10 (1987), pp. 203–214].References
- D. V. Alekseevskiĭ and A. M. Perelomov, Invariant Kähler-Einstein metrics on compact homogeneous spaces, Funktsional. Anal. i Prilozhen. 20 (1986), no. 3, 1–16, 96 (Russian). MR 868557
- Arthur L. Besse, Einstein manifolds, Classics in Mathematics, Springer-Verlag, Berlin, 2008. Reprint of the 1987 edition. MR 2371700
- Eugenio Calabi, Isometric imbedding of complex manifolds, Ann. of Math. (2) 58 (1953), 1–23. MR 57000, DOI 10.2307/1969817
- Shan Tai Chan and Ngaiming Mok, Asymptotic total geodesy of local holomorphic curves exiting a bounded symmetric domain and applications to a uniformization problem for algebraic subsets, J. Differential Geom. 120 (2022), no. 1, 1–49. MR 4362285, DOI 10.4310/jdg/1641413830
- Xiaoliang Cheng and Yihong Hao, On the non-existence of common submanifolds of Kähler manifolds and complex space forms, Ann. Global Anal. Geom. 60 (2021), no. 1, 167–180. MR 4267597, DOI 10.1007/s10455-021-09776-3
- Xiaoliang Cheng, Antonio J. Di Scala, and Yuan Yuan, Kähler submanifolds and the Umehara algebra, Internat. J. Math. 28 (2017), no. 4, 1750027, 13. MR 3639045, DOI 10.1142/S0129167X17500276
- Antonio J. Di Scala and Andrea Loi, Kähler manifolds and their relatives, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9 (2010), no. 3, 495–501. MR 2722652
- Antonio Jose Di Scala, Hideyuki Ishi, and Andrea Loi, Kähler immersions of homogeneous Kähler manifolds into complex space forms, Asian J. Math. 16 (2012), no. 3, 479–487. MR 2989231, DOI 10.4310/AJM.2012.v16.n3.a7
- Mark L. Green, Metric rigidity of holomorphic maps to Kähler manifolds, J. Differential Geometry 13 (1978), no. 2, 279–286. MR 540947
- Xiaojun Huang and Yuan Yuan, Holomorphic isometry from a Kähler manifold into a product of complex projective manifolds, Geom. Funct. Anal. 24 (2014), no. 3, 854–886. MR 3213831, DOI 10.1007/s00039-014-0278-3
- Xiaojun Huang and Yuan Yuan, Submanifolds of Hermitian symmetric spaces, Analysis and geometry, Springer Proc. Math. Stat., vol. 127, Springer, Cham, 2015, pp. 197–206. MR 3445521, DOI 10.1007/978-3-319-17443-3_{1}0
- Andrea Loi and Roberto Mossa, On holomorphic isometries into blow-ups of $\Bbb C^n$, Mediterr. J. Math. 20 (2023), no. 4, Paper No. 230, 11. MR 4597721, DOI 10.1007/s00009-023-02437-8
- Andrea Loi and Roberto Mossa, Kähler immersions of Kähler-Ricci solitons into definite or indefinite complex space forms, Proc. Amer. Math. Soc. 149 (2021), no. 11, 4931–4941. MR 4310116, DOI 10.1090/proc/15628
- Andrea Loi and Roberto Mossa, Holomorphic isometries into homogeneous bounded domains, Proc. Amer. Math. Soc. 151 (2023), no. 9, 3975–3984. MR 4607641, DOI 10.1090/proc/16335
- Andrea Loi, Roberto Mossa, and Fabio Zuddas, The log-term of the Bergman kernel of the disc bundle over a homogeneous Hodge manifold, Ann. Global Anal. Geom. 51 (2017), no. 1, 35–51. MR 3595394, DOI 10.1007/s10455-016-9522-4
- Andrea Loi, Roberto Mossa, and Fabio Zuddas, Bochner coordinates on flag manifolds, Bull. Braz. Math. Soc. (N.S.) 50 (2019), no. 2, 497–514. MR 3955252, DOI 10.1007/s00574-018-0113-9
- Andrea Loi, Filippo Salis, and Fabio Zuddas, Extremal Kähler metrics induced by finite or infinite-dimensional complex space forms, J. Geom. Anal. 31 (2021), no. 8, 7842–7865. MR 4293915, DOI 10.1007/s12220-020-00554-4
- Andrea Loi, Filippo Salis, and Fabio Zuddas, Kähler-Ricci solitons induced by infinite-dimensional complex space forms, Pacific J. Math. 316 (2022), no. 1, 183–205. MR 4388792, DOI 10.2140/pjm.2022.316.183
- Andrea Loi and Michela Zedda, Kähler immersions of Kähler manifolds into complex space forms, Lecture Notes of the Unione Matematica Italiana, vol. 23, Springer, Cham; Unione Matematica Italiana, [Bologna], 2018. MR 3838438, DOI 10.1007/978-3-319-99483-3
- Andrea Loi and Fabio Zuddas, On the Gromov width of homogeneous Kähler manifolds, Differential Geom. Appl. 47 (2016), 130–132. MR 3504923, DOI 10.1016/j.difgeo.2016.03.006
- Ngaiming Mok and Sui Chung Ng, Germs of measure-preserving holomorphic maps from bounded symmetric domains to their Cartesian products, J. Reine Angew. Math. 669 (2012), 47–73. MR 2980451, DOI 10.1515/crelle.2011.142
- Ngaiming Mok, Some recent results on holomorphic isometries of the complex unit ball into bounded symmetric domains and related problems, Geometric complex analysis, Springer Proc. Math. Stat., vol. 246, Springer, Singapore, 2018, pp. 269–290. MR 3923233, DOI 10.1007/978-981-13-1672-2
- Roberto Mossa, A bounded homogeneous domain and a projective manifold are not relatives, Riv. Math. Univ. Parma (N.S.) 4 (2013), no. 1, 55–59. MR 3137529
- Masaaki Umehara, Kaehler submanifolds of complex space forms, Tokyo J. Math. 10 (1987), no. 1, 203–214. MR 899484, DOI 10.3836/tjm/1270141804
- Harald Upmeier, Kai Wang, and Genkai Zhang, Holomorphic isometries from the unit ball into symmetric domains, Int. Math. Res. Not. IMRN 1 (2019), 55–89. MR 4009528, DOI 10.1093/imrn/rnx110
- Ming Xiao, Holomorphic isometric maps from the complex unit ball to reducible bounded symmetric domains, J. Reine Angew. Math. 789 (2022), 187–209. MR 4460168, DOI 10.1515/crelle-2022-0029
- Ming Xiao and Yuan Yuan, Holomorphic maps from the complex unit ball to type IV classical domains, J. Math. Pures Appl. (9) 133 (2020), 139–166 (English, with English and French summaries). MR 4044677, DOI 10.1016/j.matpur.2019.05.009
- Yuan Yuan, Local holomorphic isometries, old and new results, Proceedings of the Seventh International Congress of Chinese Mathematicians. Vol. II, Adv. Lect. Math. (ALM), vol. 44, Int. Press, Somerville, MA, 2019, pp. 409–419. MR 3971911
- Yuan Yuan and Yuan Zhang, Rigidity for local holomorphic isometric embeddings from $\Bbb B^n$ into $\Bbb B^{N_1}\times \dots \times \Bbb B^{N_m}$ up to conformal factors, J. Differential Geom. 90 (2012), no. 2, 329–349. MR 2899879
Bibliographic Information
- Andrea Loi
- Affiliation: Dipartimento di Matematica, Università di Cagliari, Italy
- MR Author ID: 643483
- Email: loi@unica.it
- Roberto Mossa
- Affiliation: Dipartimento di Matematica, Università di Cagliari, Italy
- MR Author ID: 920782
- ORCID: 0000-0001-9173-2386
- Email: roberto.mossa@unica.it
- Received by editor(s): June 16, 2023
- Received by editor(s) in revised form: December 6, 2023
- Published electronically: May 21, 2024
- Additional Notes: The authors were supported by INdAM and GNSAGA - Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni, by GOACT - Funded by Fondazione di Sardegna and partially funded by PNRR e.INS Ecosystem of Innovation for Next Generation Sardinia (CUP F53C22000430001, codice MUR ECS00000038).
- Communicated by: Jiaping Wang
- © Copyright 2024 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 152 (2024), 3051-3062
- MSC (2020): Primary 53C55, 32Q15, 53C24, 53C42
- DOI: https://doi.org/10.1090/proc/16754
- MathSciNet review: 4753287